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Amal Alghamdi (UT Austin), +Olalekan Babaniyi (RIT), +Joshua Chen (UT Austin), +Peng Chen (UT Austin), +Argho Datta (WashU), +Ki-Tae Kim (UC Merced), +Tom O'Leary-Roseberry (UT Austin), +Luke Lozenski (WashU), +Siddhant Wahal (UT Austin)

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© 2016-2018 The University of Texas at Austin, University of California Merced. +© 2019-2020 The University of Texas at Austin, University of California Merced, Washington University in St. Louis.

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    O. Ghattas, N. Petra (PIs), U. Villa (Co-PI), Integrating Data with Complex Predictive Models under Uncertainty: An Extensible Software Framework for Large-Scale Bayesian Inversion, NSF-SSI2, Grants No ACI-1550593, ACI-1550547 (2016-2019).

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.dropdown-toggle::after{display:none}.dropleft .dropdown-toggle::before{display:inline-block;margin-right:0.255em;vertical-align:0.255em;content:"";border-top:0.3em solid transparent;border-right:0.3em solid;border-bottom:0.3em solid transparent}.dropleft .dropdown-toggle:empty::after{margin-left:0}.dropleft .dropdown-toggle::before{vertical-align:0}.dropdown-menu[x-placement^="top"],.dropdown-menu[x-placement^="right"],.dropdown-menu[x-placement^="bottom"],.dropdown-menu[x-placement^="left"]{right:auto;bottom:auto}.dropdown-divider{height:0;margin:0.5rem 0;overflow:hidden;border-top:1px solid #eee}.dropdown-item{display:block;width:100%;padding:0.25rem 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.dropdown-toggle-split::after{margin-left:0}.dropleft .dropdown-toggle-split::before{margin-right:0}.btn-sm+.dropdown-toggle-split,.btn-group-sm>.btn+.dropdown-toggle-split{padding-right:0.375rem;padding-left:0.375rem}.btn-lg+.dropdown-toggle-split,.btn-group-lg>.btn+.dropdown-toggle-split{padding-right:0.75rem;padding-left:0.75rem}.btn-group-vertical{-webkit-box-orient:vertical;-webkit-box-direction:normal;-ms-flex-direction:column;flex-direction:column;-webkit-box-align:start;-ms-flex-align:start;align-items:flex-start;-webkit-box-pack:center;-ms-flex-pack:center;justify-content:center}.btn-group-vertical>.btn,.btn-group-vertical>.btn-group{width:100%}.btn-group-vertical>.btn:not(:first-child),.btn-group-vertical>.btn-group:not(:first-child){margin-top:-1px}.btn-group-vertical>.btn:not(:last-child):not(.dropdown-toggle),.btn-group-vertical>.btn-group:not(:last-child)>.btn{border-bottom-right-radius:0;border-bottom-left-radius:0}.btn-group-vertical>.btn:not(:first-child),.btn-group-vertical>.btn-group:not(:first-child)>.btn{border-top-left-radius:0;border-top-right-radius:0}.btn-group-toggle>.btn,.btn-group-toggle>.btn-group>.btn{margin-bottom:0}.btn-group-toggle>.btn input[type="radio"],.btn-group-toggle>.btn input[type="checkbox"],.btn-group-toggle>.btn-group>.btn input[type="radio"],.btn-group-toggle>.btn-group>.btn input[type="checkbox"]{position:absolute;clip:rect(0, 0, 0, 0);pointer-events:none}.input-group{position:relative;display:-webkit-box;display:-ms-flexbox;display:flex;-ms-flex-wrap:wrap;flex-wrap:wrap;-webkit-box-align:stretch;-ms-flex-align:stretch;align-items:stretch;width:100%}.input-group>.form-control,.input-group>.form-control-plaintext,.input-group>.custom-select,.input-group>.custom-file{position:relative;-webkit-box-flex:1;-ms-flex:1 1 0%;flex:1 1 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.custom-file-input:focus{z-index:4}.input-group>.form-control:not(:last-child),.input-group>.custom-select:not(:last-child){border-top-right-radius:0;border-bottom-right-radius:0}.input-group>.form-control:not(:first-child),.input-group>.custom-select:not(:first-child){border-top-left-radius:0;border-bottom-left-radius:0}.input-group>.custom-file{display:-webkit-box;display:-ms-flexbox;display:flex;-webkit-box-align:center;-ms-flex-align:center;align-items:center}.input-group>.custom-file:not(:last-child) .custom-file-label,.input-group>.custom-file:not(:last-child) .custom-file-label::after{border-top-right-radius:0;border-bottom-right-radius:0}.input-group>.custom-file:not(:first-child) .custom-file-label{border-top-left-radius:0;border-bottom-left-radius:0}.input-group-prepend,.input-group-append{display:-webkit-box;display:-ms-flexbox;display:flex}.input-group-prepend .btn,.input-group-append .btn{position:relative;z-index:2}.input-group-prepend .btn:focus,.input-group-append .btn:focus{z-index:3}.input-group-prepend .btn+.btn,.input-group-prepend .btn+.input-group-text,.input-group-prepend .input-group-text+.input-group-text,.input-group-prepend .input-group-text+.btn,.input-group-append .btn+.btn,.input-group-append .btn+.input-group-text,.input-group-append .input-group-text+.input-group-text,.input-group-append .input-group-text+.btn{margin-left:-1px}.input-group-prepend{margin-right:-1px}.input-group-append{margin-left:-1px}.input-group-text{display:-webkit-box;display:-ms-flexbox;display:flex;-webkit-box-align:center;-ms-flex-align:center;align-items:center;padding:0.375rem 0.75rem;margin-bottom:0;font-size:1rem;font-weight:400;line-height:1.5;color:#495057;text-align:center;white-space:nowrap;background-color:#eee;border:1px solid #ced4da;border-radius:0.25rem}.input-group-text input[type="radio"],.input-group-text input[type="checkbox"]{margin-top:0}.input-group-lg>.form-control:not(textarea),.input-group-lg>.custom-select{height:calc(1.5em + 1rem + 2px)}.input-group-lg>.form-control,.input-group-lg>.custom-select,.input-group-lg>.input-group-prepend>.input-group-text,.input-group-lg>.input-group-append>.input-group-text,.input-group-lg>.input-group-prepend>.btn,.input-group-lg>.input-group-append>.btn{padding:0.5rem 1rem;font-size:1.25rem;line-height:1.5;border-radius:0.3rem}.input-group-sm>.form-control:not(textarea),.input-group-sm>.custom-select{height:calc(1.5em + 0.5rem + 2px)}.input-group-sm>.form-control,.input-group-sm>.custom-select,.input-group-sm>.input-group-prepend>.input-group-text,.input-group-sm>.input-group-append>.input-group-text,.input-group-sm>.input-group-prepend>.btn,.input-group-sm>.input-group-append>.btn{padding:0.25rem 0.5rem;font-size:0.875rem;line-height:1.5;border-radius:0.2rem}.input-group-lg>.custom-select,.input-group-sm>.custom-select{padding-right:1.75rem}.input-group>.input-group-prepend>.btn,.input-group>.input-group-prepend>.input-group-text,.input-group>.input-group-append:not(:last-child)>.btn,.input-group>.input-group-append:not(:last-child)>.input-group-text,.input-group>.input-group-append:last-child>.btn:not(:last-child):not(.dropdown-toggle),.input-group>.input-group-append:last-child>.input-group-text:not(:last-child){border-top-right-radius:0;border-bottom-right-radius:0}.input-group>.input-group-append>.btn,.input-group>.input-group-append>.input-group-text,.input-group>.input-group-prepend:not(:first-child)>.btn,.input-group>.input-group-prepend:not(:first-child)>.input-group-text,.input-group>.input-group-prepend:first-child>.btn:not(:first-child),.input-group>.input-group-prepend:first-child>.input-group-text:not(:first-child){border-top-left-radius:0;border-bottom-left-radius:0}.custom-control{position:relative;display:block;min-height:1.5rem;padding-left:1.5rem}.custom-control-inline{display:-webkit-inline-box;display:-ms-inline-flexbox;display:inline-flex;margin-right:1rem}.custom-control-input{position:absolute;left:0;z-index:-1;width:1rem;height:1.25rem;opacity:0}.custom-control-input:checked ~ .custom-control-label::before{color:#fff;border-color:#446E9B;background-color:#446E9B}.custom-control-input:focus ~ .custom-control-label::before{-webkit-box-shadow:0 0 0 0.2rem rgba(68,110,155,0.25);box-shadow:0 0 0 0.2rem rgba(68,110,155,0.25)}.custom-control-input:focus:not(:checked) ~ .custom-control-label::before{border-color:#90aece}.custom-control-input:not(:disabled):active ~ .custom-control-label::before{color:#fff;background-color:#b4c8de;border-color:#b4c8de}.custom-control-input[disabled] ~ .custom-control-label,.custom-control-input:disabled ~ .custom-control-label{color:#777}.custom-control-input[disabled] ~ .custom-control-label::before,.custom-control-input:disabled ~ .custom-control-label::before{background-color:#eee}.custom-control-label{position:relative;margin-bottom:0;vertical-align:top}.custom-control-label::before{position:absolute;top:0.25rem;left:-1.5rem;display:block;width:1rem;height:1rem;pointer-events:none;content:"";background-color:#fff;border:#999 solid 1px}.custom-control-label::after{position:absolute;top:0.25rem;left:-1.5rem;display:block;width:1rem;height:1rem;content:"";background:no-repeat 50% / 50% 50%}.custom-checkbox .custom-control-label::before{border-radius:0.25rem}.custom-checkbox .custom-control-input:checked ~ .custom-control-label::after{background-image:url("data:image/svg+xml,%3csvg xmlns='http://www.w3.org/2000/svg' width='8' height='8' viewBox='0 0 8 8'%3e%3cpath fill='%23fff' d='M6.564.75l-3.59 3.612-1.538-1.55L0 4.26l2.974 2.99L8 2.193z'/%3e%3c/svg%3e")}.custom-checkbox .custom-control-input:indeterminate ~ .custom-control-label::before{border-color:#446E9B;background-color:#446E9B}.custom-checkbox .custom-control-input:indeterminate ~ .custom-control-label::after{background-image:url("data:image/svg+xml,%3csvg xmlns='http://www.w3.org/2000/svg' width='4' height='4' viewBox='0 0 4 4'%3e%3cpath stroke='%23fff' d='M0 2h4'/%3e%3c/svg%3e")}.custom-checkbox .custom-control-input:disabled:checked ~ .custom-control-label::before{background-color:rgba(68,110,155,0.5)}.custom-checkbox .custom-control-input:disabled:indeterminate ~ .custom-control-label::before{background-color:rgba(68,110,155,0.5)}.custom-radio .custom-control-label::before{border-radius:50%}.custom-radio .custom-control-input:checked ~ .custom-control-label::after{background-image:url("data:image/svg+xml,%3csvg xmlns='http://www.w3.org/2000/svg' width='12' height='12' viewBox='-4 -4 8 8'%3e%3ccircle r='3' fill='%23fff'/%3e%3c/svg%3e")}.custom-radio .custom-control-input:disabled:checked ~ .custom-control-label::before{background-color:rgba(68,110,155,0.5)}.custom-switch{padding-left:2.25rem}.custom-switch .custom-control-label::before{left:-2.25rem;width:1.75rem;pointer-events:all;border-radius:0.5rem}.custom-switch .custom-control-label::after{top:calc(0.25rem + 2px);left:calc(-2.25rem + 2px);width:calc(1rem - 4px);height:calc(1rem - 4px);background-color:#999;border-radius:0.5rem;-webkit-transition:background-color 0.15s ease-in-out, border-color 0.15s ease-in-out, -webkit-transform 0.15s ease-in-out, -webkit-box-shadow 0.15s ease-in-out;transition:background-color 0.15s ease-in-out, border-color 0.15s ease-in-out, -webkit-transform 0.15s ease-in-out, -webkit-box-shadow 0.15s ease-in-out;transition:transform 0.15s ease-in-out, background-color 0.15s ease-in-out, border-color 0.15s ease-in-out, box-shadow 0.15s ease-in-out;transition:transform 0.15s ease-in-out, background-color 0.15s ease-in-out, border-color 0.15s ease-in-out, box-shadow 0.15s ease-in-out, -webkit-transform 0.15s ease-in-out, -webkit-box-shadow 0.15s ease-in-out}@media (prefers-reduced-motion: reduce){.custom-switch .custom-control-label::after{-webkit-transition:none;transition:none}}.custom-switch .custom-control-input:checked ~ .custom-control-label::after{background-color:#fff;-webkit-transform:translateX(0.75rem);transform:translateX(0.75rem)}.custom-switch .custom-control-input:disabled:checked ~ .custom-control-label::before{background-color:rgba(68,110,155,0.5)}.custom-select{display:inline-block;width:100%;height:calc(1.5em + 0.75rem + 2px);padding:0.375rem 1.75rem 0.375rem 0.75rem;font-size:1rem;font-weight:400;line-height:1.5;color:#495057;vertical-align:middle;background:#fff url("data:image/svg+xml,%3csvg xmlns='http://www.w3.org/2000/svg' width='4' height='5' viewBox='0 0 4 5'%3e%3cpath fill='%23333' d='M2 0L0 2h4zm0 5L0 3h4z'/%3e%3c/svg%3e") no-repeat right 0.75rem center/8px 10px;border:1px solid #ced4da;border-radius:0.25rem;-webkit-appearance:none;-moz-appearance:none;appearance:none}.custom-select:focus{border-color:#90aece;outline:0;-webkit-box-shadow:0 0 0 0.2rem rgba(68,110,155,0.25);box-shadow:0 0 0 0.2rem rgba(68,110,155,0.25)}.custom-select:focus::-ms-value{color:#495057;background-color:#fff}.custom-select[multiple],.custom-select[size]:not([size="1"]){height:auto;padding-right:0.75rem;background-image:none}.custom-select:disabled{color:#777;background-color:#eee}.custom-select::-ms-expand{display:none}.custom-select:-moz-focusring{color:transparent;text-shadow:0 0 0 #495057}.custom-select-sm{height:calc(1.5em + 0.5rem + 2px);padding-top:0.25rem;padding-bottom:0.25rem;padding-left:0.5rem;font-size:0.875rem}.custom-select-lg{height:calc(1.5em + 1rem + 2px);padding-top:0.5rem;padding-bottom:0.5rem;padding-left:1rem;font-size:1.25rem}.custom-file{position:relative;display:inline-block;width:100%;height:calc(1.5em + 0.75rem + 2px);margin-bottom:0}.custom-file-input{position:relative;z-index:2;width:100%;height:calc(1.5em + 0.75rem + 2px);margin:0;opacity:0}.custom-file-input:focus ~ .custom-file-label{border-color:#90aece;-webkit-box-shadow:0 0 0 0.2rem rgba(68,110,155,0.25);box-shadow:0 0 0 0.2rem rgba(68,110,155,0.25)}.custom-file-input[disabled] ~ .custom-file-label,.custom-file-input:disabled ~ .custom-file-label{background-color:#eee}.custom-file-input:lang(en) ~ .custom-file-label::after{content:"Browse"}.custom-file-input ~ .custom-file-label[data-browse]::after{content:attr(data-browse)}.custom-file-label{position:absolute;top:0;right:0;left:0;z-index:1;height:calc(1.5em + 0.75rem + 2px);padding:0.375rem 0.75rem;font-weight:400;line-height:1.5;color:#495057;background-color:#fff;border:1px solid #ced4da;border-radius:0.25rem}.custom-file-label::after{position:absolute;top:0;right:0;bottom:0;z-index:3;display:block;height:calc(1.5em + 0.75rem);padding:0.375rem 0.75rem;line-height:1.5;color:#495057;content:"Browse";background-color:#eee;border-left:inherit;border-radius:0 0.25rem 0.25rem 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.page-link{margin-left:0;border-top-left-radius:0.25rem;border-bottom-left-radius:0.25rem}.page-item:last-child .page-link{border-top-right-radius:0.25rem;border-bottom-right-radius:0.25rem}.page-item.active .page-link{z-index:3;color:#fff;background-color:#446E9B;border-color:#446E9B}.page-item.disabled .page-link{color:#777;pointer-events:none;cursor:auto;background-color:#fff;border-color:#dee2e6}.pagination-lg .page-link{padding:0.75rem 1.5rem;font-size:1.25rem;line-height:1.5}.pagination-lg .page-item:first-child .page-link{border-top-left-radius:0.3rem;border-bottom-left-radius:0.3rem}.pagination-lg .page-item:last-child .page-link{border-top-right-radius:0.3rem;border-bottom-right-radius:0.3rem}.pagination-sm .page-link{padding:0.25rem 0.5rem;font-size:0.875rem;line-height:1.5}.pagination-sm .page-item:first-child .page-link{border-top-left-radius:0.2rem;border-bottom-left-radius:0.2rem}.pagination-sm .page-item:last-child .page-link{border-top-right-radius:0.2rem;border-bottom-right-radius:0.2rem}.badge{display:inline-block;padding:0.25em 0.4em;font-size:75%;font-weight:700;line-height:1;text-align:center;white-space:nowrap;vertical-align:baseline;border-radius:0.25rem;-webkit-transition:color 0.15s ease-in-out, background-color 0.15s ease-in-out, border-color 0.15s ease-in-out, -webkit-box-shadow 0.15s ease-in-out;transition:color 0.15s ease-in-out, background-color 0.15s ease-in-out, border-color 0.15s ease-in-out, -webkit-box-shadow 0.15s ease-in-out;transition:color 0.15s ease-in-out, background-color 0.15s ease-in-out, border-color 0.15s ease-in-out, box-shadow 0.15s ease-in-out;transition:color 0.15s ease-in-out, background-color 0.15s ease-in-out, border-color 0.15s ease-in-out, box-shadow 0.15s ease-in-out, -webkit-box-shadow 0.15s ease-in-out}@media (prefers-reduced-motion: reduce){.badge{-webkit-transition:none;transition:none}}a.badge:hover,a.badge:focus{text-decoration:none}.badge:empty{display:none}.btn .badge{position:relative;top:-1px}.badge-pill{padding-right:0.6em;padding-left:0.6em;border-radius:10rem}.badge-primary{color:#fff;background-color:#446E9B}a.badge-primary:hover,a.badge-primary:focus{color:#fff;background-color:#345578}a.badge-primary:focus,a.badge-primary.focus{outline:0;-webkit-box-shadow:0 0 0 0.2rem rgba(68,110,155,0.5);box-shadow:0 0 0 0.2rem rgba(68,110,155,0.5)}.badge-secondary{color:#fff;background-color:#999}a.badge-secondary:hover,a.badge-secondary:focus{color:#fff;background-color:#807f7f}a.badge-secondary:focus,a.badge-secondary.focus{outline:0;-webkit-box-shadow:0 0 0 0.2rem rgba(153,153,153,0.5);box-shadow:0 0 0 0.2rem rgba(153,153,153,0.5)}.badge-success{color:#fff;background-color:#3CB521}a.badge-success:hover,a.badge-success:focus{color:#fff;background-color:#2e8a19}a.badge-success:focus,a.badge-success.focus{outline:0;-webkit-box-shadow:0 0 0 0.2rem rgba(60,181,33,0.5);box-shadow:0 0 0 0.2rem rgba(60,181,33,0.5)}.badge-info{color:#fff;background-color:#3399F3}a.badge-info:hover,a.badge-info:focus{color:#fff;background-color:#0e80e6}a.badge-info:focus,a.badge-info.focus{outline:0;-webkit-box-shadow:0 0 0 0.2rem rgba(51,153,243,0.5);box-shadow:0 0 0 0.2rem rgba(51,153,243,0.5)}.badge-warning{color:#fff;background-color:#D47500}a.badge-warning:hover,a.badge-warning:focus{color:#fff;background-color:#a15900}a.badge-warning:focus,a.badge-warning.focus{outline:0;-webkit-box-shadow:0 0 0 0.2rem rgba(212,117,0,0.5);box-shadow:0 0 0 0.2rem rgba(212,117,0,0.5)}.badge-danger{color:#fff;background-color:#CD0200}a.badge-danger:hover,a.badge-danger:focus{color:#fff;background-color:#9a0200}a.badge-danger:focus,a.badge-danger.focus{outline:0;-webkit-box-shadow:0 0 0 0.2rem rgba(205,2,0,0.5);box-shadow:0 0 0 0.2rem rgba(205,2,0,0.5)}.badge-light{color:#2d2d2d;background-color:#eee}a.badge-light:hover,a.badge-light:focus{color:#2d2d2d;background-color:#d5d4d4}a.badge-light:focus,a.badge-light.focus{outline:0;-webkit-box-shadow:0 0 0 0.2rem rgba(238,238,238,0.5);box-shadow:0 0 0 0.2rem rgba(238,238,238,0.5)}.badge-dark{color:#fff;background-color:#333}a.badge-dark:hover,a.badge-dark:focus{color:#fff;background-color:#1a1919}a.badge-dark:focus,a.badge-dark.focus{outline:0;-webkit-box-shadow:0 0 0 0.2rem rgba(51,51,51,0.5);box-shadow:0 0 0 0.2rem rgba(51,51,51,0.5)}.jumbotron{padding:2rem 1rem;margin-bottom:2rem;background-color:#eee;border-radius:0.3rem}@media (min-width: 576px){.jumbotron{padding:4rem 2rem}}.jumbotron-fluid{padding-right:0;padding-left:0;border-radius:0}.alert{position:relative;padding:0.75rem 1.25rem;margin-bottom:1rem;border:1px solid transparent;border-radius:0.25rem}.alert-heading{color:inherit}.alert-link{font-weight:700}.alert-dismissible{padding-right:4rem}.alert-dismissible .close{position:absolute;top:0;right:0;padding:0.75rem 1.25rem;color:inherit}.alert-primary{color:#233951;background-color:#dae2eb;border-color:#cbd6e3}.alert-primary hr{border-top-color:#bac9da}.alert-primary .alert-link{color:#14202d}.alert-secondary{color:#505050;background-color:#ebebeb;border-color:#e2e2e2}.alert-secondary hr{border-top-color:#d5d5d5}.alert-secondary .alert-link{color:#373636}.alert-success{color:#1f5e11;background-color:#d8f0d3;border-color:#c8eac1}.alert-success hr{border-top-color:#b7e4ae}.alert-success .alert-link{color:#113309}.alert-info{color:#1b507e;background-color:#d6ebfd;border-color:#c6e2fc}.alert-info hr{border-top-color:#aed6fb}.alert-info .alert-link{color:#123554}.alert-warning{color:#6e3d00;background-color:#f6e3cc;border-color:#f3d8b8}.alert-warning hr{border-top-color:#efcca2}.alert-warning .alert-link{color:#3b2100}.alert-danger{color:#6b0100;background-color:#f5cccc;border-color:#f1b8b8}.alert-danger hr{border-top-color:#eda3a3}.alert-danger .alert-link{color:#380100}.alert-light{color:#7c7c7c;background-color:#fcfcfc;border-color:#fafafa}.alert-light hr{border-top-color:#ededed}.alert-light .alert-link{color:#636262}.alert-dark{color:#1b1b1b;background-color:#d6d6d6;border-color:#c6c6c6}.alert-dark hr{border-top-color:#b9b9b9}.alert-dark .alert-link{color:#020101}@-webkit-keyframes progress-bar-stripes{from{background-position:1rem 0}to{background-position:0 0}}@keyframes progress-bar-stripes{from{background-position:1rem 0}to{background-position:0 0}}.progress{display:-webkit-box;display:-ms-flexbox;display:flex;height:1rem;overflow:hidden;font-size:0.75rem;background-color:#eee;border-radius:0.25rem}.progress-bar{display:-webkit-box;display:-ms-flexbox;display:flex;-webkit-box-orient:vertical;-webkit-box-direction:normal;-ms-flex-direction:column;flex-direction:column;-webkit-box-pack:center;-ms-flex-pack:center;justify-content:center;overflow:hidden;color:#fff;text-align:center;white-space:nowrap;background-color:#446E9B;-webkit-transition:width 0.6s ease;transition:width 0.6s ease}@media (prefers-reduced-motion: reduce){.progress-bar{-webkit-transition:none;transition:none}}.progress-bar-striped{background-image:linear-gradient(45deg, rgba(255,255,255,0.15) 25%, transparent 25%, transparent 50%, rgba(255,255,255,0.15) 50%, rgba(255,255,255,0.15) 75%, transparent 75%, transparent);background-size:1rem 1rem}.progress-bar-animated{-webkit-animation:progress-bar-stripes 1s linear infinite;animation:progress-bar-stripes 1s linear 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.list-group-item+.list-group-item{border-top-width:1px;border-left-width:0}.list-group-horizontal .list-group-item+.list-group-item.active{margin-left:-1px;border-left-width:1px}@media (min-width: 576px){.list-group-horizontal-sm{-webkit-box-orient:horizontal;-webkit-box-direction:normal;-ms-flex-direction:row;flex-direction:row}.list-group-horizontal-sm .list-group-item:first-child{border-bottom-left-radius:0.25rem;border-top-right-radius:0}.list-group-horizontal-sm .list-group-item:last-child{border-top-right-radius:0.25rem;border-bottom-left-radius:0}.list-group-horizontal-sm .list-group-item.active{margin-top:0}.list-group-horizontal-sm .list-group-item+.list-group-item{border-top-width:1px;border-left-width:0}.list-group-horizontal-sm .list-group-item+.list-group-item.active{margin-left:-1px;border-left-width:1px}}@media (min-width: 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.list-group-item:last-child{border-top-right-radius:0.25rem;border-bottom-left-radius:0}.list-group-horizontal-lg .list-group-item.active{margin-top:0}.list-group-horizontal-lg .list-group-item+.list-group-item{border-top-width:1px;border-left-width:0}.list-group-horizontal-lg .list-group-item+.list-group-item.active{margin-left:-1px;border-left-width:1px}}@media (min-width: 1200px){.list-group-horizontal-xl{-webkit-box-orient:horizontal;-webkit-box-direction:normal;-ms-flex-direction:row;flex-direction:row}.list-group-horizontal-xl .list-group-item:first-child{border-bottom-left-radius:0.25rem;border-top-right-radius:0}.list-group-horizontal-xl .list-group-item:last-child{border-top-right-radius:0.25rem;border-bottom-left-radius:0}.list-group-horizontal-xl .list-group-item.active{margin-top:0}.list-group-horizontal-xl .list-group-item+.list-group-item{border-top-width:1px;border-left-width:0}.list-group-horizontal-xl 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0.75rem;color:#777;background-color:rgba(255,255,255,0.85);background-clip:padding-box;border-bottom:1px solid rgba(0,0,0,0.05)}.toast-body{padding:0.75rem}.modal-open{overflow:hidden}.modal-open .modal{overflow-x:hidden;overflow-y:auto}.modal{position:fixed;top:0;left:0;z-index:1050;display:none;width:100%;height:100%;overflow:hidden;outline:0}.modal-dialog{position:relative;width:auto;margin:0.5rem;pointer-events:none}.modal.fade .modal-dialog{-webkit-transition:-webkit-transform 0.3s ease-out;transition:-webkit-transform 0.3s ease-out;transition:transform 0.3s ease-out;transition:transform 0.3s ease-out, -webkit-transform 0.3s ease-out;-webkit-transform:translate(0, -50px);transform:translate(0, -50px)}@media (prefers-reduced-motion: reduce){.modal.fade .modal-dialog{-webkit-transition:none;transition:none}}.modal.show .modal-dialog{-webkit-transform:none;transform:none}.modal.modal-static 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nt:"\f083"}.fa-key:before{content:"\f084"}.fa-gears:before,.fa-cogs:before{content:"\f085"}.fa-comments:before{content:"\f086"}.fa-thumbs-o-up:before{content:"\f087"}.fa-thumbs-o-down:before{content:"\f088"}.fa-star-half:before{content:"\f089"}.fa-heart-o:before{content:"\f08a"}.fa-sign-out:before{content:"\f08b"}.fa-linkedin-square:before{content:"\f08c"}.fa-thumb-tack:before{content:"\f08d"}.fa-external-link:before{content:"\f08e"}.fa-sign-in:before{content:"\f090"}.fa-trophy:before{content:"\f091"}.fa-github-square:before{content:"\f092"}.fa-upload:before{content:"\f093"}.fa-lemon-o:before{content:"\f094"}.fa-phone:before{content:"\f095"}.fa-square-o:before{content:"\f096"}.fa-bookmark-o:before{content:"\f097"}.fa-phone-square:before{content:"\f098"}.fa-twitter:before{content:"\f099"}.fa-facebook-f:before,.fa-facebook:before{content:"\f09a"}.fa-github:before{content:"\f09b"}.fa-unlock:before{content:"\f09c"}.fa-credit-card:before{content:"\f09d"}.fa-feed:before,.fa-rss:before{content:"\f09e"}.fa-hdd-o:before{content:"\f0a0"}.fa-bullhorn:before{content:"\f0a1"}.fa-bell:before{content:"\f0f3"}.fa-certificate:before{content:"\f0a3"}.fa-hand-o-right:before{content:"\f0a4"}.fa-hand-o-left:before{content:"\f0a5"}.fa-hand-o-up:before{content:"\f0a6"}.fa-hand-o-down:before{content:"\f0a7"}.fa-arrow-circle-left:before{content:"\f0a8"}.fa-arrow-circle-right:before{content:"\f0a9"}.fa-arrow-circle-up:before{content:"\f0aa"}.fa-arrow-circle-down:before{content:"\f0ab"}.fa-globe:before{content:"\f0ac"}.fa-wrench:before{content:"\f0ad"}.fa-tasks:before{content:"\f0ae"}.fa-filter:before{content:"\f0b0"}.fa-briefcase:before{content:"\f0b1"}.fa-arrows-alt:before{content:"\f0b2"}.fa-group:before,.fa-users:before{content:"\f0c0"}.fa-chain:before,.fa-link:before{content:"\f0c1"}.fa-cloud:before{content:"\f0c2"}.fa-flask:before{content:"\f0c3"}.fa-cut:before,.fa-scissors:before{content:"\f0c4"}.fa-copy:before,.fa-files-o:before{content:"\f0c5"}.fa-paperclip:before{content:"\f0c6"}.fa-save:before,.fa-floppy-o:before{content:"\f0c7"}.fa-square:before{content:"\f0c8"}.fa-navicon:before,.fa-reorder:before,.fa-bars:before{content:"\f0c9"}.fa-list-ul:before{content:"\f0ca"}.fa-list-ol:before{content:"\f0cb"}.fa-strikethrough:before{content:"\f0cc"}.fa-underline:before{content:"\f0cd"}.fa-table:before{content:"\f0ce"}.fa-magic:before{content:"\f0d0"}.fa-truck:before{content:"\f0d1"}.fa-pinterest:before{content:"\f0d2"}.fa-pinterest-square:before{content:"\f0d3"}.fa-google-plus-square:before{content:"\f0d4"}.fa-google-plus:before{content:"\f0d5"}.fa-money:before{content:"\f0d6"}.fa-caret-down:before{content:"\f0d7"}.fa-caret-up:before{content:"\f0d8"}.fa-caret-left:before{content:"\f0d9"}.fa-caret-right:before{content:"\f0da"}.fa-columns:before{content:"\f0db"}.fa-unsorted:before,.fa-sort:before{content:"\f0dc"}.fa-sort-down:before,.fa-sort-desc:before{content:"\f0dd"}.fa-sort-up:before,.fa-sort-asc:before{content:"\f0de"}.fa-envelope:before{content:"\f0e0"}.fa-linkedin:before{content:"\f0e1"}.fa-rotate-left:before,.fa-undo:before{content:"\f0e2"}.fa-legal:before,.fa-gavel:before{content:"\f0e3"}.fa-dashboard:before,.fa-tachometer:before{content:"\f0e4"}.fa-comment-o:before{content:"\f0e5"}.fa-comments-o:before{content:"\f0e6"}.fa-flash:before,.fa-bolt:before{content:"\f0e7"}.fa-sitemap:before{content:"\f0e8"}.fa-umbrella:before{content:"\f0e9"}.fa-paste:before,.fa-clipboard:before{content:"\f0ea"}.fa-lightbulb-o:before{content:"\f0eb"}.fa-exchange:before{content:"\f0ec"}.fa-cloud-download:before{content:"\f0ed"}.fa-cloud-upload:before{content:"\f0ee"}.fa-user-md:before{content:"\f0f0"}.fa-stethoscope:before{content:"\f0f1"}.fa-suitcase:before{content:"\f0f2"}.fa-bell-o:before{content:"\f0a2"}.fa-coffee:before{content:"\f0f4"}.fa-cutlery:before{content:"\f0f5"}.fa-file-text-o:before{content:"\f0f6"}.fa-building-o:before{content:"\f0f7"}.fa-hospital-o:before{content:"\f0f8"}.fa-ambulance:before{content:"\f0f9"}.fa-medkit:before{content:"\f0fa"}.fa-fighter-jet:before{content:"\f0fb"}.fa-beer:before{content:"\f0fc"}.fa-h-square:before{content:"\f0fd"}.fa-plus-square:before{content:"\f0fe"}.fa-angle-double-left:before{content:"\f100"}.fa-angle-double-right:before{content:"\f101"}.fa-angle-double-up:before{content:"\f102"}.fa-angle-double-down:before{content:"\f103"}.fa-angle-left:before{content:"\f104"}.fa-angle-right:before{content:"\f105"}.fa-angle-up:before{content:"\f106"}.fa-angle-down:before{content:"\f107"}.fa-desktop:before{content:"\f108"}.fa-laptop:before{content:"\f109"}.fa-tablet:before{content:"\f10a"}.fa-mobile-phone:before,.fa-mobile:before{content:"\f10b"}.fa-circle-o:before{content:"\f10c"}.fa-quote-left:before{content:"\f10d"}.fa-quote-right:before{content:"\f10e"}.fa-spinner:before{content:"\f110"}.fa-circle:before{content:"\f111"}.fa-mail-reply:before,.fa-reply:before{content:"\f112"}.fa-github-alt:before{content:"\f113"}.fa-folder-o:before{content:"\f114"}.fa-folder-open-o:before{content:"\f115"}.fa-smile-o:before{c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0, 0, 0);border:0}.sr-only-focusable:active,.sr-only-focusable:focus{position:static;width:auto;height:auto;margin:0;overflow:visible;clip:auto} diff --git a/documentation/index.html b/documentation/index.html new file mode 100644 index 0000000..31c73a0 --- /dev/null +++ b/documentation/index.html @@ -0,0 +1,310 @@ + + + + + + + + + + + Documentation - hIPPYlib + + + + + + + + + + + +
+
+ +
+ +

Documentation

+

The complete API reference of hIPPYlib is available at readthedocs.

+

Algorithms implemented in hIPPYlib are described in

+
+

Umberto Villa, Noemi Petra, and Omar Ghattas. 2021. HIPPYlib: An Extensible Software Framework for Large-Scale Inverse Problems Governed by PDEs: Part I: Deterministic Inversion and Linearized Bayesian Inference. ACM Trans. Math. Softw. 47, 2, Article 16 (March 2021), 34 pages. arXiv

+
+

Installation of stable releases

+

The latest hIPPYlib release depends on FEniCS versions 2019.1.

+

FEniCS needs to be built with the following dependecies:

+
    +
  • numpy, scipy, matplotlib, mpi4py
  • +
  • PETSc and petsc4py (version 3.7.0 or above)
  • +
  • SLEPc and slepc4py (version 3.7.0 or above)
  • +
  • PETSc dependencies: parmetis, scotch, suitesparse, superlu_dist, ml, hypre
  • +
  • (optional): mshr, jupyter
  • +
+
+

For detailed installation instructions of the latest stable release see here.

+
+

Custom Docker Images and conda packages for FEniCS 2019.1.0

+

While hIPPYlib can be used with the Docker images and conda packages from the ufficial FEniCS packages, +a customized FEniCS docker image and customized conda packages is available.

+

hIPPYlib 2.3.0 Docker container

+

A Docker image with a working installation of hIPPYlib 2.3.0 and FEniCS 2017.2 is available here. Numerical results presented in the manuscript hIPPYlib: A Software Framework for Large-Scale Inverse Problems Governed by PDEs: Part I: Deterministic Inversion and Linearized Bayesian Inference were obtained using the software in this image.

+
+
+ + + + + + + + + + + + + diff --git a/download/index.html b/download/index.html new file mode 100644 index 0000000..30f5e87 --- /dev/null +++ b/download/index.html @@ -0,0 +1,422 @@ + + + + + + + + + + + Download - hIPPYlib + + + + + + + + + + + +
+
+
+
+ +

Download

+

Latest release

+ +

All releases

+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
FilenameVersionRelease DateNotesDocumentation
hippylib-3.1.0.zip3.1.0Dec 2022Non-Gaussian noise model and interpolation utilitieshtml
hippylib-3.0.0.zip3.0.0Feb 2020Support for FEniCS 2019.1. New features and exampleshtml
hippylib-2.3.0.zip2.3.0Sept 2019Updated exampleshtml
hippylib-2.2.1.zip2.2.1March 2019Bugfixeshtml
hippylib-2.2.0.zip2.2.0Dec 2018Enchantments and installhtml
hippylib-2.1.1.zip2.1.1Oct 2018Publication to JOSShtml
hippylib-2.1.0.tar.gz2.1.0July 2018Enchantments & Bugfixeshtml
hippylib-2.0.0.tar.gz2.0.0June 2018Major releasehtml
hippylib-1.6.0.tar.gz1.6.0May 2018Bugfix
hippylib-1.5.0.tar.gz1.5.0Jan 2018Support for FEniCS 2017.2
hippylib-1.4.0.tar.gz1.4.0Nov 2017Support for Python 3
hippylib-1.3.0.tar.gz1.3.0June 2017Support for FEniCS 2017.1; GLPv2
hippylib-1.2.0.tar.gz1.2.0Apr 2017Support for FEniCS 2016.2
hippylib-1.1.0.tar.gz1.1.0Nov 2016Support for FEniCS 2016.1
hippylib-1.0.2.tar.gz1.0.2Sept 2016Enchantment
hippylib-1.0.1.tar.gz1.0.1Aug 2016Bugfix
hippylib-1.0.0.tar.gz1.0.0Aug 2016Initial release
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hIPPYlib releases as of 1.2.0 are also archived on Zenodo.

+

See the changelog on readthedocs for further details.

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+ +
+
+
+
+ +

hIPPYlib - Inverse Problem PYthon library

+

Build Status +Doc Status +status DOI

+

hIPPYlib implements state-of-the-art scalable adjoint-based algorithms for PDE-based deterministic and Bayesian inverse problems. It builds on FEniCS for the discretization of the PDE and on PETSc for scalable and efficient linear algebra operations and solvers.

+

Features

+
    +
  • +

    Friendly, compact, near-mathematical FEniCS notation to +express, differentiate, and discretize the PDE forward model and +likelihood function

    +
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    Large scale optimization algorithms, such as globalized inexact +Newton-CG method, to solve the inverse problem

    +
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    Randomized algorithms for trace estimation, eigenvalues and singular values decomposition.

    +
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    Scalable sampling of Gaussian random fields

    +
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    Linearized Bayesian inversion with low-rank based +representation of the posterior covariance

    +
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    Hessian informed MCMC algorithms to explore the posterior + distribution

    +
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    Forward propagation of uncertainty capabilities using Monte + Carlo and Taylor expansion control variates

    +
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+

See also our tutorial and list of related publications. For additional resources and tutorials please see the teaching material for the 2018 Gene Golub SIAM Summer School on Inverse Problems: Systematic Integration of Data with Models under Uncertainty available here.

+

The complete API reference is available here.

+

Latest Release

+ +

Contact

+

Developed by the hIPPYlib team at UT Austin, UC Merced, and WUSTL.

+

Slack channel

+

The hIPPYlib slack channel is a good resource to request and receive help with using hIPPYlib. Everyone is invited to read and take part in discussions. Discussions about development of new features in hIPPYlib also take place here. You can join our Slack community by filling in this form.

+

Please cite as

+
@article{VillaPetraGhattas21,
+    author = {Villa, Umberto and Petra, Noemi and Ghattas, Omar},
+    title = "{HIPPYlib: An Extensible Software Framework for Large-Scale Inverse Problems Governed by PDEs: Part I: Deterministic Inversion and Linearized Bayesian Inference}",
+    year = {2021},
+    issue_date = {March 2021},
+    publisher = {Association for Computing Machinery},
+    address = {New York, NY, USA},
+    volume = {47},
+    number = {2},
+    issn = {0098-3500},
+    url = {https://doi.org/10.1145/3428447},
+    doi = {10.1145/3428447},
+    journal = {ACM Trans. Math. Softw.},
+    month = apr,
+    articleno = {16},
+    numpages = {34}
+    }
+
+@article{VillaPetraGhattas18,
+title = "{hIPPYlib: an Extensible Software Framework for Large-scale Deterministic and Bayesian Inverse Problems}",
+author = {Villa, U. and Petra, N. and Ghattas, O.},
+journal = {Journal of Open Source Software},
+volume = {3},
+number = {30},
+page = {940},
+doi  = {10.21105/joss.00940},
+year = {2018}
+}
+
+@article{VillaPetraGhattas16,
+title = "{hIPPYlib: an Extensible Software Framework for Large-scale Deterministic and Bayesian Inverse Problems}",
+author = {Villa, U. and Petra, N. and Ghattas, O.},
+year = {2016},
+url = {http://hippylib.github.io},
+doi = {10.5281/zenodo.596931}
+}
+
+
+
+ +
+
+

Copyright © 2016-2023 The University of Texas at Austin, University of California Merced.
Copyright © 2018-2020 The University of Texas at Austin, University of California Merced, Washington University in St. Louis.
This material is based on work partially supported by the National Science Foundation under Grants No ACI-1550593, ACI-1550547.

+

Documentation built with MkDocs.

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+ + + + + + + + + + + + + diff --git a/js/base.js b/js/base.js new file mode 100644 index 0000000..b0f4726 --- /dev/null +++ b/js/base.js @@ -0,0 +1,283 @@ +function getSearchTerm() { + var sPageURL = window.location.search.substring(1); + var sURLVariables = sPageURL.split('&'); + for (var i = 0; i < sURLVariables.length; i++) { + var sParameterName = sURLVariables[i].split('='); + if (sParameterName[0] == 'q') { + return sParameterName[1]; + } + } +} + +function applyTopPadding() { + // Update various absolute positions to match where the main container + // starts. 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Obscure keys are omitted and their use is discouraged. +var keyCodes = { + 8: 'backspace', + 9: 'tab', + 13: 'enter', + 16: 'shift', + 17: 'ctrl', + 18: 'alt', + 19: 'pause/break', + 20: 'caps lock', + 27: 'escape', + 32: 'spacebar', + 33: 'page up', + 34: 'page down', + 35: 'end', + 36: 'home', + 37: '←', + 38: '↑', + 39: '→', + 40: '↓', + 45: 'insert', + 46: 'delete', + 48: '0', + 49: '1', + 50: '2', + 51: '3', + 52: '4', + 53: '5', + 54: '6', + 55: '7', + 56: '8', + 57: '9', + 65: 'a', + 66: 'b', + 67: 'c', + 68: 'd', + 69: 'e', + 70: 'f', + 71: 'g', + 72: 'h', + 73: 'i', + 74: 'j', + 75: 'k', + 76: 'l', + 77: 'm', + 78: 'n', + 79: 'o', + 80: 'p', + 81: 'q', + 82: 'r', + 83: 's', + 84: 't', + 85: 'u', + 86: 'v', + 87: 'w', + 88: 'x', + 89: 'y', + 90: 'z', + 91: 'Left Windows Key / Left ⌘', + 92: 'Right Windows Key', + 93: 'Windows Menu / Right ⌘', + 96: 'numpad 0', + 97: 'numpad 1', + 98: 'numpad 2', + 99: 'numpad 3', + 100: 'numpad 4', + 101: 'numpad 5', + 102: 'numpad 6', + 103: 'numpad 7', + 104: 'numpad 8', + 105: 'numpad 9', + 106: 'multiply', + 107: 'add', + 109: 'subtract', + 110: 'decimal point', + 111: 'divide', + 112: 'f1', + 113: 'f2', + 114: 'f3', + 115: 'f4', + 116: 'f5', + 117: 'f6', + 118: 'f7', + 119: 'f8', + 120: 'f9', + 121: 'f10', + 122: 'f11', + 123: 'f12', + 124: 'f13', + 125: 'f14', + 126: 'f15', + 127: 'f16', + 128: 'f17', + 129: 'f18', + 130: 'f19', + 131: 'f20', + 132: 'f21', + 133: 'f22', + 134: 'f23', + 135: 'f24', + 144: 'num lock', + 145: 'scroll lock', + 186: ';', + 187: '=', + 188: ',', + 189: '‐', + 190: '.', + 191: '?', + 192: '`', + 219: '[', + 220: '\', + 221: ']', + 222: ''', +}; diff --git a/js/bootstrap.min.js b/js/bootstrap.min.js new file mode 100644 index 0000000..ca013b7 --- /dev/null +++ b/js/bootstrap.min.js @@ -0,0 +1,7 @@ +/*! + * Bootstrap v4.3.1 (https://getbootstrap.com/) + * Copyright 2011-2019 The Bootstrap Authors (https://github.com/twbs/bootstrap/graphs/contributors) + * Licensed under MIT (https://github.com/twbs/bootstrap/blob/master/LICENSE) + */ +!function(t,e){"object"==typeof exports&&"undefined"!=typeof module?e(exports,require("jquery"),require("popper.js")):"function"==typeof define&&define.amd?define(["exports","jquery","popper.js"],e):e((t=t||self).bootstrap={},t.jQuery,t.Popper)}(this,function(t,g,u){"use strict";function i(t,e){for(var n=0;nthis._items.length-1||t<0))if(this._isSliding)g(this._element).one(Q.SLID,function(){return e.to(t)});else{if(n===t)return this.pause(),void this.cycle();var i=ndocument.documentElement.clientHeight;!this._isBodyOverflowing&&t&&(this._element.style.paddingLeft=this._scrollbarWidth+"px"),this._isBodyOverflowing&&!t&&(this._element.style.paddingRight=this._scrollbarWidth+"px")},t._resetAdjustments=function(){this._element.style.paddingLeft="",this._element.style.paddingRight=""},t._checkScrollbar=function(){var t=document.body.getBoundingClientRect();this._isBodyOverflowing=t.left+t.right
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+
+
+
+ +

Selected oral presentations

+
+
+
+ +
+
+

Copyright © 2016-2023 The University of Texas at Austin, University of California Merced.
Copyright © 2018-2020 The University of Texas at Austin, University of California Merced, Washington University in St. Louis.
This material is based on work partially supported by the National Science Foundation under Grants No ACI-1550593, ACI-1550547.

+

Documentation built with MkDocs.

+
+ + + + + + + + + + + diff --git a/outreach/index.html b/outreach/index.html new file mode 100644 index 0000000..bf9c293 --- /dev/null +++ b/outreach/index.html @@ -0,0 +1,320 @@ + + + + + + + + + + + Outreach - hIPPYlib + + + + + + + + + + + +
+
+
+
+ +

Outreach

+

Schools and workshops

+ +

Graduate level courses

+
    +
  • +

    U. Villa @WUSTL, Computational Methods for Imaging Science, Spring 2020

    +
  • +
  • +

    U. Villa @WUSTL, Computational Methods for Imaging Science, Spring 2018

    +
  • +
  • +

    O. Ghattas @UT Austin, Computational and Variational Inverse Problems, Fall 2017 link

    +
  • +
  • +

    A. Alexanderian @NC State, Inverse problems, Fall 2016

    +
  • +
  • +

    G. Stadler @NYU, Advanced Topics in Numerical Analysis: Computational and Variational Methods for Inverse Problems, Spring 2016 link

    +
  • +
  • +

    N. Petra @UC Merced, Special Topics: Computational and Variational Inverse Problems, Fall 2015 link

    +
  • +
  • +

    O. Ghattas @UT Austin, Computational and Variational Inverse Problems, Fall 2015 link

    +
  • +
+
+
+ +
+
+

Copyright © 2016-2023 The University of Texas at Austin, University of California Merced.
Copyright © 2018-2020 The University of Texas at Austin, University of California Merced, Washington University in St. Louis.
This material is based on work partially supported by the National Science Foundation under Grants No ACI-1550593, ACI-1550547.

+

Documentation built with MkDocs.

+
+ + + + + + + + + + + diff --git a/postdoc_position/index.html b/postdoc_position/index.html new file mode 100644 index 0000000..a2e82fa --- /dev/null +++ b/postdoc_position/index.html @@ -0,0 +1,287 @@ + + + + + + + + + + + Openings - hIPPYlib + + + + + + + + + + + +
+
+
+
+ +

Openings

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Open Postdoc Position at UC Merced

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There is an opening for a postdoc position in Professor Noemi Petra's +research group in the School of Natural Sciences at the University of +California, Merced.

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The postdoctoral researcher will work under an +NSF-funded Collaborative Research (with UC Merced, UT Austin and MIT) +entitled: Integrating Data with Complex Predictive Models under +Uncertainty: An Extensible Software Framework for Large-Scale Bayesian +Inversion (see here for a general overview of the project).

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The postdoctoral researcher will perform research in the +field of large-scale Bayesian inverse problems, and on the +implementation of new features in hIPPYlib. The postdoc will contribute to +the research dissemination and will also help build the user/developer +community by attending and speaking at conferences, workshops and +summer schools at local and international events.

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Interested candidates should contact Noemi Petra at +npetra@ucmerced.edu.

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Apply at https://aprecruit.ucmerced.edu/apply/JPF00738.

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Copyright © 2016-2023 The University of Texas at Austin, University of California Merced.
Copyright © 2018-2020 The University of Texas at Austin, University of California Merced, Washington University in St. Louis.
This material is based on work partially supported by the National Science Foundation under Grants No ACI-1550593, ACI-1550547.

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Documentation built with MkDocs.

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Research

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Applications

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    Inversion for optical properties of biological tissues in quantitative optoacoustic tomography

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    Statistical treatment of inverse problems constrained by stochastic models

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    Accounting for model errors in inverse problems

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    Bayesian optimal experimental design for inverse problems in acoustic scattering

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    Inversion and control for CO2 sequestration with poroelastic models

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    Goal-oriented inference for reservoir models with complex features including faults

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    Joint seismic-electromagnetic inversion

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    Inference of basal boundary conditions for ice sheet flow

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    Inversion for material properties of cardiac tissue

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    Inference, prediction and optimization under uncertainty for turbulent combustion

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Selected publications

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Selected Ph.D. thesis

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Selected Honor and Master thesis

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Selected poster presentations

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Did you publish an article or give an oral/poster presentation using hIPPYlib? Please let us know by creating an issue and we will reference your work on this page.

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Copyright © 2016-2023 The University of Texas at Austin, University of California Merced.
Copyright © 2018-2020 The University of Texas at Austin, University of California Merced, Washington University in St. Louis.
This material is based on work partially supported by the National Science Foundation under Grants No ACI-1550593, ACI-1550547.

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Documentation built with MkDocs.

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+ + + + + + + + + + + diff --git a/search/lunr.js b/search/lunr.js new file mode 100644 index 0000000..aca0a16 --- /dev/null +++ b/search/lunr.js @@ -0,0 +1,3475 @@ +/** + * lunr - http://lunrjs.com - A bit like Solr, but much smaller and not as bright - 2.3.9 + * Copyright (C) 2020 Oliver Nightingale + * @license MIT + */ + +;(function(){ + +/** + * A convenience function for configuring and constructing + * a new lunr Index. + * + * A lunr.Builder instance is created and the pipeline setup + * with a trimmer, stop word filter and stemmer. + * + * This builder object is yielded to the configuration function + * that is passed as a parameter, allowing the list of fields + * and other builder parameters to be customised. + * + * All documents _must_ be added within the passed config function. + * + * @example + * var idx = lunr(function () { + * this.field('title') + * this.field('body') + * this.ref('id') + * + * documents.forEach(function (doc) { + * this.add(doc) + * }, this) + * }) + * + * @see {@link lunr.Builder} + * @see {@link lunr.Pipeline} + * @see {@link lunr.trimmer} + * @see {@link lunr.stopWordFilter} + * @see {@link lunr.stemmer} + * @namespace {function} lunr + */ +var lunr = function (config) { + var builder = new lunr.Builder + + builder.pipeline.add( + lunr.trimmer, + lunr.stopWordFilter, + lunr.stemmer + ) + + builder.searchPipeline.add( + lunr.stemmer + ) + + config.call(builder, builder) + return builder.build() +} + +lunr.version = "2.3.9" +/*! + * lunr.utils + * Copyright (C) 2020 Oliver Nightingale + */ + +/** + * A namespace containing utils for the rest of the lunr library + * @namespace lunr.utils + */ +lunr.utils = {} + +/** + * Print a warning message to the console. + * + * @param {String} message The message to be printed. + * @memberOf lunr.utils + * @function + */ +lunr.utils.warn = (function (global) { + /* eslint-disable no-console */ + return function (message) { + if (global.console && console.warn) { + console.warn(message) + } + } + /* eslint-enable no-console */ +})(this) + +/** + * Convert an object to a string. + * + * In the case of `null` and `undefined` the function returns + * the empty string, in all other cases the result of calling + * `toString` on the passed object is returned. + * + * @param {Any} obj The object to convert to a string. + * @return {String} string representation of the passed object. + * @memberOf lunr.utils + */ +lunr.utils.asString = function (obj) { + if (obj === void 0 || obj === null) { + return "" + } else { + return obj.toString() + } +} + +/** + * Clones an object. + * + * Will create a copy of an existing object such that any mutations + * on the copy cannot affect the original. + * + * Only shallow objects are supported, passing a nested object to this + * function will cause a TypeError. + * + * Objects with primitives, and arrays of primitives are supported. + * + * @param {Object} obj The object to clone. + * @return {Object} a clone of the passed object. + * @throws {TypeError} when a nested object is passed. + * @memberOf Utils + */ +lunr.utils.clone = function (obj) { + if (obj === null || obj === undefined) { + return obj + } + + var clone = Object.create(null), + keys = Object.keys(obj) + + for (var i = 0; i < keys.length; i++) { + var key = keys[i], + val = obj[key] + + if (Array.isArray(val)) { + clone[key] = val.slice() + continue + } + + if (typeof val === 'string' || + typeof val === 'number' || + typeof val === 'boolean') { + clone[key] = val + continue + } + + throw new TypeError("clone is not deep and does not support nested objects") + } + + return clone +} +lunr.FieldRef = function (docRef, fieldName, stringValue) { + this.docRef = docRef + this.fieldName = fieldName + this._stringValue = stringValue +} + +lunr.FieldRef.joiner = "/" + +lunr.FieldRef.fromString = function (s) { + var n = s.indexOf(lunr.FieldRef.joiner) + + if (n === -1) { + throw "malformed field ref string" + } + + var fieldRef = s.slice(0, n), + docRef = s.slice(n + 1) + + return new lunr.FieldRef (docRef, fieldRef, s) +} + +lunr.FieldRef.prototype.toString = function () { + if (this._stringValue == undefined) { + this._stringValue = this.fieldName + lunr.FieldRef.joiner + this.docRef + } + + return this._stringValue +} +/*! + * lunr.Set + * Copyright (C) 2020 Oliver Nightingale + */ + +/** + * A lunr set. + * + * @constructor + */ +lunr.Set = function (elements) { + this.elements = Object.create(null) + + if (elements) { + this.length = elements.length + + for (var i = 0; i < this.length; i++) { + this.elements[elements[i]] = true + } + } else { + this.length = 0 + } +} + +/** + * A complete set that contains all elements. + * + * @static + * @readonly + * @type {lunr.Set} + */ +lunr.Set.complete = { + intersect: function (other) { + return other + }, + + union: function () { + return this + }, + + contains: function () { + return true + } +} + +/** + * An empty set that contains no elements. + * + * @static + * @readonly + * @type {lunr.Set} + */ +lunr.Set.empty = { + intersect: function () { + return this + }, + + union: function (other) { + return other + }, + + contains: function () { + return false + } +} + +/** + * Returns true if this set contains the specified object. + * + * @param {object} object - Object whose presence in this set is to be tested. + * @returns {boolean} - True if this set contains the specified object. + */ +lunr.Set.prototype.contains = function (object) { + return !!this.elements[object] +} + +/** + * Returns a new set containing only the elements that are present in both + * this set and the specified set. + * + * @param {lunr.Set} other - set to intersect with this set. + * @returns {lunr.Set} a new set that is the intersection of this and the specified set. + */ + +lunr.Set.prototype.intersect = function (other) { + var a, b, elements, intersection = [] + + if (other === lunr.Set.complete) { + return this + } + + if (other === lunr.Set.empty) { + return other + } + + if (this.length < other.length) { + a = this + b = other + } else { + a = other + b = this + } + + elements = Object.keys(a.elements) + + for (var i = 0; i < elements.length; i++) { + var element = elements[i] + if (element in b.elements) { + intersection.push(element) + } + } + + return new lunr.Set (intersection) +} + +/** + * Returns a new set combining the elements of this and the specified set. + * + * @param {lunr.Set} other - set to union with this set. + * @return {lunr.Set} a new set that is the union of this and the specified set. + */ + +lunr.Set.prototype.union = function (other) { + if (other === lunr.Set.complete) { + return lunr.Set.complete + } + + if (other === lunr.Set.empty) { + return this + } + + return new lunr.Set(Object.keys(this.elements).concat(Object.keys(other.elements))) +} +/** + * A function to calculate the inverse document frequency for + * a posting. This is shared between the builder and the index + * + * @private + * @param {object} posting - The posting for a given term + * @param {number} documentCount - The total number of documents. + */ +lunr.idf = function (posting, documentCount) { + var documentsWithTerm = 0 + + for (var fieldName in posting) { + if (fieldName == '_index') continue // Ignore the term index, its not a field + documentsWithTerm += Object.keys(posting[fieldName]).length + } + + var x = (documentCount - documentsWithTerm + 0.5) / (documentsWithTerm + 0.5) + + return Math.log(1 + Math.abs(x)) +} + +/** + * A token wraps a string representation of a token + * as it is passed through the text processing pipeline. + * + * @constructor + * @param {string} [str=''] - The string token being wrapped. + * @param {object} [metadata={}] - Metadata associated with this token. + */ +lunr.Token = function (str, metadata) { + this.str = str || "" + this.metadata = metadata || {} +} + +/** + * Returns the token string that is being wrapped by this object. + * + * @returns {string} + */ +lunr.Token.prototype.toString = function () { + return this.str +} + +/** + * A token update function is used when updating or optionally + * when cloning a token. + * + * @callback lunr.Token~updateFunction + * @param {string} str - The string representation of the token. + * @param {Object} metadata - All metadata associated with this token. + */ + +/** + * Applies the given function to the wrapped string token. + * + * @example + * token.update(function (str, metadata) { + * return str.toUpperCase() + * }) + * + * @param {lunr.Token~updateFunction} fn - A function to apply to the token string. + * @returns {lunr.Token} + */ +lunr.Token.prototype.update = function (fn) { + this.str = fn(this.str, this.metadata) + return this +} + +/** + * Creates a clone of this token. Optionally a function can be + * applied to the cloned token. + * + * @param {lunr.Token~updateFunction} [fn] - An optional function to apply to the cloned token. + * @returns {lunr.Token} + */ +lunr.Token.prototype.clone = function (fn) { + fn = fn || function (s) { return s } + return new lunr.Token (fn(this.str, this.metadata), this.metadata) +} +/*! + * lunr.tokenizer + * Copyright (C) 2020 Oliver Nightingale + */ + +/** + * A function for splitting a string into tokens ready to be inserted into + * the search index. Uses `lunr.tokenizer.separator` to split strings, change + * the value of this property to change how strings are split into tokens. + * + * This tokenizer will convert its parameter to a string by calling `toString` and + * then will split this string on the character in `lunr.tokenizer.separator`. + * Arrays will have their elements converted to strings and wrapped in a lunr.Token. + * + * Optional metadata can be passed to the tokenizer, this metadata will be cloned and + * added as metadata to every token that is created from the object to be tokenized. + * + * @static + * @param {?(string|object|object[])} obj - The object to convert into tokens + * @param {?object} metadata - Optional metadata to associate with every token + * @returns {lunr.Token[]} + * @see {@link lunr.Pipeline} + */ +lunr.tokenizer = function (obj, metadata) { + if (obj == null || obj == undefined) { + return [] + } + + if (Array.isArray(obj)) { + return obj.map(function (t) { + return new lunr.Token( + lunr.utils.asString(t).toLowerCase(), + lunr.utils.clone(metadata) + ) + }) + } + + var str = obj.toString().toLowerCase(), + len = str.length, + tokens = [] + + for (var sliceEnd = 0, sliceStart = 0; sliceEnd <= len; sliceEnd++) { + var char = str.charAt(sliceEnd), + sliceLength = sliceEnd - sliceStart + + if ((char.match(lunr.tokenizer.separator) || sliceEnd == len)) { + + if (sliceLength > 0) { + var tokenMetadata = lunr.utils.clone(metadata) || {} + tokenMetadata["position"] = [sliceStart, sliceLength] + tokenMetadata["index"] = tokens.length + + tokens.push( + new lunr.Token ( + str.slice(sliceStart, sliceEnd), + tokenMetadata + ) + ) + } + + sliceStart = sliceEnd + 1 + } + + } + + return tokens +} + +/** + * The separator used to split a string into tokens. Override this property to change the behaviour of + * `lunr.tokenizer` behaviour when tokenizing strings. By default this splits on whitespace and hyphens. + * + * @static + * @see lunr.tokenizer + */ +lunr.tokenizer.separator = /[\s\-]+/ +/*! + * lunr.Pipeline + * Copyright (C) 2020 Oliver Nightingale + */ + +/** + * lunr.Pipelines maintain an ordered list of functions to be applied to all + * tokens in documents entering the search index and queries being ran against + * the index. + * + * An instance of lunr.Index created with the lunr shortcut will contain a + * pipeline with a stop word filter and an English language stemmer. Extra + * functions can be added before or after either of these functions or these + * default functions can be removed. + * + * When run the pipeline will call each function in turn, passing a token, the + * index of that token in the original list of all tokens and finally a list of + * all the original tokens. + * + * The output of functions in the pipeline will be passed to the next function + * in the pipeline. To exclude a token from entering the index the function + * should return undefined, the rest of the pipeline will not be called with + * this token. + * + * For serialisation of pipelines to work, all functions used in an instance of + * a pipeline should be registered with lunr.Pipeline. Registered functions can + * then be loaded. If trying to load a serialised pipeline that uses functions + * that are not registered an error will be thrown. + * + * If not planning on serialising the pipeline then registering pipeline functions + * is not necessary. + * + * @constructor + */ +lunr.Pipeline = function () { + this._stack = [] +} + +lunr.Pipeline.registeredFunctions = Object.create(null) + +/** + * A pipeline function maps lunr.Token to lunr.Token. A lunr.Token contains the token + * string as well as all known metadata. A pipeline function can mutate the token string + * or mutate (or add) metadata for a given token. + * + * A pipeline function can indicate that the passed token should be discarded by returning + * null, undefined or an empty string. This token will not be passed to any downstream pipeline + * functions and will not be added to the index. + * + * Multiple tokens can be returned by returning an array of tokens. Each token will be passed + * to any downstream pipeline functions and all will returned tokens will be added to the index. + * + * Any number of pipeline functions may be chained together using a lunr.Pipeline. + * + * @interface lunr.PipelineFunction + * @param {lunr.Token} token - A token from the document being processed. + * @param {number} i - The index of this token in the complete list of tokens for this document/field. + * @param {lunr.Token[]} tokens - All tokens for this document/field. + * @returns {(?lunr.Token|lunr.Token[])} + */ + +/** + * Register a function with the pipeline. + * + * Functions that are used in the pipeline should be registered if the pipeline + * needs to be serialised, or a serialised pipeline needs to be loaded. + * + * Registering a function does not add it to a pipeline, functions must still be + * added to instances of the pipeline for them to be used when running a pipeline. + * + * @param {lunr.PipelineFunction} fn - The function to check for. + * @param {String} label - The label to register this function with + */ +lunr.Pipeline.registerFunction = function (fn, label) { + if (label in this.registeredFunctions) { + lunr.utils.warn('Overwriting existing registered function: ' + label) + } + + fn.label = label + lunr.Pipeline.registeredFunctions[fn.label] = fn +} + +/** + * Warns if the function is not registered as a Pipeline function. + * + * @param {lunr.PipelineFunction} fn - The function to check for. + * @private + */ +lunr.Pipeline.warnIfFunctionNotRegistered = function (fn) { + var isRegistered = fn.label && (fn.label in this.registeredFunctions) + + if (!isRegistered) { + lunr.utils.warn('Function is not registered with pipeline. This may cause problems when serialising the index.\n', fn) + } +} + +/** + * Loads a previously serialised pipeline. + * + * All functions to be loaded must already be registered with lunr.Pipeline. + * If any function from the serialised data has not been registered then an + * error will be thrown. + * + * @param {Object} serialised - The serialised pipeline to load. + * @returns {lunr.Pipeline} + */ +lunr.Pipeline.load = function (serialised) { + var pipeline = new lunr.Pipeline + + serialised.forEach(function (fnName) { + var fn = lunr.Pipeline.registeredFunctions[fnName] + + if (fn) { + pipeline.add(fn) + } else { + throw new Error('Cannot load unregistered function: ' + fnName) + } + }) + + return pipeline +} + +/** + * Adds new functions to the end of the pipeline. + * + * Logs a warning if the function has not been registered. + * + * @param {lunr.PipelineFunction[]} functions - Any number of functions to add to the pipeline. + */ +lunr.Pipeline.prototype.add = function () { + var fns = Array.prototype.slice.call(arguments) + + fns.forEach(function (fn) { + lunr.Pipeline.warnIfFunctionNotRegistered(fn) + this._stack.push(fn) + }, this) +} + +/** + * Adds a single function after a function that already exists in the + * pipeline. + * + * Logs a warning if the function has not been registered. + * + * @param {lunr.PipelineFunction} existingFn - A function that already exists in the pipeline. + * @param {lunr.PipelineFunction} newFn - The new function to add to the pipeline. + */ +lunr.Pipeline.prototype.after = function (existingFn, newFn) { + lunr.Pipeline.warnIfFunctionNotRegistered(newFn) + + var pos = this._stack.indexOf(existingFn) + if (pos == -1) { + throw new Error('Cannot find existingFn') + } + + pos = pos + 1 + this._stack.splice(pos, 0, newFn) +} + +/** + * Adds a single function before a function that already exists in the + * pipeline. + * + * Logs a warning if the function has not been registered. + * + * @param {lunr.PipelineFunction} existingFn - A function that already exists in the pipeline. + * @param {lunr.PipelineFunction} newFn - The new function to add to the pipeline. + */ +lunr.Pipeline.prototype.before = function (existingFn, newFn) { + lunr.Pipeline.warnIfFunctionNotRegistered(newFn) + + var pos = this._stack.indexOf(existingFn) + if (pos == -1) { + throw new Error('Cannot find existingFn') + } + + this._stack.splice(pos, 0, newFn) +} + +/** + * Removes a function from the pipeline. + * + * @param {lunr.PipelineFunction} fn The function to remove from the pipeline. + */ +lunr.Pipeline.prototype.remove = function (fn) { + var pos = this._stack.indexOf(fn) + if (pos == -1) { + return + } + + this._stack.splice(pos, 1) +} + +/** + * Runs the current list of functions that make up the pipeline against the + * passed tokens. + * + * @param {Array} tokens The tokens to run through the pipeline. + * @returns {Array} + */ +lunr.Pipeline.prototype.run = function (tokens) { + var stackLength = this._stack.length + + for (var i = 0; i < stackLength; i++) { + var fn = this._stack[i] + var memo = [] + + for (var j = 0; j < tokens.length; j++) { + var result = fn(tokens[j], j, tokens) + + if (result === null || result === void 0 || result === '') continue + + if (Array.isArray(result)) { + for (var k = 0; k < result.length; k++) { + memo.push(result[k]) + } + } else { + memo.push(result) + } + } + + tokens = memo + } + + return tokens +} + +/** + * Convenience method for passing a string through a pipeline and getting + * strings out. This method takes care of wrapping the passed string in a + * token and mapping the resulting tokens back to strings. + * + * @param {string} str - The string to pass through the pipeline. + * @param {?object} metadata - Optional metadata to associate with the token + * passed to the pipeline. + * @returns {string[]} + */ +lunr.Pipeline.prototype.runString = function (str, metadata) { + var token = new lunr.Token (str, metadata) + + return this.run([token]).map(function (t) { + return t.toString() + }) +} + +/** + * Resets the pipeline by removing any existing processors. + * + */ +lunr.Pipeline.prototype.reset = function () { + this._stack = [] +} + +/** + * Returns a representation of the pipeline ready for serialisation. + * + * Logs a warning if the function has not been registered. + * + * @returns {Array} + */ +lunr.Pipeline.prototype.toJSON = function () { + return this._stack.map(function (fn) { + lunr.Pipeline.warnIfFunctionNotRegistered(fn) + + return fn.label + }) +} +/*! + * lunr.Vector + * Copyright (C) 2020 Oliver Nightingale + */ + +/** + * A vector is used to construct the vector space of documents and queries. These + * vectors support operations to determine the similarity between two documents or + * a document and a query. + * + * Normally no parameters are required for initializing a vector, but in the case of + * loading a previously dumped vector the raw elements can be provided to the constructor. + * + * For performance reasons vectors are implemented with a flat array, where an elements + * index is immediately followed by its value. E.g. [index, value, index, value]. This + * allows the underlying array to be as sparse as possible and still offer decent + * performance when being used for vector calculations. + * + * @constructor + * @param {Number[]} [elements] - The flat list of element index and element value pairs. + */ +lunr.Vector = function (elements) { + this._magnitude = 0 + this.elements = elements || [] +} + + +/** + * Calculates the position within the vector to insert a given index. + * + * This is used internally by insert and upsert. If there are duplicate indexes then + * the position is returned as if the value for that index were to be updated, but it + * is the callers responsibility to check whether there is a duplicate at that index + * + * @param {Number} insertIdx - The index at which the element should be inserted. + * @returns {Number} + */ +lunr.Vector.prototype.positionForIndex = function (index) { + // For an empty vector the tuple can be inserted at the beginning + if (this.elements.length == 0) { + return 0 + } + + var start = 0, + end = this.elements.length / 2, + sliceLength = end - start, + pivotPoint = Math.floor(sliceLength / 2), + pivotIndex = this.elements[pivotPoint * 2] + + while (sliceLength > 1) { + if (pivotIndex < index) { + start = pivotPoint + } + + if (pivotIndex > index) { + end = pivotPoint + } + + if (pivotIndex == index) { + break + } + + sliceLength = end - start + pivotPoint = start + Math.floor(sliceLength / 2) + pivotIndex = this.elements[pivotPoint * 2] + } + + if (pivotIndex == index) { + return pivotPoint * 2 + } + + if (pivotIndex > index) { + return pivotPoint * 2 + } + + if (pivotIndex < index) { + return (pivotPoint + 1) * 2 + } +} + +/** + * Inserts an element at an index within the vector. + * + * Does not allow duplicates, will throw an error if there is already an entry + * for this index. + * + * @param {Number} insertIdx - The index at which the element should be inserted. + * @param {Number} val - The value to be inserted into the vector. + */ +lunr.Vector.prototype.insert = function (insertIdx, val) { + this.upsert(insertIdx, val, function () { + throw "duplicate index" + }) +} + +/** + * Inserts or updates an existing index within the vector. + * + * @param {Number} insertIdx - The index at which the element should be inserted. + * @param {Number} val - The value to be inserted into the vector. + * @param {function} fn - A function that is called for updates, the existing value and the + * requested value are passed as arguments + */ +lunr.Vector.prototype.upsert = function (insertIdx, val, fn) { + this._magnitude = 0 + var position = this.positionForIndex(insertIdx) + + if (this.elements[position] == insertIdx) { + this.elements[position + 1] = fn(this.elements[position + 1], val) + } else { + this.elements.splice(position, 0, insertIdx, val) + } +} + +/** + * Calculates the magnitude of this vector. + * + * @returns {Number} + */ +lunr.Vector.prototype.magnitude = function () { + if (this._magnitude) return this._magnitude + + var sumOfSquares = 0, + elementsLength = this.elements.length + + for (var i = 1; i < elementsLength; i += 2) { + var val = this.elements[i] + sumOfSquares += val * val + } + + return this._magnitude = Math.sqrt(sumOfSquares) +} + +/** + * Calculates the dot product of this vector and another vector. + * + * @param {lunr.Vector} otherVector - The vector to compute the dot product with. + * @returns {Number} + */ +lunr.Vector.prototype.dot = function (otherVector) { + var dotProduct = 0, + a = this.elements, b = otherVector.elements, + aLen = a.length, bLen = b.length, + aVal = 0, bVal = 0, + i = 0, j = 0 + + while (i < aLen && j < bLen) { + aVal = a[i], bVal = b[j] + if (aVal < bVal) { + i += 2 + } else if (aVal > bVal) { + j += 2 + } else if (aVal == bVal) { + dotProduct += a[i + 1] * b[j + 1] + i += 2 + j += 2 + } + } + + return dotProduct +} + +/** + * Calculates the similarity between this vector and another vector. + * + * @param {lunr.Vector} otherVector - The other vector to calculate the + * similarity with. + * @returns {Number} + */ +lunr.Vector.prototype.similarity = function (otherVector) { + return this.dot(otherVector) / this.magnitude() || 0 +} + +/** + * Converts the vector to an array of the elements within the vector. + * + * @returns {Number[]} + */ +lunr.Vector.prototype.toArray = function () { + var output = new Array (this.elements.length / 2) + + for (var i = 1, j = 0; i < this.elements.length; i += 2, j++) { + output[j] = this.elements[i] + } + + return output +} + +/** + * A JSON serializable representation of the vector. + * + * @returns {Number[]} + */ +lunr.Vector.prototype.toJSON = function () { + return this.elements +} +/* eslint-disable */ +/*! + * lunr.stemmer + * Copyright (C) 2020 Oliver Nightingale + * Includes code from - http://tartarus.org/~martin/PorterStemmer/js.txt + */ + +/** + * lunr.stemmer is an english language stemmer, this is a JavaScript + * implementation of the PorterStemmer taken from http://tartarus.org/~martin + * + * @static + * @implements {lunr.PipelineFunction} + * @param {lunr.Token} token - The string to stem + * @returns {lunr.Token} + * @see {@link lunr.Pipeline} + * @function + */ +lunr.stemmer = (function(){ + var step2list = { + "ational" : "ate", + "tional" : "tion", + "enci" : "ence", + "anci" : "ance", + "izer" : "ize", + "bli" : "ble", + "alli" : "al", + "entli" : "ent", + "eli" : "e", + "ousli" : "ous", + "ization" : "ize", + "ation" : "ate", + "ator" : "ate", + "alism" : "al", + "iveness" : "ive", + "fulness" : "ful", + "ousness" : "ous", + "aliti" : "al", + "iviti" : "ive", + "biliti" : "ble", + "logi" : "log" + }, + + step3list = { + "icate" : "ic", + "ative" : "", + "alize" : "al", + "iciti" : "ic", + "ical" : "ic", + "ful" : "", + "ness" : "" + }, + + c = "[^aeiou]", // consonant + v = "[aeiouy]", // vowel + C = c + "[^aeiouy]*", // consonant sequence + V = v + "[aeiou]*", // vowel sequence + + mgr0 = "^(" + C + ")?" + V + C, // [C]VC... is m>0 + meq1 = "^(" + C + ")?" + V + C + "(" + V + ")?$", // [C]VC[V] is m=1 + mgr1 = "^(" + C + ")?" + V + C + V + C, // [C]VCVC... is m>1 + s_v = "^(" + C + ")?" + v; // vowel in stem + + var re_mgr0 = new RegExp(mgr0); + var re_mgr1 = new RegExp(mgr1); + var re_meq1 = new RegExp(meq1); + var re_s_v = new RegExp(s_v); + + var re_1a = /^(.+?)(ss|i)es$/; + var re2_1a = /^(.+?)([^s])s$/; + var re_1b = /^(.+?)eed$/; + var re2_1b = /^(.+?)(ed|ing)$/; + var re_1b_2 = /.$/; + var re2_1b_2 = /(at|bl|iz)$/; + var re3_1b_2 = new RegExp("([^aeiouylsz])\\1$"); + var re4_1b_2 = new RegExp("^" + C + v + "[^aeiouwxy]$"); + + var re_1c = /^(.+?[^aeiou])y$/; + var re_2 = /^(.+?)(ational|tional|enci|anci|izer|bli|alli|entli|eli|ousli|ization|ation|ator|alism|iveness|fulness|ousness|aliti|iviti|biliti|logi)$/; + + var re_3 = /^(.+?)(icate|ative|alize|iciti|ical|ful|ness)$/; + + var re_4 = /^(.+?)(al|ance|ence|er|ic|able|ible|ant|ement|ment|ent|ou|ism|ate|iti|ous|ive|ize)$/; + var re2_4 = /^(.+?)(s|t)(ion)$/; + + var re_5 = /^(.+?)e$/; + var re_5_1 = /ll$/; + var re3_5 = new RegExp("^" + C + v + "[^aeiouwxy]$"); + + var porterStemmer = function porterStemmer(w) { + var stem, + suffix, + firstch, + re, + re2, + re3, + re4; + + if (w.length < 3) { return w; } + + firstch = w.substr(0,1); + if (firstch == "y") { + w = firstch.toUpperCase() + w.substr(1); + } + + // Step 1a + re = re_1a + re2 = re2_1a; + + if (re.test(w)) { w = w.replace(re,"$1$2"); } + else if (re2.test(w)) { w = w.replace(re2,"$1$2"); } + + // Step 1b + re = re_1b; + re2 = re2_1b; + if (re.test(w)) { + var fp = re.exec(w); + re = re_mgr0; + if (re.test(fp[1])) { + re = re_1b_2; + w = w.replace(re,""); + } + } else if (re2.test(w)) { + var fp = re2.exec(w); + stem = fp[1]; + re2 = re_s_v; + if (re2.test(stem)) { + w = stem; + re2 = re2_1b_2; + re3 = re3_1b_2; + re4 = re4_1b_2; + if (re2.test(w)) { w = w + "e"; } + else if (re3.test(w)) { re = re_1b_2; w = w.replace(re,""); } + else if (re4.test(w)) { w = w + "e"; } + } + } + + // Step 1c - replace suffix y or Y by i if preceded by a non-vowel which is not the first letter of the word (so cry -> cri, by -> by, say -> say) + re = re_1c; + if (re.test(w)) { + var fp = re.exec(w); + stem = fp[1]; + w = stem + "i"; + } + + // Step 2 + re = re_2; + if (re.test(w)) { + var fp = re.exec(w); + stem = fp[1]; + suffix = fp[2]; + re = re_mgr0; + if (re.test(stem)) { + w = stem + step2list[suffix]; + } + } + + // Step 3 + re = re_3; + if (re.test(w)) { + var fp = re.exec(w); + stem = fp[1]; + suffix = fp[2]; + re = re_mgr0; + if (re.test(stem)) { + w = stem + step3list[suffix]; + } + } + + // Step 4 + re = re_4; + re2 = re2_4; + if (re.test(w)) { + var fp = re.exec(w); + stem = fp[1]; + re = re_mgr1; + if (re.test(stem)) { + w = stem; + } + } else if (re2.test(w)) { + var fp = re2.exec(w); + stem = fp[1] + fp[2]; + re2 = re_mgr1; + if (re2.test(stem)) { + w = stem; + } + } + + // Step 5 + re = re_5; + if (re.test(w)) { + var fp = re.exec(w); + stem = fp[1]; + re = re_mgr1; + re2 = re_meq1; + re3 = re3_5; + if (re.test(stem) || (re2.test(stem) && !(re3.test(stem)))) { + w = stem; + } + } + + re = re_5_1; + re2 = re_mgr1; + if (re.test(w) && re2.test(w)) { + re = re_1b_2; + w = w.replace(re,""); + } + + // and turn initial Y back to y + + if (firstch == "y") { + w = firstch.toLowerCase() + w.substr(1); + } + + return w; + }; + + return function (token) { + return token.update(porterStemmer); + } +})(); + +lunr.Pipeline.registerFunction(lunr.stemmer, 'stemmer') +/*! + * lunr.stopWordFilter + * Copyright (C) 2020 Oliver Nightingale + */ + +/** + * lunr.generateStopWordFilter builds a stopWordFilter function from the provided + * list of stop words. + * + * The built in lunr.stopWordFilter is built using this generator and can be used + * to generate custom stopWordFilters for applications or non English languages. + * + * @function + * @param {Array} token The token to pass through the filter + * @returns {lunr.PipelineFunction} + * @see lunr.Pipeline + * @see lunr.stopWordFilter + */ +lunr.generateStopWordFilter = function (stopWords) { + var words = stopWords.reduce(function (memo, stopWord) { + memo[stopWord] = stopWord + return memo + }, {}) + + return function (token) { + if (token && words[token.toString()] !== token.toString()) return token + } +} + +/** + * lunr.stopWordFilter is an English language stop word list filter, any words + * contained in the list will not be passed through the filter. + * + * This is intended to be used in the Pipeline. If the token does not pass the + * filter then undefined will be returned. + * + * @function + * @implements {lunr.PipelineFunction} + * @params {lunr.Token} token - A token to check for being a stop word. + * @returns {lunr.Token} + * @see {@link lunr.Pipeline} + */ +lunr.stopWordFilter = lunr.generateStopWordFilter([ + 'a', + 'able', + 'about', + 'across', + 'after', + 'all', + 'almost', + 'also', + 'am', + 'among', + 'an', + 'and', + 'any', + 'are', + 'as', + 'at', + 'be', + 'because', + 'been', + 'but', + 'by', + 'can', + 'cannot', + 'could', + 'dear', + 'did', + 'do', + 'does', + 'either', + 'else', + 'ever', + 'every', + 'for', + 'from', + 'get', + 'got', + 'had', + 'has', + 'have', + 'he', + 'her', + 'hers', + 'him', + 'his', + 'how', + 'however', + 'i', + 'if', + 'in', + 'into', + 'is', + 'it', + 'its', + 'just', + 'least', + 'let', + 'like', + 'likely', + 'may', + 'me', + 'might', + 'most', + 'must', + 'my', + 'neither', + 'no', + 'nor', + 'not', + 'of', + 'off', + 'often', + 'on', + 'only', + 'or', + 'other', + 'our', + 'own', + 'rather', + 'said', + 'say', + 'says', + 'she', + 'should', + 'since', + 'so', + 'some', + 'than', + 'that', + 'the', + 'their', + 'them', + 'then', + 'there', + 'these', + 'they', + 'this', + 'tis', + 'to', + 'too', + 'twas', + 'us', + 'wants', + 'was', + 'we', + 'were', + 'what', + 'when', + 'where', + 'which', + 'while', + 'who', + 'whom', + 'why', + 'will', + 'with', + 'would', + 'yet', + 'you', + 'your' +]) + +lunr.Pipeline.registerFunction(lunr.stopWordFilter, 'stopWordFilter') +/*! + * lunr.trimmer + * Copyright (C) 2020 Oliver Nightingale + */ + +/** + * lunr.trimmer is a pipeline function for trimming non word + * characters from the beginning and end of tokens before they + * enter the index. + * + * This implementation may not work correctly for non latin + * characters and should either be removed or adapted for use + * with languages with non-latin characters. + * + * @static + * @implements {lunr.PipelineFunction} + * @param {lunr.Token} token The token to pass through the filter + * @returns {lunr.Token} + * @see lunr.Pipeline + */ +lunr.trimmer = function (token) { + return token.update(function (s) { + return s.replace(/^\W+/, '').replace(/\W+$/, '') + }) +} + +lunr.Pipeline.registerFunction(lunr.trimmer, 'trimmer') +/*! + * lunr.TokenSet + * Copyright (C) 2020 Oliver Nightingale + */ + +/** + * A token set is used to store the unique list of all tokens + * within an index. Token sets are also used to represent an + * incoming query to the index, this query token set and index + * token set are then intersected to find which tokens to look + * up in the inverted index. + * + * A token set can hold multiple tokens, as in the case of the + * index token set, or it can hold a single token as in the + * case of a simple query token set. + * + * Additionally token sets are used to perform wildcard matching. + * Leading, contained and trailing wildcards are supported, and + * from this edit distance matching can also be provided. + * + * Token sets are implemented as a minimal finite state automata, + * where both common prefixes and suffixes are shared between tokens. + * This helps to reduce the space used for storing the token set. + * + * @constructor + */ +lunr.TokenSet = function () { + this.final = false + this.edges = {} + this.id = lunr.TokenSet._nextId + lunr.TokenSet._nextId += 1 +} + +/** + * Keeps track of the next, auto increment, identifier to assign + * to a new tokenSet. + * + * TokenSets require a unique identifier to be correctly minimised. + * + * @private + */ +lunr.TokenSet._nextId = 1 + +/** + * Creates a TokenSet instance from the given sorted array of words. + * + * @param {String[]} arr - A sorted array of strings to create the set from. + * @returns {lunr.TokenSet} + * @throws Will throw an error if the input array is not sorted. + */ +lunr.TokenSet.fromArray = function (arr) { + var builder = new lunr.TokenSet.Builder + + for (var i = 0, len = arr.length; i < len; i++) { + builder.insert(arr[i]) + } + + builder.finish() + return builder.root +} + +/** + * Creates a token set from a query clause. + * + * @private + * @param {Object} clause - A single clause from lunr.Query. + * @param {string} clause.term - The query clause term. + * @param {number} [clause.editDistance] - The optional edit distance for the term. + * @returns {lunr.TokenSet} + */ +lunr.TokenSet.fromClause = function (clause) { + if ('editDistance' in clause) { + return lunr.TokenSet.fromFuzzyString(clause.term, clause.editDistance) + } else { + return lunr.TokenSet.fromString(clause.term) + } +} + +/** + * Creates a token set representing a single string with a specified + * edit distance. + * + * Insertions, deletions, substitutions and transpositions are each + * treated as an edit distance of 1. + * + * Increasing the allowed edit distance will have a dramatic impact + * on the performance of both creating and intersecting these TokenSets. + * It is advised to keep the edit distance less than 3. + * + * @param {string} str - The string to create the token set from. + * @param {number} editDistance - The allowed edit distance to match. + * @returns {lunr.Vector} + */ +lunr.TokenSet.fromFuzzyString = function (str, editDistance) { + var root = new lunr.TokenSet + + var stack = [{ + node: root, + editsRemaining: editDistance, + str: str + }] + + while (stack.length) { + var frame = stack.pop() + + // no edit + if (frame.str.length > 0) { + var char = frame.str.charAt(0), + noEditNode + + if (char in frame.node.edges) { + noEditNode = frame.node.edges[char] + } else { + noEditNode = new lunr.TokenSet + frame.node.edges[char] = noEditNode + } + + if (frame.str.length == 1) { + noEditNode.final = true + } + + stack.push({ + node: noEditNode, + editsRemaining: frame.editsRemaining, + str: frame.str.slice(1) + }) + } + + if (frame.editsRemaining == 0) { + continue + } + + // insertion + if ("*" in frame.node.edges) { + var insertionNode = frame.node.edges["*"] + } else { + var insertionNode = new lunr.TokenSet + frame.node.edges["*"] = insertionNode + } + + if (frame.str.length == 0) { + insertionNode.final = true + } + + stack.push({ + node: insertionNode, + editsRemaining: frame.editsRemaining - 1, + str: frame.str + }) + + // deletion + // can only do a deletion if we have enough edits remaining + // and if there are characters left to delete in the string + if (frame.str.length > 1) { + stack.push({ + node: frame.node, + editsRemaining: frame.editsRemaining - 1, + str: frame.str.slice(1) + }) + } + + // deletion + // just removing the last character from the str + if (frame.str.length == 1) { + frame.node.final = true + } + + // substitution + // can only do a substitution if we have enough edits remaining + // and if there are characters left to substitute + if (frame.str.length >= 1) { + if ("*" in frame.node.edges) { + var substitutionNode = frame.node.edges["*"] + } else { + var substitutionNode = new lunr.TokenSet + frame.node.edges["*"] = substitutionNode + } + + if (frame.str.length == 1) { + substitutionNode.final = true + } + + stack.push({ + node: substitutionNode, + editsRemaining: frame.editsRemaining - 1, + str: frame.str.slice(1) + }) + } + + // transposition + // can only do a transposition if there are edits remaining + // and there are enough characters to transpose + if (frame.str.length > 1) { + var charA = frame.str.charAt(0), + charB = frame.str.charAt(1), + transposeNode + + if (charB in frame.node.edges) { + transposeNode = frame.node.edges[charB] + } else { + transposeNode = new lunr.TokenSet + frame.node.edges[charB] = transposeNode + } + + if (frame.str.length == 1) { + transposeNode.final = true + } + + stack.push({ + node: transposeNode, + editsRemaining: frame.editsRemaining - 1, + str: charA + frame.str.slice(2) + }) + } + } + + return root +} + +/** + * Creates a TokenSet from a string. + * + * The string may contain one or more wildcard characters (*) + * that will allow wildcard matching when intersecting with + * another TokenSet. + * + * @param {string} str - The string to create a TokenSet from. + * @returns {lunr.TokenSet} + */ +lunr.TokenSet.fromString = function (str) { + var node = new lunr.TokenSet, + root = node + + /* + * Iterates through all characters within the passed string + * appending a node for each character. + * + * When a wildcard character is found then a self + * referencing edge is introduced to continually match + * any number of any characters. + */ + for (var i = 0, len = str.length; i < len; i++) { + var char = str[i], + final = (i == len - 1) + + if (char == "*") { + node.edges[char] = node + node.final = final + + } else { + var next = new lunr.TokenSet + next.final = final + + node.edges[char] = next + node = next + } + } + + return root +} + +/** + * Converts this TokenSet into an array of strings + * contained within the TokenSet. + * + * This is not intended to be used on a TokenSet that + * contains wildcards, in these cases the results are + * undefined and are likely to cause an infinite loop. + * + * @returns {string[]} + */ +lunr.TokenSet.prototype.toArray = function () { + var words = [] + + var stack = [{ + prefix: "", + node: this + }] + + while (stack.length) { + var frame = stack.pop(), + edges = Object.keys(frame.node.edges), + len = edges.length + + if (frame.node.final) { + /* In Safari, at this point the prefix is sometimes corrupted, see: + * https://github.com/olivernn/lunr.js/issues/279 Calling any + * String.prototype method forces Safari to "cast" this string to what + * it's supposed to be, fixing the bug. */ + frame.prefix.charAt(0) + words.push(frame.prefix) + } + + for (var i = 0; i < len; i++) { + var edge = edges[i] + + stack.push({ + prefix: frame.prefix.concat(edge), + node: frame.node.edges[edge] + }) + } + } + + return words +} + +/** + * Generates a string representation of a TokenSet. + * + * This is intended to allow TokenSets to be used as keys + * in objects, largely to aid the construction and minimisation + * of a TokenSet. As such it is not designed to be a human + * friendly representation of the TokenSet. + * + * @returns {string} + */ +lunr.TokenSet.prototype.toString = function () { + // NOTE: Using Object.keys here as this.edges is very likely + // to enter 'hash-mode' with many keys being added + // + // avoiding a for-in loop here as it leads to the function + // being de-optimised (at least in V8). From some simple + // benchmarks the performance is comparable, but allowing + // V8 to optimize may mean easy performance wins in the future. + + if (this._str) { + return this._str + } + + var str = this.final ? '1' : '0', + labels = Object.keys(this.edges).sort(), + len = labels.length + + for (var i = 0; i < len; i++) { + var label = labels[i], + node = this.edges[label] + + str = str + label + node.id + } + + return str +} + +/** + * Returns a new TokenSet that is the intersection of + * this TokenSet and the passed TokenSet. + * + * This intersection will take into account any wildcards + * contained within the TokenSet. + * + * @param {lunr.TokenSet} b - An other TokenSet to intersect with. + * @returns {lunr.TokenSet} + */ +lunr.TokenSet.prototype.intersect = function (b) { + var output = new lunr.TokenSet, + frame = undefined + + var stack = [{ + qNode: b, + output: output, + node: this + }] + + while (stack.length) { + frame = stack.pop() + + // NOTE: As with the #toString method, we are using + // Object.keys and a for loop instead of a for-in loop + // as both of these objects enter 'hash' mode, causing + // the function to be de-optimised in V8 + var qEdges = Object.keys(frame.qNode.edges), + qLen = qEdges.length, + nEdges = Object.keys(frame.node.edges), + nLen = nEdges.length + + for (var q = 0; q < qLen; q++) { + var qEdge = qEdges[q] + + for (var n = 0; n < nLen; n++) { + var nEdge = nEdges[n] + + if (nEdge == qEdge || qEdge == '*') { + var node = frame.node.edges[nEdge], + qNode = frame.qNode.edges[qEdge], + final = node.final && qNode.final, + next = undefined + + if (nEdge in frame.output.edges) { + // an edge already exists for this character + // no need to create a new node, just set the finality + // bit unless this node is already final + next = frame.output.edges[nEdge] + next.final = next.final || final + + } else { + // no edge exists yet, must create one + // set the finality bit and insert it + // into the output + next = new lunr.TokenSet + next.final = final + frame.output.edges[nEdge] = next + } + + stack.push({ + qNode: qNode, + output: next, + node: node + }) + } + } + } + } + + return output +} +lunr.TokenSet.Builder = function () { + this.previousWord = "" + this.root = new lunr.TokenSet + this.uncheckedNodes = [] + this.minimizedNodes = {} +} + +lunr.TokenSet.Builder.prototype.insert = function (word) { + var node, + commonPrefix = 0 + + if (word < this.previousWord) { + throw new Error ("Out of order word insertion") + } + + for (var i = 0; i < word.length && i < this.previousWord.length; i++) { + if (word[i] != this.previousWord[i]) break + commonPrefix++ + } + + this.minimize(commonPrefix) + + if (this.uncheckedNodes.length == 0) { + node = this.root + } else { + node = this.uncheckedNodes[this.uncheckedNodes.length - 1].child + } + + for (var i = commonPrefix; i < word.length; i++) { + var nextNode = new lunr.TokenSet, + char = word[i] + + node.edges[char] = nextNode + + this.uncheckedNodes.push({ + parent: node, + char: char, + child: nextNode + }) + + node = nextNode + } + + node.final = true + this.previousWord = word +} + +lunr.TokenSet.Builder.prototype.finish = function () { + this.minimize(0) +} + +lunr.TokenSet.Builder.prototype.minimize = function (downTo) { + for (var i = this.uncheckedNodes.length - 1; i >= downTo; i--) { + var node = this.uncheckedNodes[i], + childKey = node.child.toString() + + if (childKey in this.minimizedNodes) { + node.parent.edges[node.char] = this.minimizedNodes[childKey] + } else { + // Cache the key for this node since + // we know it can't change anymore + node.child._str = childKey + + this.minimizedNodes[childKey] = node.child + } + + this.uncheckedNodes.pop() + } +} +/*! + * lunr.Index + * Copyright (C) 2020 Oliver Nightingale + */ + +/** + * An index contains the built index of all documents and provides a query interface + * to the index. + * + * Usually instances of lunr.Index will not be created using this constructor, instead + * lunr.Builder should be used to construct new indexes, or lunr.Index.load should be + * used to load previously built and serialized indexes. + * + * @constructor + * @param {Object} attrs - The attributes of the built search index. + * @param {Object} attrs.invertedIndex - An index of term/field to document reference. + * @param {Object} attrs.fieldVectors - Field vectors + * @param {lunr.TokenSet} attrs.tokenSet - An set of all corpus tokens. + * @param {string[]} attrs.fields - The names of indexed document fields. + * @param {lunr.Pipeline} attrs.pipeline - The pipeline to use for search terms. + */ +lunr.Index = function (attrs) { + this.invertedIndex = attrs.invertedIndex + this.fieldVectors = attrs.fieldVectors + this.tokenSet = attrs.tokenSet + this.fields = attrs.fields + this.pipeline = attrs.pipeline +} + +/** + * A result contains details of a document matching a search query. + * @typedef {Object} lunr.Index~Result + * @property {string} ref - The reference of the document this result represents. + * @property {number} score - A number between 0 and 1 representing how similar this document is to the query. + * @property {lunr.MatchData} matchData - Contains metadata about this match including which term(s) caused the match. + */ + +/** + * Although lunr provides the ability to create queries using lunr.Query, it also provides a simple + * query language which itself is parsed into an instance of lunr.Query. + * + * For programmatically building queries it is advised to directly use lunr.Query, the query language + * is best used for human entered text rather than program generated text. + * + * At its simplest queries can just be a single term, e.g. `hello`, multiple terms are also supported + * and will be combined with OR, e.g `hello world` will match documents that contain either 'hello' + * or 'world', though those that contain both will rank higher in the results. + * + * Wildcards can be included in terms to match one or more unspecified characters, these wildcards can + * be inserted anywhere within the term, and more than one wildcard can exist in a single term. Adding + * wildcards will increase the number of documents that will be found but can also have a negative + * impact on query performance, especially with wildcards at the beginning of a term. + * + * Terms can be restricted to specific fields, e.g. `title:hello`, only documents with the term + * hello in the title field will match this query. Using a field not present in the index will lead + * to an error being thrown. + * + * Modifiers can also be added to terms, lunr supports edit distance and boost modifiers on terms. A term + * boost will make documents matching that term score higher, e.g. `foo^5`. Edit distance is also supported + * to provide fuzzy matching, e.g. 'hello~2' will match documents with hello with an edit distance of 2. + * Avoid large values for edit distance to improve query performance. + * + * Each term also supports a presence modifier. By default a term's presence in document is optional, however + * this can be changed to either required or prohibited. For a term's presence to be required in a document the + * term should be prefixed with a '+', e.g. `+foo bar` is a search for documents that must contain 'foo' and + * optionally contain 'bar'. Conversely a leading '-' sets the terms presence to prohibited, i.e. it must not + * appear in a document, e.g. `-foo bar` is a search for documents that do not contain 'foo' but may contain 'bar'. + * + * To escape special characters the backslash character '\' can be used, this allows searches to include + * characters that would normally be considered modifiers, e.g. `foo\~2` will search for a term "foo~2" instead + * of attempting to apply a boost of 2 to the search term "foo". + * + * @typedef {string} lunr.Index~QueryString + * @example Simple single term query + * hello + * @example Multiple term query + * hello world + * @example term scoped to a field + * title:hello + * @example term with a boost of 10 + * hello^10 + * @example term with an edit distance of 2 + * hello~2 + * @example terms with presence modifiers + * -foo +bar baz + */ + +/** + * Performs a search against the index using lunr query syntax. + * + * Results will be returned sorted by their score, the most relevant results + * will be returned first. For details on how the score is calculated, please see + * the {@link https://lunrjs.com/guides/searching.html#scoring|guide}. + * + * For more programmatic querying use lunr.Index#query. + * + * @param {lunr.Index~QueryString} queryString - A string containing a lunr query. + * @throws {lunr.QueryParseError} If the passed query string cannot be parsed. + * @returns {lunr.Index~Result[]} + */ +lunr.Index.prototype.search = function (queryString) { + return this.query(function (query) { + var parser = new lunr.QueryParser(queryString, query) + parser.parse() + }) +} + +/** + * A query builder callback provides a query object to be used to express + * the query to perform on the index. + * + * @callback lunr.Index~queryBuilder + * @param {lunr.Query} query - The query object to build up. + * @this lunr.Query + */ + +/** + * Performs a query against the index using the yielded lunr.Query object. + * + * If performing programmatic queries against the index, this method is preferred + * over lunr.Index#search so as to avoid the additional query parsing overhead. + * + * A query object is yielded to the supplied function which should be used to + * express the query to be run against the index. + * + * Note that although this function takes a callback parameter it is _not_ an + * asynchronous operation, the callback is just yielded a query object to be + * customized. + * + * @param {lunr.Index~queryBuilder} fn - A function that is used to build the query. + * @returns {lunr.Index~Result[]} + */ +lunr.Index.prototype.query = function (fn) { + // for each query clause + // * process terms + // * expand terms from token set + // * find matching documents and metadata + // * get document vectors + // * score documents + + var query = new lunr.Query(this.fields), + matchingFields = Object.create(null), + queryVectors = Object.create(null), + termFieldCache = Object.create(null), + requiredMatches = Object.create(null), + prohibitedMatches = Object.create(null) + + /* + * To support field level boosts a query vector is created per + * field. An empty vector is eagerly created to support negated + * queries. + */ + for (var i = 0; i < this.fields.length; i++) { + queryVectors[this.fields[i]] = new lunr.Vector + } + + fn.call(query, query) + + for (var i = 0; i < query.clauses.length; i++) { + /* + * Unless the pipeline has been disabled for this term, which is + * the case for terms with wildcards, we need to pass the clause + * term through the search pipeline. A pipeline returns an array + * of processed terms. Pipeline functions may expand the passed + * term, which means we may end up performing multiple index lookups + * for a single query term. + */ + var clause = query.clauses[i], + terms = null, + clauseMatches = lunr.Set.empty + + if (clause.usePipeline) { + terms = this.pipeline.runString(clause.term, { + fields: clause.fields + }) + } else { + terms = [clause.term] + } + + for (var m = 0; m < terms.length; m++) { + var term = terms[m] + + /* + * Each term returned from the pipeline needs to use the same query + * clause object, e.g. the same boost and or edit distance. The + * simplest way to do this is to re-use the clause object but mutate + * its term property. + */ + clause.term = term + + /* + * From the term in the clause we create a token set which will then + * be used to intersect the indexes token set to get a list of terms + * to lookup in the inverted index + */ + var termTokenSet = lunr.TokenSet.fromClause(clause), + expandedTerms = this.tokenSet.intersect(termTokenSet).toArray() + + /* + * If a term marked as required does not exist in the tokenSet it is + * impossible for the search to return any matches. We set all the field + * scoped required matches set to empty and stop examining any further + * clauses. + */ + if (expandedTerms.length === 0 && clause.presence === lunr.Query.presence.REQUIRED) { + for (var k = 0; k < clause.fields.length; k++) { + var field = clause.fields[k] + requiredMatches[field] = lunr.Set.empty + } + + break + } + + for (var j = 0; j < expandedTerms.length; j++) { + /* + * For each term get the posting and termIndex, this is required for + * building the query vector. + */ + var expandedTerm = expandedTerms[j], + posting = this.invertedIndex[expandedTerm], + termIndex = posting._index + + for (var k = 0; k < clause.fields.length; k++) { + /* + * For each field that this query term is scoped by (by default + * all fields are in scope) we need to get all the document refs + * that have this term in that field. + * + * The posting is the entry in the invertedIndex for the matching + * term from above. + */ + var field = clause.fields[k], + fieldPosting = posting[field], + matchingDocumentRefs = Object.keys(fieldPosting), + termField = expandedTerm + "/" + field, + matchingDocumentsSet = new lunr.Set(matchingDocumentRefs) + + /* + * if the presence of this term is required ensure that the matching + * documents are added to the set of required matches for this clause. + * + */ + if (clause.presence == lunr.Query.presence.REQUIRED) { + clauseMatches = clauseMatches.union(matchingDocumentsSet) + + if (requiredMatches[field] === undefined) { + requiredMatches[field] = lunr.Set.complete + } + } + + /* + * if the presence of this term is prohibited ensure that the matching + * documents are added to the set of prohibited matches for this field, + * creating that set if it does not yet exist. + */ + if (clause.presence == lunr.Query.presence.PROHIBITED) { + if (prohibitedMatches[field] === undefined) { + prohibitedMatches[field] = lunr.Set.empty + } + + prohibitedMatches[field] = prohibitedMatches[field].union(matchingDocumentsSet) + + /* + * Prohibited matches should not be part of the query vector used for + * similarity scoring and no metadata should be extracted so we continue + * to the next field + */ + continue + } + + /* + * The query field vector is populated using the termIndex found for + * the term and a unit value with the appropriate boost applied. + * Using upsert because there could already be an entry in the vector + * for the term we are working with. In that case we just add the scores + * together. + */ + queryVectors[field].upsert(termIndex, clause.boost, function (a, b) { return a + b }) + + /** + * If we've already seen this term, field combo then we've already collected + * the matching documents and metadata, no need to go through all that again + */ + if (termFieldCache[termField]) { + continue + } + + for (var l = 0; l < matchingDocumentRefs.length; l++) { + /* + * All metadata for this term/field/document triple + * are then extracted and collected into an instance + * of lunr.MatchData ready to be returned in the query + * results + */ + var matchingDocumentRef = matchingDocumentRefs[l], + matchingFieldRef = new lunr.FieldRef (matchingDocumentRef, field), + metadata = fieldPosting[matchingDocumentRef], + fieldMatch + + if ((fieldMatch = matchingFields[matchingFieldRef]) === undefined) { + matchingFields[matchingFieldRef] = new lunr.MatchData (expandedTerm, field, metadata) + } else { + fieldMatch.add(expandedTerm, field, metadata) + } + + } + + termFieldCache[termField] = true + } + } + } + + /** + * If the presence was required we need to update the requiredMatches field sets. + * We do this after all fields for the term have collected their matches because + * the clause terms presence is required in _any_ of the fields not _all_ of the + * fields. + */ + if (clause.presence === lunr.Query.presence.REQUIRED) { + for (var k = 0; k < clause.fields.length; k++) { + var field = clause.fields[k] + requiredMatches[field] = requiredMatches[field].intersect(clauseMatches) + } + } + } + + /** + * Need to combine the field scoped required and prohibited + * matching documents into a global set of required and prohibited + * matches + */ + var allRequiredMatches = lunr.Set.complete, + allProhibitedMatches = lunr.Set.empty + + for (var i = 0; i < this.fields.length; i++) { + var field = this.fields[i] + + if (requiredMatches[field]) { + allRequiredMatches = allRequiredMatches.intersect(requiredMatches[field]) + } + + if (prohibitedMatches[field]) { + allProhibitedMatches = allProhibitedMatches.union(prohibitedMatches[field]) + } + } + + var matchingFieldRefs = Object.keys(matchingFields), + results = [], + matches = Object.create(null) + + /* + * If the query is negated (contains only prohibited terms) + * we need to get _all_ fieldRefs currently existing in the + * index. This is only done when we know that the query is + * entirely prohibited terms to avoid any cost of getting all + * fieldRefs unnecessarily. + * + * Additionally, blank MatchData must be created to correctly + * populate the results. + */ + if (query.isNegated()) { + matchingFieldRefs = Object.keys(this.fieldVectors) + + for (var i = 0; i < matchingFieldRefs.length; i++) { + var matchingFieldRef = matchingFieldRefs[i] + var fieldRef = lunr.FieldRef.fromString(matchingFieldRef) + matchingFields[matchingFieldRef] = new lunr.MatchData + } + } + + for (var i = 0; i < matchingFieldRefs.length; i++) { + /* + * Currently we have document fields that match the query, but we + * need to return documents. The matchData and scores are combined + * from multiple fields belonging to the same document. + * + * Scores are calculated by field, using the query vectors created + * above, and combined into a final document score using addition. + */ + var fieldRef = lunr.FieldRef.fromString(matchingFieldRefs[i]), + docRef = fieldRef.docRef + + if (!allRequiredMatches.contains(docRef)) { + continue + } + + if (allProhibitedMatches.contains(docRef)) { + continue + } + + var fieldVector = this.fieldVectors[fieldRef], + score = queryVectors[fieldRef.fieldName].similarity(fieldVector), + docMatch + + if ((docMatch = matches[docRef]) !== undefined) { + docMatch.score += score + docMatch.matchData.combine(matchingFields[fieldRef]) + } else { + var match = { + ref: docRef, + score: score, + matchData: matchingFields[fieldRef] + } + matches[docRef] = match + results.push(match) + } + } + + /* + * Sort the results objects by score, highest first. + */ + return results.sort(function (a, b) { + return b.score - a.score + }) +} + +/** + * Prepares the index for JSON serialization. + * + * The schema for this JSON blob will be described in a + * separate JSON schema file. + * + * @returns {Object} + */ +lunr.Index.prototype.toJSON = function () { + var invertedIndex = Object.keys(this.invertedIndex) + .sort() + .map(function (term) { + return [term, this.invertedIndex[term]] + }, this) + + var fieldVectors = Object.keys(this.fieldVectors) + .map(function (ref) { + return [ref, this.fieldVectors[ref].toJSON()] + }, this) + + return { + version: lunr.version, + fields: this.fields, + fieldVectors: fieldVectors, + invertedIndex: invertedIndex, + pipeline: this.pipeline.toJSON() + } +} + +/** + * Loads a previously serialized lunr.Index + * + * @param {Object} serializedIndex - A previously serialized lunr.Index + * @returns {lunr.Index} + */ +lunr.Index.load = function (serializedIndex) { + var attrs = {}, + fieldVectors = {}, + serializedVectors = serializedIndex.fieldVectors, + invertedIndex = Object.create(null), + serializedInvertedIndex = serializedIndex.invertedIndex, + tokenSetBuilder = new lunr.TokenSet.Builder, + pipeline = lunr.Pipeline.load(serializedIndex.pipeline) + + if (serializedIndex.version != lunr.version) { + lunr.utils.warn("Version mismatch when loading serialised index. Current version of lunr '" + lunr.version + "' does not match serialized index '" + serializedIndex.version + "'") + } + + for (var i = 0; i < serializedVectors.length; i++) { + var tuple = serializedVectors[i], + ref = tuple[0], + elements = tuple[1] + + fieldVectors[ref] = new lunr.Vector(elements) + } + + for (var i = 0; i < serializedInvertedIndex.length; i++) { + var tuple = serializedInvertedIndex[i], + term = tuple[0], + posting = tuple[1] + + tokenSetBuilder.insert(term) + invertedIndex[term] = posting + } + + tokenSetBuilder.finish() + + attrs.fields = serializedIndex.fields + + attrs.fieldVectors = fieldVectors + attrs.invertedIndex = invertedIndex + attrs.tokenSet = tokenSetBuilder.root + attrs.pipeline = pipeline + + return new lunr.Index(attrs) +} +/*! + * lunr.Builder + * Copyright (C) 2020 Oliver Nightingale + */ + +/** + * lunr.Builder performs indexing on a set of documents and + * returns instances of lunr.Index ready for querying. + * + * All configuration of the index is done via the builder, the + * fields to index, the document reference, the text processing + * pipeline and document scoring parameters are all set on the + * builder before indexing. + * + * @constructor + * @property {string} _ref - Internal reference to the document reference field. + * @property {string[]} _fields - Internal reference to the document fields to index. + * @property {object} invertedIndex - The inverted index maps terms to document fields. + * @property {object} documentTermFrequencies - Keeps track of document term frequencies. + * @property {object} documentLengths - Keeps track of the length of documents added to the index. + * @property {lunr.tokenizer} tokenizer - Function for splitting strings into tokens for indexing. + * @property {lunr.Pipeline} pipeline - The pipeline performs text processing on tokens before indexing. + * @property {lunr.Pipeline} searchPipeline - A pipeline for processing search terms before querying the index. + * @property {number} documentCount - Keeps track of the total number of documents indexed. + * @property {number} _b - A parameter to control field length normalization, setting this to 0 disabled normalization, 1 fully normalizes field lengths, the default value is 0.75. + * @property {number} _k1 - A parameter to control how quickly an increase in term frequency results in term frequency saturation, the default value is 1.2. + * @property {number} termIndex - A counter incremented for each unique term, used to identify a terms position in the vector space. + * @property {array} metadataWhitelist - A list of metadata keys that have been whitelisted for entry in the index. + */ +lunr.Builder = function () { + this._ref = "id" + this._fields = Object.create(null) + this._documents = Object.create(null) + this.invertedIndex = Object.create(null) + this.fieldTermFrequencies = {} + this.fieldLengths = {} + this.tokenizer = lunr.tokenizer + this.pipeline = new lunr.Pipeline + this.searchPipeline = new lunr.Pipeline + this.documentCount = 0 + this._b = 0.75 + this._k1 = 1.2 + this.termIndex = 0 + this.metadataWhitelist = [] +} + +/** + * Sets the document field used as the document reference. Every document must have this field. + * The type of this field in the document should be a string, if it is not a string it will be + * coerced into a string by calling toString. + * + * The default ref is 'id'. + * + * The ref should _not_ be changed during indexing, it should be set before any documents are + * added to the index. Changing it during indexing can lead to inconsistent results. + * + * @param {string} ref - The name of the reference field in the document. + */ +lunr.Builder.prototype.ref = function (ref) { + this._ref = ref +} + +/** + * A function that is used to extract a field from a document. + * + * Lunr expects a field to be at the top level of a document, if however the field + * is deeply nested within a document an extractor function can be used to extract + * the right field for indexing. + * + * @callback fieldExtractor + * @param {object} doc - The document being added to the index. + * @returns {?(string|object|object[])} obj - The object that will be indexed for this field. + * @example Extracting a nested field + * function (doc) { return doc.nested.field } + */ + +/** + * Adds a field to the list of document fields that will be indexed. Every document being + * indexed should have this field. Null values for this field in indexed documents will + * not cause errors but will limit the chance of that document being retrieved by searches. + * + * All fields should be added before adding documents to the index. Adding fields after + * a document has been indexed will have no effect on already indexed documents. + * + * Fields can be boosted at build time. This allows terms within that field to have more + * importance when ranking search results. Use a field boost to specify that matches within + * one field are more important than other fields. + * + * @param {string} fieldName - The name of a field to index in all documents. + * @param {object} attributes - Optional attributes associated with this field. + * @param {number} [attributes.boost=1] - Boost applied to all terms within this field. + * @param {fieldExtractor} [attributes.extractor] - Function to extract a field from a document. + * @throws {RangeError} fieldName cannot contain unsupported characters '/' + */ +lunr.Builder.prototype.field = function (fieldName, attributes) { + if (/\//.test(fieldName)) { + throw new RangeError ("Field '" + fieldName + "' contains illegal character '/'") + } + + this._fields[fieldName] = attributes || {} +} + +/** + * A parameter to tune the amount of field length normalisation that is applied when + * calculating relevance scores. A value of 0 will completely disable any normalisation + * and a value of 1 will fully normalise field lengths. The default is 0.75. Values of b + * will be clamped to the range 0 - 1. + * + * @param {number} number - The value to set for this tuning parameter. + */ +lunr.Builder.prototype.b = function (number) { + if (number < 0) { + this._b = 0 + } else if (number > 1) { + this._b = 1 + } else { + this._b = number + } +} + +/** + * A parameter that controls the speed at which a rise in term frequency results in term + * frequency saturation. The default value is 1.2. Setting this to a higher value will give + * slower saturation levels, a lower value will result in quicker saturation. + * + * @param {number} number - The value to set for this tuning parameter. + */ +lunr.Builder.prototype.k1 = function (number) { + this._k1 = number +} + +/** + * Adds a document to the index. + * + * Before adding fields to the index the index should have been fully setup, with the document + * ref and all fields to index already having been specified. + * + * The document must have a field name as specified by the ref (by default this is 'id') and + * it should have all fields defined for indexing, though null or undefined values will not + * cause errors. + * + * Entire documents can be boosted at build time. Applying a boost to a document indicates that + * this document should rank higher in search results than other documents. + * + * @param {object} doc - The document to add to the index. + * @param {object} attributes - Optional attributes associated with this document. + * @param {number} [attributes.boost=1] - Boost applied to all terms within this document. + */ +lunr.Builder.prototype.add = function (doc, attributes) { + var docRef = doc[this._ref], + fields = Object.keys(this._fields) + + this._documents[docRef] = attributes || {} + this.documentCount += 1 + + for (var i = 0; i < fields.length; i++) { + var fieldName = fields[i], + extractor = this._fields[fieldName].extractor, + field = extractor ? extractor(doc) : doc[fieldName], + tokens = this.tokenizer(field, { + fields: [fieldName] + }), + terms = this.pipeline.run(tokens), + fieldRef = new lunr.FieldRef (docRef, fieldName), + fieldTerms = Object.create(null) + + this.fieldTermFrequencies[fieldRef] = fieldTerms + this.fieldLengths[fieldRef] = 0 + + // store the length of this field for this document + this.fieldLengths[fieldRef] += terms.length + + // calculate term frequencies for this field + for (var j = 0; j < terms.length; j++) { + var term = terms[j] + + if (fieldTerms[term] == undefined) { + fieldTerms[term] = 0 + } + + fieldTerms[term] += 1 + + // add to inverted index + // create an initial posting if one doesn't exist + if (this.invertedIndex[term] == undefined) { + var posting = Object.create(null) + posting["_index"] = this.termIndex + this.termIndex += 1 + + for (var k = 0; k < fields.length; k++) { + posting[fields[k]] = Object.create(null) + } + + this.invertedIndex[term] = posting + } + + // add an entry for this term/fieldName/docRef to the invertedIndex + if (this.invertedIndex[term][fieldName][docRef] == undefined) { + this.invertedIndex[term][fieldName][docRef] = Object.create(null) + } + + // store all whitelisted metadata about this token in the + // inverted index + for (var l = 0; l < this.metadataWhitelist.length; l++) { + var metadataKey = this.metadataWhitelist[l], + metadata = term.metadata[metadataKey] + + if (this.invertedIndex[term][fieldName][docRef][metadataKey] == undefined) { + this.invertedIndex[term][fieldName][docRef][metadataKey] = [] + } + + this.invertedIndex[term][fieldName][docRef][metadataKey].push(metadata) + } + } + + } +} + +/** + * Calculates the average document length for this index + * + * @private + */ +lunr.Builder.prototype.calculateAverageFieldLengths = function () { + + var fieldRefs = Object.keys(this.fieldLengths), + numberOfFields = fieldRefs.length, + accumulator = {}, + documentsWithField = {} + + for (var i = 0; i < numberOfFields; i++) { + var fieldRef = lunr.FieldRef.fromString(fieldRefs[i]), + field = fieldRef.fieldName + + documentsWithField[field] || (documentsWithField[field] = 0) + documentsWithField[field] += 1 + + accumulator[field] || (accumulator[field] = 0) + accumulator[field] += this.fieldLengths[fieldRef] + } + + var fields = Object.keys(this._fields) + + for (var i = 0; i < fields.length; i++) { + var fieldName = fields[i] + accumulator[fieldName] = accumulator[fieldName] / documentsWithField[fieldName] + } + + this.averageFieldLength = accumulator +} + +/** + * Builds a vector space model of every document using lunr.Vector + * + * @private + */ +lunr.Builder.prototype.createFieldVectors = function () { + var fieldVectors = {}, + fieldRefs = Object.keys(this.fieldTermFrequencies), + fieldRefsLength = fieldRefs.length, + termIdfCache = Object.create(null) + + for (var i = 0; i < fieldRefsLength; i++) { + var fieldRef = lunr.FieldRef.fromString(fieldRefs[i]), + fieldName = fieldRef.fieldName, + fieldLength = this.fieldLengths[fieldRef], + fieldVector = new lunr.Vector, + termFrequencies = this.fieldTermFrequencies[fieldRef], + terms = Object.keys(termFrequencies), + termsLength = terms.length + + + var fieldBoost = this._fields[fieldName].boost || 1, + docBoost = this._documents[fieldRef.docRef].boost || 1 + + for (var j = 0; j < termsLength; j++) { + var term = terms[j], + tf = termFrequencies[term], + termIndex = this.invertedIndex[term]._index, + idf, score, scoreWithPrecision + + if (termIdfCache[term] === undefined) { + idf = lunr.idf(this.invertedIndex[term], this.documentCount) + termIdfCache[term] = idf + } else { + idf = termIdfCache[term] + } + + score = idf * ((this._k1 + 1) * tf) / (this._k1 * (1 - this._b + this._b * (fieldLength / this.averageFieldLength[fieldName])) + tf) + score *= fieldBoost + score *= docBoost + scoreWithPrecision = Math.round(score * 1000) / 1000 + // Converts 1.23456789 to 1.234. + // Reducing the precision so that the vectors take up less + // space when serialised. Doing it now so that they behave + // the same before and after serialisation. Also, this is + // the fastest approach to reducing a number's precision in + // JavaScript. + + fieldVector.insert(termIndex, scoreWithPrecision) + } + + fieldVectors[fieldRef] = fieldVector + } + + this.fieldVectors = fieldVectors +} + +/** + * Creates a token set of all tokens in the index using lunr.TokenSet + * + * @private + */ +lunr.Builder.prototype.createTokenSet = function () { + this.tokenSet = lunr.TokenSet.fromArray( + Object.keys(this.invertedIndex).sort() + ) +} + +/** + * Builds the index, creating an instance of lunr.Index. + * + * This completes the indexing process and should only be called + * once all documents have been added to the index. + * + * @returns {lunr.Index} + */ +lunr.Builder.prototype.build = function () { + this.calculateAverageFieldLengths() + this.createFieldVectors() + this.createTokenSet() + + return new lunr.Index({ + invertedIndex: this.invertedIndex, + fieldVectors: this.fieldVectors, + tokenSet: this.tokenSet, + fields: Object.keys(this._fields), + pipeline: this.searchPipeline + }) +} + +/** + * Applies a plugin to the index builder. + * + * A plugin is a function that is called with the index builder as its context. + * Plugins can be used to customise or extend the behaviour of the index + * in some way. A plugin is just a function, that encapsulated the custom + * behaviour that should be applied when building the index. + * + * The plugin function will be called with the index builder as its argument, additional + * arguments can also be passed when calling use. The function will be called + * with the index builder as its context. + * + * @param {Function} plugin The plugin to apply. + */ +lunr.Builder.prototype.use = function (fn) { + var args = Array.prototype.slice.call(arguments, 1) + args.unshift(this) + fn.apply(this, args) +} +/** + * Contains and collects metadata about a matching document. + * A single instance of lunr.MatchData is returned as part of every + * lunr.Index~Result. + * + * @constructor + * @param {string} term - The term this match data is associated with + * @param {string} field - The field in which the term was found + * @param {object} metadata - The metadata recorded about this term in this field + * @property {object} metadata - A cloned collection of metadata associated with this document. + * @see {@link lunr.Index~Result} + */ +lunr.MatchData = function (term, field, metadata) { + var clonedMetadata = Object.create(null), + metadataKeys = Object.keys(metadata || {}) + + // Cloning the metadata to prevent the original + // being mutated during match data combination. + // Metadata is kept in an array within the inverted + // index so cloning the data can be done with + // Array#slice + for (var i = 0; i < metadataKeys.length; i++) { + var key = metadataKeys[i] + clonedMetadata[key] = metadata[key].slice() + } + + this.metadata = Object.create(null) + + if (term !== undefined) { + this.metadata[term] = Object.create(null) + this.metadata[term][field] = clonedMetadata + } +} + +/** + * An instance of lunr.MatchData will be created for every term that matches a + * document. However only one instance is required in a lunr.Index~Result. This + * method combines metadata from another instance of lunr.MatchData with this + * objects metadata. + * + * @param {lunr.MatchData} otherMatchData - Another instance of match data to merge with this one. + * @see {@link lunr.Index~Result} + */ +lunr.MatchData.prototype.combine = function (otherMatchData) { + var terms = Object.keys(otherMatchData.metadata) + + for (var i = 0; i < terms.length; i++) { + var term = terms[i], + fields = Object.keys(otherMatchData.metadata[term]) + + if (this.metadata[term] == undefined) { + this.metadata[term] = Object.create(null) + } + + for (var j = 0; j < fields.length; j++) { + var field = fields[j], + keys = Object.keys(otherMatchData.metadata[term][field]) + + if (this.metadata[term][field] == undefined) { + this.metadata[term][field] = Object.create(null) + } + + for (var k = 0; k < keys.length; k++) { + var key = keys[k] + + if (this.metadata[term][field][key] == undefined) { + this.metadata[term][field][key] = otherMatchData.metadata[term][field][key] + } else { + this.metadata[term][field][key] = this.metadata[term][field][key].concat(otherMatchData.metadata[term][field][key]) + } + + } + } + } +} + +/** + * Add metadata for a term/field pair to this instance of match data. + * + * @param {string} term - The term this match data is associated with + * @param {string} field - The field in which the term was found + * @param {object} metadata - The metadata recorded about this term in this field + */ +lunr.MatchData.prototype.add = function (term, field, metadata) { + if (!(term in this.metadata)) { + this.metadata[term] = Object.create(null) + this.metadata[term][field] = metadata + return + } + + if (!(field in this.metadata[term])) { + this.metadata[term][field] = metadata + return + } + + var metadataKeys = Object.keys(metadata) + + for (var i = 0; i < metadataKeys.length; i++) { + var key = metadataKeys[i] + + if (key in this.metadata[term][field]) { + this.metadata[term][field][key] = this.metadata[term][field][key].concat(metadata[key]) + } else { + this.metadata[term][field][key] = metadata[key] + } + } +} +/** + * A lunr.Query provides a programmatic way of defining queries to be performed + * against a {@link lunr.Index}. + * + * Prefer constructing a lunr.Query using the {@link lunr.Index#query} method + * so the query object is pre-initialized with the right index fields. + * + * @constructor + * @property {lunr.Query~Clause[]} clauses - An array of query clauses. + * @property {string[]} allFields - An array of all available fields in a lunr.Index. + */ +lunr.Query = function (allFields) { + this.clauses = [] + this.allFields = allFields +} + +/** + * Constants for indicating what kind of automatic wildcard insertion will be used when constructing a query clause. + * + * This allows wildcards to be added to the beginning and end of a term without having to manually do any string + * concatenation. + * + * The wildcard constants can be bitwise combined to select both leading and trailing wildcards. + * + * @constant + * @default + * @property {number} wildcard.NONE - The term will have no wildcards inserted, this is the default behaviour + * @property {number} wildcard.LEADING - Prepend the term with a wildcard, unless a leading wildcard already exists + * @property {number} wildcard.TRAILING - Append a wildcard to the term, unless a trailing wildcard already exists + * @see lunr.Query~Clause + * @see lunr.Query#clause + * @see lunr.Query#term + * @example query term with trailing wildcard + * query.term('foo', { wildcard: lunr.Query.wildcard.TRAILING }) + * @example query term with leading and trailing wildcard + * query.term('foo', { + * wildcard: lunr.Query.wildcard.LEADING | lunr.Query.wildcard.TRAILING + * }) + */ + +lunr.Query.wildcard = new String ("*") +lunr.Query.wildcard.NONE = 0 +lunr.Query.wildcard.LEADING = 1 +lunr.Query.wildcard.TRAILING = 2 + +/** + * Constants for indicating what kind of presence a term must have in matching documents. + * + * @constant + * @enum {number} + * @see lunr.Query~Clause + * @see lunr.Query#clause + * @see lunr.Query#term + * @example query term with required presence + * query.term('foo', { presence: lunr.Query.presence.REQUIRED }) + */ +lunr.Query.presence = { + /** + * Term's presence in a document is optional, this is the default value. + */ + OPTIONAL: 1, + + /** + * Term's presence in a document is required, documents that do not contain + * this term will not be returned. + */ + REQUIRED: 2, + + /** + * Term's presence in a document is prohibited, documents that do contain + * this term will not be returned. + */ + PROHIBITED: 3 +} + +/** + * A single clause in a {@link lunr.Query} contains a term and details on how to + * match that term against a {@link lunr.Index}. + * + * @typedef {Object} lunr.Query~Clause + * @property {string[]} fields - The fields in an index this clause should be matched against. + * @property {number} [boost=1] - Any boost that should be applied when matching this clause. + * @property {number} [editDistance] - Whether the term should have fuzzy matching applied, and how fuzzy the match should be. + * @property {boolean} [usePipeline] - Whether the term should be passed through the search pipeline. + * @property {number} [wildcard=lunr.Query.wildcard.NONE] - Whether the term should have wildcards appended or prepended. + * @property {number} [presence=lunr.Query.presence.OPTIONAL] - The terms presence in any matching documents. + */ + +/** + * Adds a {@link lunr.Query~Clause} to this query. + * + * Unless the clause contains the fields to be matched all fields will be matched. In addition + * a default boost of 1 is applied to the clause. + * + * @param {lunr.Query~Clause} clause - The clause to add to this query. + * @see lunr.Query~Clause + * @returns {lunr.Query} + */ +lunr.Query.prototype.clause = function (clause) { + if (!('fields' in clause)) { + clause.fields = this.allFields + } + + if (!('boost' in clause)) { + clause.boost = 1 + } + + if (!('usePipeline' in clause)) { + clause.usePipeline = true + } + + if (!('wildcard' in clause)) { + clause.wildcard = lunr.Query.wildcard.NONE + } + + if ((clause.wildcard & lunr.Query.wildcard.LEADING) && (clause.term.charAt(0) != lunr.Query.wildcard)) { + clause.term = "*" + clause.term + } + + if ((clause.wildcard & lunr.Query.wildcard.TRAILING) && (clause.term.slice(-1) != lunr.Query.wildcard)) { + clause.term = "" + clause.term + "*" + } + + if (!('presence' in clause)) { + clause.presence = lunr.Query.presence.OPTIONAL + } + + this.clauses.push(clause) + + return this +} + +/** + * A negated query is one in which every clause has a presence of + * prohibited. These queries require some special processing to return + * the expected results. + * + * @returns boolean + */ +lunr.Query.prototype.isNegated = function () { + for (var i = 0; i < this.clauses.length; i++) { + if (this.clauses[i].presence != lunr.Query.presence.PROHIBITED) { + return false + } + } + + return true +} + +/** + * Adds a term to the current query, under the covers this will create a {@link lunr.Query~Clause} + * to the list of clauses that make up this query. + * + * The term is used as is, i.e. no tokenization will be performed by this method. Instead conversion + * to a token or token-like string should be done before calling this method. + * + * The term will be converted to a string by calling `toString`. Multiple terms can be passed as an + * array, each term in the array will share the same options. + * + * @param {object|object[]} term - The term(s) to add to the query. + * @param {object} [options] - Any additional properties to add to the query clause. + * @returns {lunr.Query} + * @see lunr.Query#clause + * @see lunr.Query~Clause + * @example adding a single term to a query + * query.term("foo") + * @example adding a single term to a query and specifying search fields, term boost and automatic trailing wildcard + * query.term("foo", { + * fields: ["title"], + * boost: 10, + * wildcard: lunr.Query.wildcard.TRAILING + * }) + * @example using lunr.tokenizer to convert a string to tokens before using them as terms + * query.term(lunr.tokenizer("foo bar")) + */ +lunr.Query.prototype.term = function (term, options) { + if (Array.isArray(term)) { + term.forEach(function (t) { this.term(t, lunr.utils.clone(options)) }, this) + return this + } + + var clause = options || {} + clause.term = term.toString() + + this.clause(clause) + + return this +} +lunr.QueryParseError = function (message, start, end) { + this.name = "QueryParseError" + this.message = message + this.start = start + this.end = end +} + +lunr.QueryParseError.prototype = new Error +lunr.QueryLexer = function (str) { + this.lexemes = [] + this.str = str + this.length = str.length + this.pos = 0 + this.start = 0 + this.escapeCharPositions = [] +} + +lunr.QueryLexer.prototype.run = function () { + var state = lunr.QueryLexer.lexText + + while (state) { + state = state(this) + } +} + +lunr.QueryLexer.prototype.sliceString = function () { + var subSlices = [], + sliceStart = this.start, + sliceEnd = this.pos + + for (var i = 0; i < this.escapeCharPositions.length; i++) { + sliceEnd = this.escapeCharPositions[i] + subSlices.push(this.str.slice(sliceStart, sliceEnd)) + sliceStart = sliceEnd + 1 + } + + subSlices.push(this.str.slice(sliceStart, this.pos)) + this.escapeCharPositions.length = 0 + + return subSlices.join('') +} + +lunr.QueryLexer.prototype.emit = function (type) { + this.lexemes.push({ + type: type, + str: this.sliceString(), + start: this.start, + end: this.pos + }) + + this.start = this.pos +} + +lunr.QueryLexer.prototype.escapeCharacter = function () { + this.escapeCharPositions.push(this.pos - 1) + this.pos += 1 +} + +lunr.QueryLexer.prototype.next = function () { + if (this.pos >= this.length) { + return lunr.QueryLexer.EOS + } + + var char = this.str.charAt(this.pos) + this.pos += 1 + return char +} + +lunr.QueryLexer.prototype.width = function () { + return this.pos - this.start +} + +lunr.QueryLexer.prototype.ignore = function () { + if (this.start == this.pos) { + this.pos += 1 + } + + this.start = this.pos +} + +lunr.QueryLexer.prototype.backup = function () { + this.pos -= 1 +} + +lunr.QueryLexer.prototype.acceptDigitRun = function () { + var char, charCode + + do { + char = this.next() + charCode = char.charCodeAt(0) + } while (charCode > 47 && charCode < 58) + + if (char != lunr.QueryLexer.EOS) { + this.backup() + } +} + +lunr.QueryLexer.prototype.more = function () { + return this.pos < this.length +} + +lunr.QueryLexer.EOS = 'EOS' +lunr.QueryLexer.FIELD = 'FIELD' +lunr.QueryLexer.TERM = 'TERM' +lunr.QueryLexer.EDIT_DISTANCE = 'EDIT_DISTANCE' +lunr.QueryLexer.BOOST = 'BOOST' +lunr.QueryLexer.PRESENCE = 'PRESENCE' + +lunr.QueryLexer.lexField = function (lexer) { + lexer.backup() + lexer.emit(lunr.QueryLexer.FIELD) + lexer.ignore() + return lunr.QueryLexer.lexText +} + +lunr.QueryLexer.lexTerm = function (lexer) { + if (lexer.width() > 1) { + lexer.backup() + lexer.emit(lunr.QueryLexer.TERM) + } + + lexer.ignore() + + if (lexer.more()) { + return lunr.QueryLexer.lexText + } +} + +lunr.QueryLexer.lexEditDistance = function (lexer) { + lexer.ignore() + lexer.acceptDigitRun() + lexer.emit(lunr.QueryLexer.EDIT_DISTANCE) + return lunr.QueryLexer.lexText +} + +lunr.QueryLexer.lexBoost = function (lexer) { + lexer.ignore() + lexer.acceptDigitRun() + lexer.emit(lunr.QueryLexer.BOOST) + return lunr.QueryLexer.lexText +} + +lunr.QueryLexer.lexEOS = function (lexer) { + if (lexer.width() > 0) { + lexer.emit(lunr.QueryLexer.TERM) + } +} + +// This matches the separator used when tokenising fields +// within a document. These should match otherwise it is +// not possible to search for some tokens within a document. +// +// It is possible for the user to change the separator on the +// tokenizer so it _might_ clash with any other of the special +// characters already used within the search string, e.g. :. +// +// This means that it is possible to change the separator in +// such a way that makes some words unsearchable using a search +// string. +lunr.QueryLexer.termSeparator = lunr.tokenizer.separator + +lunr.QueryLexer.lexText = function (lexer) { + while (true) { + var char = lexer.next() + + if (char == lunr.QueryLexer.EOS) { + return lunr.QueryLexer.lexEOS + } + + // Escape character is '\' + if (char.charCodeAt(0) == 92) { + lexer.escapeCharacter() + continue + } + + if (char == ":") { + return lunr.QueryLexer.lexField + } + + if (char == "~") { + lexer.backup() + if (lexer.width() > 0) { + lexer.emit(lunr.QueryLexer.TERM) + } + return lunr.QueryLexer.lexEditDistance + } + + if (char == "^") { + lexer.backup() + if (lexer.width() > 0) { + lexer.emit(lunr.QueryLexer.TERM) + } + return lunr.QueryLexer.lexBoost + } + + // "+" indicates term presence is required + // checking for length to ensure that only + // leading "+" are considered + if (char == "+" && lexer.width() === 1) { + lexer.emit(lunr.QueryLexer.PRESENCE) + return lunr.QueryLexer.lexText + } + + // "-" indicates term presence is prohibited + // checking for length to ensure that only + // leading "-" are considered + if (char == "-" && lexer.width() === 1) { + lexer.emit(lunr.QueryLexer.PRESENCE) + return lunr.QueryLexer.lexText + } + + if (char.match(lunr.QueryLexer.termSeparator)) { + return lunr.QueryLexer.lexTerm + } + } +} + +lunr.QueryParser = function (str, query) { + this.lexer = new lunr.QueryLexer (str) + this.query = query + this.currentClause = {} + this.lexemeIdx = 0 +} + +lunr.QueryParser.prototype.parse = function () { + this.lexer.run() + this.lexemes = this.lexer.lexemes + + var state = lunr.QueryParser.parseClause + + while (state) { + state = state(this) + } + + return this.query +} + +lunr.QueryParser.prototype.peekLexeme = function () { + return this.lexemes[this.lexemeIdx] +} + +lunr.QueryParser.prototype.consumeLexeme = function () { + var lexeme = this.peekLexeme() + this.lexemeIdx += 1 + return lexeme +} + +lunr.QueryParser.prototype.nextClause = function () { + var completedClause = this.currentClause + this.query.clause(completedClause) + this.currentClause = {} +} + +lunr.QueryParser.parseClause = function (parser) { + var lexeme = parser.peekLexeme() + + if (lexeme == undefined) { + return + } + + switch (lexeme.type) { + case lunr.QueryLexer.PRESENCE: + return lunr.QueryParser.parsePresence + case lunr.QueryLexer.FIELD: + return lunr.QueryParser.parseField + case lunr.QueryLexer.TERM: + return lunr.QueryParser.parseTerm + default: + var errorMessage = "expected either a field or a term, found " + lexeme.type + + if (lexeme.str.length >= 1) { + errorMessage += " with value '" + lexeme.str + "'" + } + + throw new lunr.QueryParseError (errorMessage, lexeme.start, lexeme.end) + } +} + +lunr.QueryParser.parsePresence = function (parser) { + var lexeme = parser.consumeLexeme() + + if (lexeme == undefined) { + return + } + + switch (lexeme.str) { + case "-": + parser.currentClause.presence = lunr.Query.presence.PROHIBITED + break + case "+": + parser.currentClause.presence = lunr.Query.presence.REQUIRED + break + default: + var errorMessage = "unrecognised presence operator'" + lexeme.str + "'" + throw new lunr.QueryParseError (errorMessage, lexeme.start, lexeme.end) + } + + var nextLexeme = parser.peekLexeme() + + if (nextLexeme == undefined) { + var errorMessage = "expecting term or field, found nothing" + throw new lunr.QueryParseError (errorMessage, lexeme.start, lexeme.end) + } + + switch (nextLexeme.type) { + case lunr.QueryLexer.FIELD: + return lunr.QueryParser.parseField + case lunr.QueryLexer.TERM: + return lunr.QueryParser.parseTerm + default: + var errorMessage = "expecting term or field, found '" + nextLexeme.type + "'" + throw new lunr.QueryParseError (errorMessage, nextLexeme.start, nextLexeme.end) + } +} + +lunr.QueryParser.parseField = function (parser) { + var lexeme = parser.consumeLexeme() + + if (lexeme == undefined) { + return + } + + if (parser.query.allFields.indexOf(lexeme.str) == -1) { + var possibleFields = parser.query.allFields.map(function (f) { return "'" + f + "'" }).join(', '), + errorMessage = "unrecognised field '" + lexeme.str + "', possible fields: " + possibleFields + + throw new lunr.QueryParseError (errorMessage, lexeme.start, lexeme.end) + } + + parser.currentClause.fields = [lexeme.str] + + var nextLexeme = parser.peekLexeme() + + if (nextLexeme == undefined) { + var errorMessage = "expecting term, found nothing" + throw new lunr.QueryParseError (errorMessage, lexeme.start, lexeme.end) + } + + switch (nextLexeme.type) { + case lunr.QueryLexer.TERM: + return lunr.QueryParser.parseTerm + default: + var errorMessage = "expecting term, found '" + nextLexeme.type + "'" + throw new lunr.QueryParseError (errorMessage, nextLexeme.start, nextLexeme.end) + } +} + +lunr.QueryParser.parseTerm = function (parser) { + var lexeme = parser.consumeLexeme() + + if (lexeme == undefined) { + return + } + + parser.currentClause.term = lexeme.str.toLowerCase() + + if (lexeme.str.indexOf("*") != -1) { + parser.currentClause.usePipeline = false + } + + var nextLexeme = parser.peekLexeme() + + if (nextLexeme == undefined) { + parser.nextClause() + return + } + + switch (nextLexeme.type) { + case lunr.QueryLexer.TERM: + parser.nextClause() + return lunr.QueryParser.parseTerm + case lunr.QueryLexer.FIELD: + parser.nextClause() + return lunr.QueryParser.parseField + case lunr.QueryLexer.EDIT_DISTANCE: + return lunr.QueryParser.parseEditDistance + case lunr.QueryLexer.BOOST: + return lunr.QueryParser.parseBoost + case lunr.QueryLexer.PRESENCE: + parser.nextClause() + return lunr.QueryParser.parsePresence + default: + var errorMessage = "Unexpected lexeme type '" + nextLexeme.type + "'" + throw new lunr.QueryParseError (errorMessage, nextLexeme.start, nextLexeme.end) + } +} + +lunr.QueryParser.parseEditDistance = function (parser) { + var lexeme = parser.consumeLexeme() + + if (lexeme == undefined) { + return + } + + var editDistance = parseInt(lexeme.str, 10) + + if (isNaN(editDistance)) { + var errorMessage = "edit distance must be numeric" + throw new lunr.QueryParseError (errorMessage, lexeme.start, lexeme.end) + } + + parser.currentClause.editDistance = editDistance + + var nextLexeme = parser.peekLexeme() + + if (nextLexeme == undefined) { + parser.nextClause() + return + } + + switch (nextLexeme.type) { + case lunr.QueryLexer.TERM: + parser.nextClause() + return lunr.QueryParser.parseTerm + case lunr.QueryLexer.FIELD: + parser.nextClause() + return lunr.QueryParser.parseField + case lunr.QueryLexer.EDIT_DISTANCE: + return lunr.QueryParser.parseEditDistance + case lunr.QueryLexer.BOOST: + return lunr.QueryParser.parseBoost + case lunr.QueryLexer.PRESENCE: + parser.nextClause() + return lunr.QueryParser.parsePresence + default: + var errorMessage = "Unexpected lexeme type '" + nextLexeme.type + "'" + throw new lunr.QueryParseError (errorMessage, nextLexeme.start, nextLexeme.end) + } +} + +lunr.QueryParser.parseBoost = function (parser) { + var lexeme = parser.consumeLexeme() + + if (lexeme == undefined) { + return + } + + var boost = parseInt(lexeme.str, 10) + + if (isNaN(boost)) { + var errorMessage = "boost must be numeric" + throw new lunr.QueryParseError (errorMessage, lexeme.start, lexeme.end) + } + + parser.currentClause.boost = boost + + var nextLexeme = parser.peekLexeme() + + if (nextLexeme == undefined) { + parser.nextClause() + return + } + + switch (nextLexeme.type) { + case lunr.QueryLexer.TERM: + parser.nextClause() + return lunr.QueryParser.parseTerm + case lunr.QueryLexer.FIELD: + parser.nextClause() + return lunr.QueryParser.parseField + case lunr.QueryLexer.EDIT_DISTANCE: + return lunr.QueryParser.parseEditDistance + case lunr.QueryLexer.BOOST: + return lunr.QueryParser.parseBoost + case lunr.QueryLexer.PRESENCE: + parser.nextClause() + return lunr.QueryParser.parsePresence + default: + var errorMessage = "Unexpected lexeme type '" + nextLexeme.type + "'" + throw new lunr.QueryParseError (errorMessage, nextLexeme.start, nextLexeme.end) + } +} + + /** + * export the module via AMD, CommonJS or as a browser global + * Export code from https://github.com/umdjs/umd/blob/master/returnExports.js + */ + ;(function (root, factory) { + if (typeof define === 'function' && define.amd) { + // AMD. Register as an anonymous module. + define(factory) + } else if (typeof exports === 'object') { + /** + * Node. Does not work with strict CommonJS, but + * only CommonJS-like environments that support module.exports, + * like Node. + */ + module.exports = factory() + } else { + // Browser globals (root is window) + root.lunr = factory() + } + }(this, function () { + /** + * Just return a value to define the module export. + * This example returns an object, but the module + * can return a function as the exported value. + */ + return lunr + })) +})(); diff --git a/search/main.js b/search/main.js new file mode 100644 index 0000000..a5e469d --- /dev/null +++ b/search/main.js @@ -0,0 +1,109 @@ +function getSearchTermFromLocation() { + var sPageURL = window.location.search.substring(1); + var sURLVariables = sPageURL.split('&'); + for (var i = 0; i < sURLVariables.length; i++) { + var sParameterName = sURLVariables[i].split('='); + if (sParameterName[0] == 'q') { + return decodeURIComponent(sParameterName[1].replace(/\+/g, '%20')); + } + } +} + +function joinUrl (base, path) { + if (path.substring(0, 1) === "/") { + // path starts with `/`. 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It builds on FEniCS for the discretization of the PDE and on PETSc for scalable and efficient linear algebra operations and solvers. Features Friendly, compact, near-mathematical FEniCS notation to express, differentiate, and discretize the PDE forward model and likelihood function Large scale optimization algorithms, such as globalized inexact Newton-CG method, to solve the inverse problem Randomized algorithms for trace estimation, eigenvalues and singular values decomposition. Scalable sampling of Gaussian random fields Linearized Bayesian inversion with low-rank based representation of the posterior covariance Hessian informed MCMC algorithms to explore the posterior distribution Forward propagation of uncertainty capabilities using Monte Carlo and Taylor expansion control variates See also our tutorial and list of related publications . For additional resources and tutorials please see the teaching material for the 2018 Gene Golub SIAM Summer School on Inverse Problems: Systematic Integration of Data with Models under Uncertainty available here . The complete API reference is available here . Latest Release Development version Download hippylib-3.0.0.zip Previous releases Contact Developed by the hIPPYlib team at UT Austin , UC Merced , and WUSTL . Slack channel The hIPPYlib slack channel is a good resource to request and receive help with using hIPPYlib. Everyone is invited to read and take part in discussions. Discussions about development of new features in hIPPYlib also take place here. You can join our Slack community by filling in this form . Please cite as @article{VillaPetraGhattas21, author = {Villa, Umberto and Petra, Noemi and Ghattas, Omar}, title = \"{HIPPYlib: An Extensible Software Framework for Large-Scale Inverse Problems Governed by PDEs: Part I: Deterministic Inversion and Linearized Bayesian Inference}\", year = {2021}, issue_date = {March 2021}, publisher = {Association for Computing Machinery}, address = {New York, NY, USA}, volume = {47}, number = {2}, issn = {0098-3500}, url = {https://doi.org/10.1145/3428447}, doi = {10.1145/3428447}, journal = {ACM Trans. Math. Softw.}, month = apr, articleno = {16}, numpages = {34} } @article{VillaPetraGhattas18, title = \"{hIPPYlib: an Extensible Software Framework for Large-scale Deterministic and Bayesian Inverse Problems}\", author = {Villa, U. and Petra, N. and Ghattas, O.}, journal = {Journal of Open Source Software}, volume = {3}, number = {30}, page = {940}, doi = {10.21105/joss.00940}, year = {2018} } @article{VillaPetraGhattas16, title = \"{hIPPYlib: an Extensible Software Framework for Large-scale Deterministic and Bayesian Inverse Problems}\", author = {Villa, U. and Petra, N. and Ghattas, O.}, year = {2016}, url = {http://hippylib.github.io}, doi = {10.5281/zenodo.596931} }","title":"Home"},{"location":"#hippylib-inverse-problem-python-library","text":"hIPPYlib implements state-of-the-art scalable adjoint-based algorithms for PDE-based deterministic and Bayesian inverse problems . It builds on FEniCS for the discretization of the PDE and on PETSc for scalable and efficient linear algebra operations and solvers.","title":"hIPPYlib - Inverse Problem PYthon library"},{"location":"#features","text":"Friendly, compact, near-mathematical FEniCS notation to express, differentiate, and discretize the PDE forward model and likelihood function Large scale optimization algorithms, such as globalized inexact Newton-CG method, to solve the inverse problem Randomized algorithms for trace estimation, eigenvalues and singular values decomposition. Scalable sampling of Gaussian random fields Linearized Bayesian inversion with low-rank based representation of the posterior covariance Hessian informed MCMC algorithms to explore the posterior distribution Forward propagation of uncertainty capabilities using Monte Carlo and Taylor expansion control variates See also our tutorial and list of related publications . For additional resources and tutorials please see the teaching material for the 2018 Gene Golub SIAM Summer School on Inverse Problems: Systematic Integration of Data with Models under Uncertainty available here . The complete API reference is available here .","title":"Features"},{"location":"#latest-release","text":"Development version Download hippylib-3.0.0.zip Previous releases","title":"Latest Release"},{"location":"#contact","text":"Developed by the hIPPYlib team at UT Austin , UC Merced , and WUSTL .","title":"Contact"},{"location":"#slack-channel","text":"The hIPPYlib slack channel is a good resource to request and receive help with using hIPPYlib. Everyone is invited to read and take part in discussions. Discussions about development of new features in hIPPYlib also take place here. You can join our Slack community by filling in this form . Please cite as @article{VillaPetraGhattas21, author = {Villa, Umberto and Petra, Noemi and Ghattas, Omar}, title = \"{HIPPYlib: An Extensible Software Framework for Large-Scale Inverse Problems Governed by PDEs: Part I: Deterministic Inversion and Linearized Bayesian Inference}\", year = {2021}, issue_date = {March 2021}, publisher = {Association for Computing Machinery}, address = {New York, NY, USA}, volume = {47}, number = {2}, issn = {0098-3500}, url = {https://doi.org/10.1145/3428447}, doi = {10.1145/3428447}, journal = {ACM Trans. Math. Softw.}, month = apr, articleno = {16}, numpages = {34} } @article{VillaPetraGhattas18, title = \"{hIPPYlib: an Extensible Software Framework for Large-scale Deterministic and Bayesian Inverse Problems}\", author = {Villa, U. and Petra, N. and Ghattas, O.}, journal = {Journal of Open Source Software}, volume = {3}, number = {30}, page = {940}, doi = {10.21105/joss.00940}, year = {2018} } @article{VillaPetraGhattas16, title = \"{hIPPYlib: an Extensible Software Framework for Large-scale Deterministic and Bayesian Inverse Problems}\", author = {Villa, U. and Petra, N. and Ghattas, O.}, year = {2016}, url = {http://hippylib.github.io}, doi = {10.5281/zenodo.596931} }","title":"Slack channel"},{"location":"about/","text":"About hIPPYlib Authors Umberto Villa Noemi Petra Omar Ghattas Current contributors Amal Alghamdi (UT Austin), Olalekan Babaniyi (RIT), Joshua Chen (UT Austin), Peng Chen (UT Austin), Argho Datta (WashU), Ki-Tae Kim (UC Merced), Tom O'Leary-Roseberry (UT Austin), Luke Lozenski (WashU), Siddhant Wahal (UT Austin) Past contributors Ilona Ambartsumyan , Ben Crestel, Eldar Khattatov , Vishwas Rao Copyright \u00a9 2016-2018 The University of Texas at Austin, University of California Merced. \u00a9 2019-2020 The University of Texas at Austin, University of California Merced, Washington University in St. Louis. Support O. Ghattas, N. Petra (PIs), U. Villa (Co-PI), Integrating Data with Complex Predictive Models under Uncertainty: An Extensible Software Framework for Large-Scale Bayesian Inversion , NSF-SSI2, Grants No ACI-1550593, ACI-1550547 (2016-2019). O. Ghattas (PI); G. Biros and Y. Marzouk (Co-PIs), Bayesian Optimal Experimental Design for Inverse Scattering , Air Force Office of Scientific Research, Computational Mathematics program, grant FA9550-12-1-81243 (2017\u20132020). O. Ghattas (PI), G. Biros, M. Girolami, M. Heinkenschloss, R. Moser, A. Philpott, A. Stuart, and K. Willcox (Co-PIs), Inference, Simulation, and Optimization of Complex Systems Under Uncertainty: Theory, Algorithms, and Applications to Turbulent Combustion , DARPA/ARO, Grants No W911NF-15-2-0121 (2016-2017). O. Ghattas (PI), M. Hesse (Co-PI), CDS&E: A Bayesian inference framework for management of CO2 sequestration , National Science Foundation, Division Of Chemical, Bioengineering, Environmental, & Transport Systems, grant CBET-1508713 (2015-2017). License GNU General Public License version 2 (GPL) Older Releases (1.2.0 or older): GNU General Public License version 3 (GPL) Website built with MkDocs , bootstrap , bootswatch , and MathJax . Hosted on GitHub .","title":"About"},{"location":"about/#about-hippylib","text":"","title":"About hIPPYlib"},{"location":"about/#authors","text":"Umberto Villa Noemi Petra Omar Ghattas","title":"Authors"},{"location":"about/#current-contributors","text":"Amal Alghamdi (UT Austin), Olalekan Babaniyi (RIT), Joshua Chen (UT Austin), Peng Chen (UT Austin), Argho Datta (WashU), Ki-Tae Kim (UC Merced), Tom O'Leary-Roseberry (UT Austin), Luke Lozenski (WashU), Siddhant Wahal (UT Austin)","title":"Current contributors"},{"location":"about/#past-contributors","text":"Ilona Ambartsumyan , Ben Crestel, Eldar Khattatov , Vishwas Rao","title":"Past contributors"},{"location":"about/#copyright","text":"\u00a9 2016-2018 The University of Texas at Austin, University of California Merced. \u00a9 2019-2020 The University of Texas at Austin, University of California Merced, Washington University in St. Louis.","title":"Copyright"},{"location":"about/#support","text":"O. Ghattas, N. Petra (PIs), U. Villa (Co-PI), Integrating Data with Complex Predictive Models under Uncertainty: An Extensible Software Framework for Large-Scale Bayesian Inversion , NSF-SSI2, Grants No ACI-1550593, ACI-1550547 (2016-2019). O. Ghattas (PI); G. Biros and Y. Marzouk (Co-PIs), Bayesian Optimal Experimental Design for Inverse Scattering , Air Force Office of Scientific Research, Computational Mathematics program, grant FA9550-12-1-81243 (2017\u20132020). O. Ghattas (PI), G. Biros, M. Girolami, M. Heinkenschloss, R. Moser, A. Philpott, A. Stuart, and K. Willcox (Co-PIs), Inference, Simulation, and Optimization of Complex Systems Under Uncertainty: Theory, Algorithms, and Applications to Turbulent Combustion , DARPA/ARO, Grants No W911NF-15-2-0121 (2016-2017). O. Ghattas (PI), M. Hesse (Co-PI), CDS&E: A Bayesian inference framework for management of CO2 sequestration , National Science Foundation, Division Of Chemical, Bioengineering, Environmental, & Transport Systems, grant CBET-1508713 (2015-2017).","title":"Support"},{"location":"about/#license","text":"GNU General Public License version 2 (GPL) Older Releases (1.2.0 or older): GNU General Public License version 3 (GPL) Website built with MkDocs , bootstrap , bootswatch , and MathJax . Hosted on GitHub .","title":"License"},{"location":"documentation/","text":"Documentation The complete API reference of hIPPYlib is available at readthedocs . Algorithms implemented in hIPPYlib are described in Umberto Villa, Noemi Petra, and Omar Ghattas. 2021. HIPPYlib: An Extensible Software Framework for Large-Scale Inverse Problems Governed by PDEs: Part I: Deterministic Inversion and Linearized Bayesian Inference . ACM Trans. Math. Softw. 47, 2, Article 16 (March 2021), 34 pages. arXiv Installation of stable releases The latest hIPPYlib release depends on FEniCS versions 2019.1. FEniCS needs to be built with the following dependecies: numpy , scipy , matplotlib , mpi4py PETSc and petsc4py (version 3.7.0 or above) SLEPc and slepc4py (version 3.7.0 or above) PETSc dependencies: parmetis , scotch , suitesparse , superlu_dist , ml , hypre (optional): mshr , jupyter For detailed installation instructions of the latest stable release see here . Custom Docker Images and conda packages for FEniCS 2019.1.0 While hIPPYlib can be used with the Docker images and conda packages from the ufficial FEniCS packages, a customized FEniCS docker image and customized conda packages is available. hIPPYlib 2.3.0 Docker container A Docker image with a working installation of hIPPYlib 2.3.0 and FEniCS 2017.2 is available here . Numerical results presented in the manuscript hIPPYlib: A Software Framework for Large-Scale Inverse Problems Governed by PDEs: Part I: Deterministic Inversion and Linearized Bayesian Inference were obtained using the software in this image.","title":"Documentation"},{"location":"documentation/#documentation","text":"The complete API reference of hIPPYlib is available at readthedocs . Algorithms implemented in hIPPYlib are described in Umberto Villa, Noemi Petra, and Omar Ghattas. 2021. HIPPYlib: An Extensible Software Framework for Large-Scale Inverse Problems Governed by PDEs: Part I: Deterministic Inversion and Linearized Bayesian Inference . ACM Trans. Math. Softw. 47, 2, Article 16 (March 2021), 34 pages. arXiv","title":"Documentation"},{"location":"documentation/#installation-of-stable-releases","text":"The latest hIPPYlib release depends on FEniCS versions 2019.1. FEniCS needs to be built with the following dependecies: numpy , scipy , matplotlib , mpi4py PETSc and petsc4py (version 3.7.0 or above) SLEPc and slepc4py (version 3.7.0 or above) PETSc dependencies: parmetis , scotch , suitesparse , superlu_dist , ml , hypre (optional): mshr , jupyter For detailed installation instructions of the latest stable release see here .","title":"Installation of stable releases"},{"location":"documentation/#custom-docker-images-and-conda-packages-for-fenics-201910","text":"While hIPPYlib can be used with the Docker images and conda packages from the ufficial FEniCS packages, a customized FEniCS docker image and customized conda packages is available.","title":"Custom Docker Images and conda packages for FEniCS 2019.1.0"},{"location":"documentation/#hippylib-230-docker-container","text":"A Docker image with a working installation of hIPPYlib 2.3.0 and FEniCS 2017.2 is available here . Numerical results presented in the manuscript hIPPYlib: A Software Framework for Large-Scale Inverse Problems Governed by PDEs: Part I: Deterministic Inversion and Linearized Bayesian Inference were obtained using the software in this image.","title":"hIPPYlib 2.3.0 Docker container"},{"location":"download/","text":"Download Latest release Download hippylib-3.1.0.zip All releases Filename Version Release Date Notes Documentation hippylib-3.1.0.zip 3.1.0 Dec 2022 Non-Gaussian noise model and interpolation utilities html hippylib-3.0.0.zip 3.0.0 Feb 2020 Support for FEniCS 2019.1. 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See the changelog on readthedocs for further details.","title":"Download"},{"location":"download/#download","text":"","title":"Download"},{"location":"download/#latest-release","text":"Download hippylib-3.1.0.zip","title":"Latest release"},{"location":"download/#all-releases","text":"Filename Version Release Date Notes Documentation hippylib-3.1.0.zip 3.1.0 Dec 2022 Non-Gaussian noise model and interpolation utilities html hippylib-3.0.0.zip 3.0.0 Feb 2020 Support for FEniCS 2019.1. New features and examples html hippylib-2.3.0.zip 2.3.0 Sept 2019 Updated examples html hippylib-2.2.1.zip 2.2.1 March 2019 Bugfixes html hippylib-2.2.0.zip 2.2.0 Dec 2018 Enchantments and install html hippylib-2.1.1.zip 2.1.1 Oct 2018 Publication to JOSS html hippylib-2.1.0.tar.gz 2.1.0 July 2018 Enchantments & Bugfixes html hippylib-2.0.0.tar.gz 2.0.0 June 2018 Major release html hippylib-1.6.0.tar.gz 1.6.0 May 2018 Bugfix hippylib-1.5.0.tar.gz 1.5.0 Jan 2018 Support for FEniCS 2017.2 hippylib-1.4.0.tar.gz 1.4.0 Nov 2017 Support for Python 3 hippylib-1.3.0.tar.gz 1.3.0 June 2017 Support for FEniCS 2017.1; GLPv2 hippylib-1.2.0.tar.gz 1.2.0 Apr 2017 Support for FEniCS 2016.2 hippylib-1.1.0.tar.gz 1.1.0 Nov 2016 Support for FEniCS 2016.1 hippylib-1.0.2.tar.gz 1.0.2 Sept 2016 Enchantment hippylib-1.0.1.tar.gz 1.0.1 Aug 2016 Bugfix hippylib-1.0.0.tar.gz 1.0.0 Aug 2016 Initial release hIPPYlib releases as of 1.2.0 are also archived on Zenodo . See the changelog on readthedocs for further details.","title":"All releases"},{"location":"oral_presentations/","text":"Selected oral presentations N. Petra, Inferring the basal sliding coefficient for the Stokes ice sheet model under rheological uncertainty , Mathematical Modelling in Glaciology, Banff International Research Station, Banff, AB, Candada I. Ambartsumyan, O. Ghattas , Fast methods for Bayesian inverse problems governed by PDE forward models with random coefficient fields , Applied Inverse Problems Conference, July 8-12, 2019, Grenoble, France U. Villa , O. Ghattas, Scalable optimal experimental design for large scale non-linear Bayesian inverse problems , Applied Inverse Problems Conference, July 8-12, 2019, Grenoble, France O. Babaniyi , O. Ghattas, N. Petra and U. Villa, hIPPYlib: An Extensible Software Framework for Large-Scale Inverse Problems , 2019 FEniCS Conference, Carnegie Institution for Science Department of Terrestrial Magnetism (DTM), Washington DC,June 12-14, 2019. Best postdoc presentation award N. Petra, G. Stadler , Inverse Problems: Integrating Data with Models under Uncertainty , SIAM CSE, Feb 25-March 1, 2019, Spokane, Wa, US K. Wu , P. Chen, O. Ghattas, A Stein Variational Newton Method for Optimal Experiment Design Problems , SIAM CSE, Feb 25-March 1, 2019, Spokane, Wa, US P. Chen , O. Ghattas, U. Villa, Large-scale Optimal Experimental Design for Bayesian Nonlinear Inverse Problems , SIAM CSE, Feb 25-March 1, 2019, Spokane, Wa, US T. O'Leary-Roseberry , J. Chen, P. Chen, U. Villa, O. Ghattas, Large-scale Optimization in Deep Learning for PDE Representation abstract , SIAM CSE, Feb 25-March 1, 2019, Spokane, Wa, US J. J. Lee , O. Ghattas, T. Bui-Thanh, U. Villa, Derivative Informed MCMC Methods for Subsurface Models with Faults , SIAM CSE, Feb 25-March 1, 2019, Spokane, Wa, US U. Villa , O. Ghattas, Scalable Methods for Bayesian Optimal Experimental Design Using Laplace Approximation , SIAM CSE, Feb 25-March 1, 2019, Spokane, Wa, US U. Villa , Learning from data through the lens of mathematical models , Analysis Seminar, Dept. of Mathematics & Statistics, Washington University, January 28, 2019, St Louis, MO, US U. Villa , Large Scale Inverse Problems and Uncertainty Quantification: Computational Tools and Imaging Applications , Electrical & Systems Engineering Seminar, Washington University, January 24, 2019, St Louis, MO, US N. Petra , hIPPYlib: An Extensible Software Framework for Large-Scale Inverse Problems , Optimization Seminar, University of California, Merced, October 19, 2018, Merced, CA, US U. Villa , O. Ghattas, Maximize the Expected Information Gain in Bayesian Experimental Design Problems: A Fast Optimization Algorithm Based on Laplace Approximation and Randomized Eigensolvers , SIAM UQ, April 16-19, 2018, Garden Grove, CA, US T. O\u2019Leary-Roseberry , A PDE Constrained Optimization Approach to the Solution of the Stefan Problem , Texas Applied Mathematics and Engineering Symposium, Sept. 21-23, 2017, Austin, TX, US U. Villa , hIPPYlib: An Extensible Software Framework for Large-Scale Deterministic and Linearized Bayesian Inverse , Texas Applied Mathematics and Engineering Symposium, Sept. 21-23, 2017, Austin, TX, US U. Villa , Taylor Approximation for PDE-constrained Optimal Control Problems under High-dimensional Uncertainty , SIAM Control, July 10 - 12, 2017, Pittsburgh, Pa, US U. Villa , Derivative-Informed MCMC for Bayesian Calibration of Stochastic PDE Models , SIAM Annual, July 10 - 14, 2017, Pittsburgh, Pa, US U. Villa , Hessian-based sampling techniques for Bayesian inverse problems with stochastic PDE forward model , Applied Inverse Problems, May 29 - June 2, 2017, Hangzhou, China U. Villa , Bayesian Calibration of Inadequate Stochastic PDE Models , SIAM CSE, Feb 27-March 3, 2017, Atlanta, GA, US P. Chen , Taylor Approximation for PDE-Constrained Optimal Control Problems Under High-Dimensional Uncertainty: Application to a Turbulence Model , SIAM CSE, Feb 27-March 3, 2017, Atlanta, GA, US B. Crestel , Scalable Solvers for Joint Inversion with Several Structural Coupling Terms , SIAM CSE, Feb 27-March 3, 2017, Atlanta, GA, US Amal Alghamdi , Bayesian Inversion for Subsurface Properties from Poroelastic Forward Models and Surface Deformation Data , SIAM CSE, Feb 27-March 3, 2017, Atlanta, GA, US U. Villa , An Analytical Technique for Forward and Inverse Propagation of Uncertainty , SIAM UQ, April 5-8, 2016, Lausanne, Switzerland","title":"Oral presentations"},{"location":"oral_presentations/#selected-oral-presentations","text":"N. Petra, Inferring the basal sliding coefficient for the Stokes ice sheet model under rheological uncertainty , Mathematical Modelling in Glaciology, Banff International Research Station, Banff, AB, Candada I. Ambartsumyan, O. Ghattas , Fast methods for Bayesian inverse problems governed by PDE forward models with random coefficient fields , Applied Inverse Problems Conference, July 8-12, 2019, Grenoble, France U. Villa , O. Ghattas, Scalable optimal experimental design for large scale non-linear Bayesian inverse problems , Applied Inverse Problems Conference, July 8-12, 2019, Grenoble, France O. Babaniyi , O. Ghattas, N. Petra and U. Villa, hIPPYlib: An Extensible Software Framework for Large-Scale Inverse Problems , 2019 FEniCS Conference, Carnegie Institution for Science Department of Terrestrial Magnetism (DTM), Washington DC,June 12-14, 2019. Best postdoc presentation award N. Petra, G. Stadler , Inverse Problems: Integrating Data with Models under Uncertainty , SIAM CSE, Feb 25-March 1, 2019, Spokane, Wa, US K. Wu , P. Chen, O. Ghattas, A Stein Variational Newton Method for Optimal Experiment Design Problems , SIAM CSE, Feb 25-March 1, 2019, Spokane, Wa, US P. Chen , O. Ghattas, U. Villa, Large-scale Optimal Experimental Design for Bayesian Nonlinear Inverse Problems , SIAM CSE, Feb 25-March 1, 2019, Spokane, Wa, US T. O'Leary-Roseberry , J. Chen, P. Chen, U. Villa, O. Ghattas, Large-scale Optimization in Deep Learning for PDE Representation abstract , SIAM CSE, Feb 25-March 1, 2019, Spokane, Wa, US J. J. Lee , O. Ghattas, T. Bui-Thanh, U. Villa, Derivative Informed MCMC Methods for Subsurface Models with Faults , SIAM CSE, Feb 25-March 1, 2019, Spokane, Wa, US U. Villa , O. Ghattas, Scalable Methods for Bayesian Optimal Experimental Design Using Laplace Approximation , SIAM CSE, Feb 25-March 1, 2019, Spokane, Wa, US U. Villa , Learning from data through the lens of mathematical models , Analysis Seminar, Dept. of Mathematics & Statistics, Washington University, January 28, 2019, St Louis, MO, US U. Villa , Large Scale Inverse Problems and Uncertainty Quantification: Computational Tools and Imaging Applications , Electrical & Systems Engineering Seminar, Washington University, January 24, 2019, St Louis, MO, US N. Petra , hIPPYlib: An Extensible Software Framework for Large-Scale Inverse Problems , Optimization Seminar, University of California, Merced, October 19, 2018, Merced, CA, US U. Villa , O. Ghattas, Maximize the Expected Information Gain in Bayesian Experimental Design Problems: A Fast Optimization Algorithm Based on Laplace Approximation and Randomized Eigensolvers , SIAM UQ, April 16-19, 2018, Garden Grove, CA, US T. O\u2019Leary-Roseberry , A PDE Constrained Optimization Approach to the Solution of the Stefan Problem , Texas Applied Mathematics and Engineering Symposium, Sept. 21-23, 2017, Austin, TX, US U. Villa , hIPPYlib: An Extensible Software Framework for Large-Scale Deterministic and Linearized Bayesian Inverse , Texas Applied Mathematics and Engineering Symposium, Sept. 21-23, 2017, Austin, TX, US U. Villa , Taylor Approximation for PDE-constrained Optimal Control Problems under High-dimensional Uncertainty , SIAM Control, July 10 - 12, 2017, Pittsburgh, Pa, US U. Villa , Derivative-Informed MCMC for Bayesian Calibration of Stochastic PDE Models , SIAM Annual, July 10 - 14, 2017, Pittsburgh, Pa, US U. Villa , Hessian-based sampling techniques for Bayesian inverse problems with stochastic PDE forward model , Applied Inverse Problems, May 29 - June 2, 2017, Hangzhou, China U. Villa , Bayesian Calibration of Inadequate Stochastic PDE Models , SIAM CSE, Feb 27-March 3, 2017, Atlanta, GA, US P. Chen , Taylor Approximation for PDE-Constrained Optimal Control Problems Under High-Dimensional Uncertainty: Application to a Turbulence Model , SIAM CSE, Feb 27-March 3, 2017, Atlanta, GA, US B. Crestel , Scalable Solvers for Joint Inversion with Several Structural Coupling Terms , SIAM CSE, Feb 27-March 3, 2017, Atlanta, GA, US Amal Alghamdi , Bayesian Inversion for Subsurface Properties from Poroelastic Forward Models and Surface Deformation Data , SIAM CSE, Feb 27-March 3, 2017, Atlanta, GA, US U. Villa , An Analytical Technique for Forward and Inverse Propagation of Uncertainty , SIAM UQ, April 5-8, 2016, Lausanne, Switzerland","title":"Selected oral presentations"},{"location":"outreach/","text":"Outreach Schools and workshops O. Ghattas, Y. Marzouk, M. Parno, N. Petra, G. Stadler, U. Villa, Inverse Problems: Systematic Integration of Data with Models under Uncertainty , 2018 Gene Golub SIAM Summer School, June 17-30, 2018. Breckenridge, Colorado, USA. See also announcement flyer N. Petra, SAMSI Optimization Program Summer School , Research Triangle Park, NC, August 8-12, 2016 N. Petra and O. Ghattas, IDEALab: Inverse Problems and Uncertainty Quantification , Brown University, Providence, RD, July 6-10, 2015 Graduate level courses U. Villa @WUSTL, Computational Methods for Imaging Science , Spring 2020 U. Villa @WUSTL, Computational Methods for Imaging Science , Spring 2018 O. Ghattas @UT Austin, Computational and Variational Inverse Problems , Fall 2017 link A. Alexanderian @NC State, Inverse problems , Fall 2016 G. Stadler @NYU, Advanced Topics in Numerical Analysis: Computational and Variational Methods for Inverse Problems , Spring 2016 link N. Petra @UC Merced, Special Topics: Computational and Variational Inverse Problems , Fall 2015 link O. Ghattas @UT Austin, Computational and Variational Inverse Problems , Fall 2015 link","title":"Outreach"},{"location":"outreach/#outreach","text":"","title":"Outreach"},{"location":"outreach/#schools-and-workshops","text":"O. Ghattas, Y. Marzouk, M. Parno, N. Petra, G. Stadler, U. Villa, Inverse Problems: Systematic Integration of Data with Models under Uncertainty , 2018 Gene Golub SIAM Summer School, June 17-30, 2018. Breckenridge, Colorado, USA. See also announcement flyer N. Petra, SAMSI Optimization Program Summer School , Research Triangle Park, NC, August 8-12, 2016 N. Petra and O. Ghattas, IDEALab: Inverse Problems and Uncertainty Quantification , Brown University, Providence, RD, July 6-10, 2015","title":"Schools and workshops"},{"location":"outreach/#graduate-level-courses","text":"U. Villa @WUSTL, Computational Methods for Imaging Science , Spring 2020 U. Villa @WUSTL, Computational Methods for Imaging Science , Spring 2018 O. Ghattas @UT Austin, Computational and Variational Inverse Problems , Fall 2017 link A. Alexanderian @NC State, Inverse problems , Fall 2016 G. Stadler @NYU, Advanced Topics in Numerical Analysis: Computational and Variational Methods for Inverse Problems , Spring 2016 link N. Petra @UC Merced, Special Topics: Computational and Variational Inverse Problems , Fall 2015 link O. Ghattas @UT Austin, Computational and Variational Inverse Problems , Fall 2015 link","title":"Graduate level courses"},{"location":"postdoc_position/","text":"Openings Open Postdoc Position at UC Merced There is an opening for a postdoc position in Professor Noemi Petra 's research group in the School of Natural Sciences at the University of California, Merced . The postdoctoral researcher will work under an NSF-funded Collaborative Research (with UC Merced, UT Austin and MIT) entitled: Integrating Data with Complex Predictive Models under Uncertainty: An Extensible Software Framework for Large-Scale Bayesian Inversion (see here for a general overview of the project). The postdoctoral researcher will perform research in the field of large-scale Bayesian inverse problems, and on the implementation of new features in hIPPYlib. The postdoc will contribute to the research dissemination and will also help build the user/developer community by attending and speaking at conferences, workshops and summer schools at local and international events. Interested candidates should contact Noemi Petra at npetra@ucmerced.edu . Apply at https://aprecruit.ucmerced.edu/apply/JPF00738 .","title":"Openings"},{"location":"postdoc_position/#openings","text":"","title":"Openings"},{"location":"postdoc_position/#open-postdoc-position-at-uc-merced","text":"There is an opening for a postdoc position in Professor Noemi Petra 's research group in the School of Natural Sciences at the University of California, Merced . The postdoctoral researcher will work under an NSF-funded Collaborative Research (with UC Merced, UT Austin and MIT) entitled: Integrating Data with Complex Predictive Models under Uncertainty: An Extensible Software Framework for Large-Scale Bayesian Inversion (see here for a general overview of the project). The postdoctoral researcher will perform research in the field of large-scale Bayesian inverse problems, and on the implementation of new features in hIPPYlib. The postdoc will contribute to the research dissemination and will also help build the user/developer community by attending and speaking at conferences, workshops and summer schools at local and international events. Interested candidates should contact Noemi Petra at npetra@ucmerced.edu . Apply at https://aprecruit.ucmerced.edu/apply/JPF00738 .","title":"Open Postdoc Position at UC Merced"},{"location":"research/","text":"Research Applications Inversion for optical properties of biological tissues in quantitative optoacoustic tomography Statistical treatment of inverse problems constrained by stochastic models Accounting for model errors in inverse problems Bayesian optimal experimental design for inverse problems in acoustic scattering Inversion and control for CO 2 sequestration with poroelastic models Goal-oriented inference for reservoir models with complex features including faults Joint seismic-electromagnetic inversion Inference of basal boundary conditions for ice sheet flow Inversion for material properties of cardiac tissue Inference, prediction and optimization under uncertainty for turbulent combustion Selected publications T. O'Leary-Roseberry, U. Villa, P. Chen, and O. Ghattas: Derivative-Informed Projected Neural Networks for High-Dimensional Parametric Maps Governed by PDEs , Computer Methods in Applied Mechanics and Engineering, accepted, 2021 O. Babaniyi, R. Nicholson, U. Villa, and N. Petra: Inferring the basal sliding coefficient field for the Stokes ice sheet model under rheological uncertainty , The Cryosphere Discuss. [preprint], accepted, 2021. U. Villa, N. Petra, O. Ghattas, hIPPYlib: An extensible software framework for large-scale inverse problems; Part I: Deterministic inversion and linearized Bayesian inference , ACM Trans. Math. Softw. 47, 2, Article 16 (March 2021), 34 pages, 2021 A. Alghamdi, M.~A. Hesse, J. Chen, O. Ghattas, Bayesian Poroelastic Aquifer Characterization from InSAR Surface Deformation Data. Part I: Maximum A Posteriori Estimate , Water Resources Research, e2020WR027391, 2020 K. Koval, A. Alexanderian, G. Stadler, Optimal experimental design under irreducible uncertainty for inverse problems governed by PDEs , arXiv, 2019 S. Wahal, G. Biros, BIMC: The Bayesian Inverse Monte Carlo method for goal-oriented uncertainty quantification. Part I , arXiv, 2019 S. Lan, Adaptive dimension reduction to accelerate infinite-dimensional geometric Markov Chain Monte Carlo , Journal of Computational Physics, 392:71-95, 2019 P. Chen, U. Villa, O. Ghattas, Taylor approximation and variance reduction for PDE-constrained optimal control under uncertainty , Journal of Computational Physics, 385:163--186, 2019 B. Crestel, G. Stadler and O. Ghattas, A comparative study of structural similarity and regularization for joint inverse problems governed by PDEs , Inverse Problems, 35:024003, 2018 A. Attia, A. Alexanderian, A. K. Saibaba, Goal-oriented optimal design of experiments for large-scale Bayesian linear inverse problems , Inverse Problems, 34:095009, 2018 R. Nicholson, N. Petra and Jari P Kaipio. Estimation of the Robin coefficient field in a Poisson problem with uncertain conductivity field , Inverse Problems, Volume 34, Number 11, 2018. P. Chen, K. Wu, J. Chen, T. O'Leary-Roseberry, O. Ghattas, Projected Stein Variational Newton: A Fast and Scalable Bayesian Inference Method in High Dimensions , arXiv, 2018 E. M. Constantinescu, N. Petra, J. Bessac, C. G. Petra, Statistical Treatment of Inverse Problems Constrained by Differential Equations-Based Models with Stochastic Terms , arXiv, 2018 U. Villa, N. Petra, O. Ghattas, hIPPYlib: An extensible software framework for large-scale inverse problems , Journal of Open Source Software (JOSS), 3(30):940, 2018 P. Chen, U. Villa, O. Ghattas, Taylor approximation for PDE-constrained optimization under uncertainty: Application to turbulent jet flow , Proceedings in Applied Mathematics and Mechanics - 89th GAMM Annual Meeting, 18:e201800466, 2018 P. Chen, U. Villa, O. Ghattas, Hessian-based adaptive sparse quadrature for infinite-dimensional Bayesian inverse problems , Computer Methods in Applied Mechanics and Engineering, 327:147-172, 2017 S. Fatehiboroujeni, N. Petra and S. Goyal. Towards Adjoint-Based Inversion of the Lam\u00e9 Parameter Field for Slender Structures With Cantilever Loading , ASME 2016 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference, Volume 8: 28th Conference on Mechanical Vibration and Noise Charlotte, North Carolina, USA, August 21\u201324, 2016. Selected Ph.D. thesis T. O'Leary-Roseberry, Efficient and Dimension Independent Methods for Neural Network Surrogate Construction and Training , The University of Texas at Austin, 2020. Adviser O. Ghattas & P. Heimbach A. Alghamdi, Bayesian Inverse Problems for Quasi-Static Poroelasticity with Application to Ground Water Aquifer Characterization from Geodetic Data , The University of Texas at Austin, 2020. Adviser O. Ghattas & M. Hesse S. Fatehiboroujeni, Inverse Approaches for Identification of Constitutive Laws of Slender Structures Motivated by Application to Biological Filaments , University of California, Merced, 2018. Adviser S. Goyal K. A. McCormack, Earthquakes, groundwater and surface deformation: exploring the poroelastic response to megathrust earthquakes , The University of Texas at Austin, 2018. Adviser M. Hesse B. Crestel, Advanced techniques for multi-source, multi-parameter, and multi-physics inverse problems , The University of Texas at Austin, 2017. Adviser O. Ghattas Selected Honor and Master thesis B. Saleh, Scientific Machine Learning: A Neural Network-Based Estimator for Forward Uncertainty Quantification , The University of Texas at Austin, 2018. Adviser O. Ghattas G. Gao, hIPPYLearn: An inexact Newton-CG method for training neural networks with analysis of the Hessian , The University of Texas at Austin, 2017. Supervisor O. Ghattas D. Liu, hIPPYLearn: An inexact stochastic Newton-CG method for training neural networks , The University of Texas at Austin, 2017. Supervisor O. Ghattas Selected poster presentations O. Ghattas, K. Kim, Y. Marzouk, M. Parno, N. Petra, U. Villa, Integrating Data with Complex Predictive Models under Uncertainty: An Extensible Software Framework for Large-Scale Bayesian Inversion , NSF-CSSI PI meeting, Feb. 13-14, Seattle, Wa, US I. Ambartsumyan, T. Bui-Thanh, O. Ghattas, E. Khattatov, An Edge-preserving Method for Joint Bayesian Inversion with Non-Gaussian Priors , SIAM CSE, Feb 25- March 1, 2019, Spokane, Wa, US E. Khattatov, O. Ghattas, T. Bui-Thanh, and I. Ambartsumyan, U. Villa, Bayesian Inversion of Fault Properties in Two-phase Flow in Fractured Media , SIAM CSE, Feb 25- March 1, 2019, Spokane, Wa, US A. O. Babaniyi, O. Ghattas, N. Petra, U. Villa, hIPPYlib: An Extensible Software Framework for Large-scale Inverse Problems , SIAM CSE, Feb 25- March 1, 2019, Spokane, Wa, US O. Ghattas, Y. Marzouk, M. Parno, N. Petra, U. Villa, Integrating Data with Complex Predictive Models under Uncertainty: An Extensible Software Framework for Large-Scale Bayesian Inversion , NSF-SI2 PI meeting, Apr. 30- May 1, 2018, Washington, DC, US K. Koval, G. Stadler, Computational Approaches for Linear Goal-Oriented Bayesian Inverse Problems , SIAM Annual, July 10 - 14, 2017, Pittsburgh, Pa, US J. Chen, A. Drach, U. Villa, R. Avazmohammadi, D. Li, O. Ghattas and M. Sacks, Identification of Mechanical Properties of 3D Myocardial Tissue: An Inverse Modeling and Optimal Experimental Design Problem , FEniCS'17, June 12-14, 2017, University of Luxembourg, Luxembourg T. O\u2019Leary-Roseberry, U. Villa, O. Ghattas, P. Heimbach, An Adjoint Capable Solver for the Stefan Problem: a Bilevel Optimization and Level Set Approach , SIAM CSE, Feb. 27 - March 3, 2017, Atlanta, GA, US O. Ghattas, Y. Marzouk, M. Parno, N. Petra, U. Villa, Integrating Data with Complex Predictive Models under Uncertainty: An Extensible Software Framework for Large-Scale Bayesian Inversion , NSF-SI2 PI meeting, Feb. 21-22, 2017, Arlington, VA, US Did you publish an article or give an oral/poster presentation using hIPPYlib ? Please let us know by creating an issue and we will reference your work on this page.","title":"Research"},{"location":"research/#research","text":"","title":"Research"},{"location":"research/#applications","text":"Inversion for optical properties of biological tissues in quantitative optoacoustic tomography Statistical treatment of inverse problems constrained by stochastic models Accounting for model errors in inverse problems Bayesian optimal experimental design for inverse problems in acoustic scattering Inversion and control for CO 2 sequestration with poroelastic models Goal-oriented inference for reservoir models with complex features including faults Joint seismic-electromagnetic inversion Inference of basal boundary conditions for ice sheet flow Inversion for material properties of cardiac tissue Inference, prediction and optimization under uncertainty for turbulent combustion","title":"Applications"},{"location":"research/#selected-publications","text":"T. O'Leary-Roseberry, U. Villa, P. Chen, and O. Ghattas: Derivative-Informed Projected Neural Networks for High-Dimensional Parametric Maps Governed by PDEs , Computer Methods in Applied Mechanics and Engineering, accepted, 2021 O. Babaniyi, R. Nicholson, U. Villa, and N. Petra: Inferring the basal sliding coefficient field for the Stokes ice sheet model under rheological uncertainty , The Cryosphere Discuss. [preprint], accepted, 2021. U. Villa, N. Petra, O. Ghattas, hIPPYlib: An extensible software framework for large-scale inverse problems; Part I: Deterministic inversion and linearized Bayesian inference , ACM Trans. Math. Softw. 47, 2, Article 16 (March 2021), 34 pages, 2021 A. Alghamdi, M.~A. Hesse, J. Chen, O. Ghattas, Bayesian Poroelastic Aquifer Characterization from InSAR Surface Deformation Data. Part I: Maximum A Posteriori Estimate , Water Resources Research, e2020WR027391, 2020 K. Koval, A. Alexanderian, G. Stadler, Optimal experimental design under irreducible uncertainty for inverse problems governed by PDEs , arXiv, 2019 S. Wahal, G. Biros, BIMC: The Bayesian Inverse Monte Carlo method for goal-oriented uncertainty quantification. Part I , arXiv, 2019 S. Lan, Adaptive dimension reduction to accelerate infinite-dimensional geometric Markov Chain Monte Carlo , Journal of Computational Physics, 392:71-95, 2019 P. Chen, U. Villa, O. Ghattas, Taylor approximation and variance reduction for PDE-constrained optimal control under uncertainty , Journal of Computational Physics, 385:163--186, 2019 B. Crestel, G. Stadler and O. Ghattas, A comparative study of structural similarity and regularization for joint inverse problems governed by PDEs , Inverse Problems, 35:024003, 2018 A. Attia, A. Alexanderian, A. K. Saibaba, Goal-oriented optimal design of experiments for large-scale Bayesian linear inverse problems , Inverse Problems, 34:095009, 2018 R. Nicholson, N. Petra and Jari P Kaipio. Estimation of the Robin coefficient field in a Poisson problem with uncertain conductivity field , Inverse Problems, Volume 34, Number 11, 2018. P. Chen, K. Wu, J. Chen, T. O'Leary-Roseberry, O. Ghattas, Projected Stein Variational Newton: A Fast and Scalable Bayesian Inference Method in High Dimensions , arXiv, 2018 E. M. Constantinescu, N. Petra, J. Bessac, C. G. Petra, Statistical Treatment of Inverse Problems Constrained by Differential Equations-Based Models with Stochastic Terms , arXiv, 2018 U. Villa, N. Petra, O. Ghattas, hIPPYlib: An extensible software framework for large-scale inverse problems , Journal of Open Source Software (JOSS), 3(30):940, 2018 P. Chen, U. Villa, O. Ghattas, Taylor approximation for PDE-constrained optimization under uncertainty: Application to turbulent jet flow , Proceedings in Applied Mathematics and Mechanics - 89th GAMM Annual Meeting, 18:e201800466, 2018 P. Chen, U. Villa, O. Ghattas, Hessian-based adaptive sparse quadrature for infinite-dimensional Bayesian inverse problems , Computer Methods in Applied Mechanics and Engineering, 327:147-172, 2017 S. Fatehiboroujeni, N. Petra and S. Goyal. Towards Adjoint-Based Inversion of the Lam\u00e9 Parameter Field for Slender Structures With Cantilever Loading , ASME 2016 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference, Volume 8: 28th Conference on Mechanical Vibration and Noise Charlotte, North Carolina, USA, August 21\u201324, 2016.","title":"Selected publications"},{"location":"research/#selected-phd-thesis","text":"T. O'Leary-Roseberry, Efficient and Dimension Independent Methods for Neural Network Surrogate Construction and Training , The University of Texas at Austin, 2020. Adviser O. Ghattas & P. Heimbach A. Alghamdi, Bayesian Inverse Problems for Quasi-Static Poroelasticity with Application to Ground Water Aquifer Characterization from Geodetic Data , The University of Texas at Austin, 2020. Adviser O. Ghattas & M. Hesse S. Fatehiboroujeni, Inverse Approaches for Identification of Constitutive Laws of Slender Structures Motivated by Application to Biological Filaments , University of California, Merced, 2018. Adviser S. Goyal K. A. McCormack, Earthquakes, groundwater and surface deformation: exploring the poroelastic response to megathrust earthquakes , The University of Texas at Austin, 2018. Adviser M. Hesse B. Crestel, Advanced techniques for multi-source, multi-parameter, and multi-physics inverse problems , The University of Texas at Austin, 2017. Adviser O. Ghattas","title":"Selected Ph.D. thesis"},{"location":"research/#selected-honor-and-master-thesis","text":"B. Saleh, Scientific Machine Learning: A Neural Network-Based Estimator for Forward Uncertainty Quantification , The University of Texas at Austin, 2018. Adviser O. Ghattas G. Gao, hIPPYLearn: An inexact Newton-CG method for training neural networks with analysis of the Hessian , The University of Texas at Austin, 2017. Supervisor O. Ghattas D. Liu, hIPPYLearn: An inexact stochastic Newton-CG method for training neural networks , The University of Texas at Austin, 2017. Supervisor O. Ghattas","title":"Selected Honor and Master thesis"},{"location":"research/#selected-poster-presentations","text":"O. Ghattas, K. Kim, Y. Marzouk, M. Parno, N. Petra, U. Villa, Integrating Data with Complex Predictive Models under Uncertainty: An Extensible Software Framework for Large-Scale Bayesian Inversion , NSF-CSSI PI meeting, Feb. 13-14, Seattle, Wa, US I. Ambartsumyan, T. Bui-Thanh, O. Ghattas, E. Khattatov, An Edge-preserving Method for Joint Bayesian Inversion with Non-Gaussian Priors , SIAM CSE, Feb 25- March 1, 2019, Spokane, Wa, US E. Khattatov, O. Ghattas, T. Bui-Thanh, and I. Ambartsumyan, U. Villa, Bayesian Inversion of Fault Properties in Two-phase Flow in Fractured Media , SIAM CSE, Feb 25- March 1, 2019, Spokane, Wa, US A. O. Babaniyi, O. Ghattas, N. Petra, U. Villa, hIPPYlib: An Extensible Software Framework for Large-scale Inverse Problems , SIAM CSE, Feb 25- March 1, 2019, Spokane, Wa, US O. Ghattas, Y. Marzouk, M. Parno, N. Petra, U. Villa, Integrating Data with Complex Predictive Models under Uncertainty: An Extensible Software Framework for Large-Scale Bayesian Inversion , NSF-SI2 PI meeting, Apr. 30- May 1, 2018, Washington, DC, US K. Koval, G. Stadler, Computational Approaches for Linear Goal-Oriented Bayesian Inverse Problems , SIAM Annual, July 10 - 14, 2017, Pittsburgh, Pa, US J. Chen, A. Drach, U. Villa, R. Avazmohammadi, D. Li, O. Ghattas and M. Sacks, Identification of Mechanical Properties of 3D Myocardial Tissue: An Inverse Modeling and Optimal Experimental Design Problem , FEniCS'17, June 12-14, 2017, University of Luxembourg, Luxembourg T. O\u2019Leary-Roseberry, U. Villa, O. Ghattas, P. Heimbach, An Adjoint Capable Solver for the Stefan Problem: a Bilevel Optimization and Level Set Approach , SIAM CSE, Feb. 27 - March 3, 2017, Atlanta, GA, US O. Ghattas, Y. Marzouk, M. Parno, N. Petra, U. Villa, Integrating Data with Complex Predictive Models under Uncertainty: An Extensible Software Framework for Large-Scale Bayesian Inversion , NSF-SI2 PI meeting, Feb. 21-22, 2017, Arlington, VA, US Did you publish an article or give an oral/poster presentation using hIPPYlib ? Please let us know by creating an issue and we will reference your work on this page.","title":"Selected poster presentations"},{"location":"tutorial_v1.6.0/","text":"Tutorial These tutorials are the best place to learn about the basic features and the algorithms in hIPPYlib . FEniCS101 notebook illustrates the use of FEniCS for the solution of a linear boundary value problem. Poisson Deterministic notebook illustrates how to compute gradient/Hessian information and solve a non-linear parameter inversion for the Poisson equation in a deterministic setting. Subsurface Bayesian notebook illustrates how to solve a non-linear parameter inversion for the Poisson equation in a Bayesian setting. Advection-Diffusion Bayesian notebook illustrates how to solve a time-dependent linear inverse problem in a Bayesian setting. Hessian Spectrum notebook illustrates the spectral property of the Hessian operator for a linear source inversion problem. The interactive ipython notebooks are located in the tutorial folder of the hIPPYlib release. To run the notebooks follow these instructions. Open a FEniCS terminal and type $ cd tutorial $ jupyter notebook A new tab will open in your web-brower showing the notebooks. Click on the notebook you would like to use. To run all the code in the notebook simply click on Cell --> Run All. For more information on installing ipython and using notebooks see here .","title":"README"},{"location":"tutorial_v1.6.0/#tutorial","text":"These tutorials are the best place to learn about the basic features and the algorithms in hIPPYlib . FEniCS101 notebook illustrates the use of FEniCS for the solution of a linear boundary value problem. Poisson Deterministic notebook illustrates how to compute gradient/Hessian information and solve a non-linear parameter inversion for the Poisson equation in a deterministic setting. Subsurface Bayesian notebook illustrates how to solve a non-linear parameter inversion for the Poisson equation in a Bayesian setting. Advection-Diffusion Bayesian notebook illustrates how to solve a time-dependent linear inverse problem in a Bayesian setting. Hessian Spectrum notebook illustrates the spectral property of the Hessian operator for a linear source inversion problem. The interactive ipython notebooks are located in the tutorial folder of the hIPPYlib release. To run the notebooks follow these instructions. Open a FEniCS terminal and type $ cd tutorial $ jupyter notebook A new tab will open in your web-brower showing the notebooks. Click on the notebook you would like to use. To run all the code in the notebook simply click on Cell --> Run All. For more information on installing ipython and using notebooks see here .","title":"Tutorial"},{"location":"tutorial_v2.3.0/","text":"Tutorial hIPPYlib 2.3.0 These tutorials are the best place to learn about the basic features and the algorithms in hIPPYlib . For the complete API reference click here . FEniCS101 notebook illustrates the use of FEniCS for the solution of a linear boundary value problem. Poisson Deterministic notebook illustrates how to compute gradient/Hessian information and solve a non-linear parameter inversion for the Poisson equation in a deterministic setting. Subsurface Bayesian notebook illustrates how to solve a non-linear parameter inversion for the Poisson equation in a Bayesian setting Advection-Diffusion Bayesian notebook illustrates how to solve a time-dependent linear inverse problem in a Bayesian setting. Hessian Spectrum notebook illustrates the spectral property of the Hessian operator for a linear source inversion problem. Tutorials 3 and 4 contain a line-by-line explanation of the model problems presented in hIPPYlib: An Extensible Software Framework for Large-Scale Inverse Problems Governed by PDEs; Part I: Deterministic Inversion and Linearized Bayesian Inference . Interactive tutorials hIPPYlib 2.3.0 is available as source code or prebuilt docker image . The interactive ipython notebooks are located in the tutorial folder of the hIPPYlib release. To run the notebooks follow these instructions. Open a FEniCS terminal and type $ cd tutorial $ jupyter notebook A new tab will open in your web-brower showing the notebooks. Click on the notebook you would like to use. To run all the code in the notebook simply click on Cell --> Run All. For more information on installing ipython and using notebooks see here . Additional resources For additional resources and tutorials please see the teaching material for the 2018 Gene Golub SIAM Summer School on Inverse Problems: Systematic Integration of Data with Models under Uncertainty available here .","title":"README"},{"location":"tutorial_v2.3.0/#tutorial-hippylib-230","text":"These tutorials are the best place to learn about the basic features and the algorithms in hIPPYlib . For the complete API reference click here . FEniCS101 notebook illustrates the use of FEniCS for the solution of a linear boundary value problem. Poisson Deterministic notebook illustrates how to compute gradient/Hessian information and solve a non-linear parameter inversion for the Poisson equation in a deterministic setting. Subsurface Bayesian notebook illustrates how to solve a non-linear parameter inversion for the Poisson equation in a Bayesian setting Advection-Diffusion Bayesian notebook illustrates how to solve a time-dependent linear inverse problem in a Bayesian setting. Hessian Spectrum notebook illustrates the spectral property of the Hessian operator for a linear source inversion problem. Tutorials 3 and 4 contain a line-by-line explanation of the model problems presented in hIPPYlib: An Extensible Software Framework for Large-Scale Inverse Problems Governed by PDEs; Part I: Deterministic Inversion and Linearized Bayesian Inference .","title":"Tutorial hIPPYlib 2.3.0"},{"location":"tutorial_v2.3.0/#interactive-tutorials","text":"hIPPYlib 2.3.0 is available as source code or prebuilt docker image . The interactive ipython notebooks are located in the tutorial folder of the hIPPYlib release. To run the notebooks follow these instructions. Open a FEniCS terminal and type $ cd tutorial $ jupyter notebook A new tab will open in your web-brower showing the notebooks. Click on the notebook you would like to use. To run all the code in the notebook simply click on Cell --> Run All. For more information on installing ipython and using notebooks see here .","title":"Interactive tutorials"},{"location":"tutorial_v2.3.0/#additional-resources","text":"For additional resources and tutorials please see the teaching material for the 2018 Gene Golub SIAM Summer School on Inverse Problems: Systematic Integration of Data with Models under Uncertainty available here .","title":"Additional resources"},{"location":"tutorial_v3.0.0/","text":"Tutorial - Version 3.0.0 These tutorials are the best place to learn about the basic features and the algorithms in hIPPYlib . For the complete API reference click here . FEniCS101 notebook illustrates the use of FEniCS for the solution of a linear boundary value problem. Poisson Deterministic notebook illustrates how to compute gradient/Hessian information and solve a non-linear parameter inversion for the Poisson equation in a deterministic setting. Subsurface Bayesian notebook illustrates how to solve a non-linear parameter inversion for the Poisson equation in a Bayesian setting. Advection-Diffusion Bayesian notebook illustrates how to solve a time-dependent linear inverse problem in a Bayesian setting. Hessian Spectrum notebook illustrates the spectral property of the Hessian operator for a linear source inversion problem. Interactive tutorials The interactive ipython notebooks are located in the tutorial folder of the hIPPYlib repository. To run the notebooks follow these instructions. Open a FEniCS terminal and type $ cd tutorial $ jupyter notebook A new tab will open in your web-brower showing the notebooks. Click on the notebook you would like to use. To run all the code in the notebook simply click on Cell --> Run All. For more information on installing ipython and using notebooks see here . Additional resources For additional resources and tutorials please see the teaching material for the 2018 Gene Golub SIAM Summer School on Inverse Problems: Systematic Integration of Data with Models under Uncertainty available here . These tutorials require hIPPYlib versions 2.1.1 - 2.3.0.","title":"README"},{"location":"tutorial_v3.0.0/#tutorial-version-300","text":"These tutorials are the best place to learn about the basic features and the algorithms in hIPPYlib . For the complete API reference click here . FEniCS101 notebook illustrates the use of FEniCS for the solution of a linear boundary value problem. Poisson Deterministic notebook illustrates how to compute gradient/Hessian information and solve a non-linear parameter inversion for the Poisson equation in a deterministic setting. Subsurface Bayesian notebook illustrates how to solve a non-linear parameter inversion for the Poisson equation in a Bayesian setting. Advection-Diffusion Bayesian notebook illustrates how to solve a time-dependent linear inverse problem in a Bayesian setting. Hessian Spectrum notebook illustrates the spectral property of the Hessian operator for a linear source inversion problem.","title":"Tutorial - Version 3.0.0"},{"location":"tutorial_v3.0.0/#interactive-tutorials","text":"The interactive ipython notebooks are located in the tutorial folder of the hIPPYlib repository. To run the notebooks follow these instructions. Open a FEniCS terminal and type $ cd tutorial $ jupyter notebook A new tab will open in your web-brower showing the notebooks. Click on the notebook you would like to use. To run all the code in the notebook simply click on Cell --> Run All. For more information on installing ipython and using notebooks see here .","title":"Interactive tutorials"},{"location":"tutorial_v3.0.0/#additional-resources","text":"For additional resources and tutorials please see the teaching material for the 2018 Gene Golub SIAM Summer School on Inverse Problems: Systematic Integration of Data with Models under Uncertainty available here . These tutorials require hIPPYlib versions 2.1.1 - 2.3.0.","title":"Additional resources"},{"location":"tutorials_v1.6.0/1_FEniCS101/","text":"FEniCS101 Tutorial In this tutorial we consider the boundary value problem (BVP) \\begin{eqnarray*} - \\nabla \\cdot (k \\nabla u) = f & \\text{ in } \\Omega,\\\\ u = u_0 & \\text{ on } \\Gamma_D = \\Gamma_{\\rm left} \\bigcup \\Gamma_{\\rm right},\\\\ k \\frac{\\partial u}{\\partial {\\bf{n}}} = \\sigma & \\text{ on } \\Gamma_N = \\Gamma_{\\rm top} \\bigcup \\Gamma_{\\rm bottom}, \\end{eqnarray*} where \\Omega = (0,1) \\times (0,1) , \\Gamma_D and and \\Gamma_N are the union of the left and right, and top and bottom boundaries of \\Omega , respectively. Here \\begin{eqnarray*} k(x,y) = 1 & \\text{ on } \\Omega\\\\ f(x,y) = \\left(4\\pi^2+\\frac{\\pi^2}{4}\\right)\\sin(2 \\pi x) \\sin\\left(\\frac{\\pi}{2} y\\right) & \\text{ on } \\Omega\\\\ u_0(x,y) = 0 & \\text{ on } \\Gamma_D, \\\\ \\sigma(x) = \\left\\{ \\begin{array}{l} -\\frac{\\pi}{2}\\sin(2 \\pi x) \\\\ 0 \\end{array}\\right. & \\begin{array}{l} \\text{ on } \\Gamma_{\\rm bottom},\\\\ \\text{ on } \\Gamma_{\\rm top}.\\end{array} \\end{eqnarray*} The exact solution is u_e(x,y) = \\sin(2\\pi x)\\sin\\left(\\frac{\\pi}{2}y\\right). Weak formulation Let us define the Hilbert spaces V_{u_0}, V_0 \\in \\Omega as V_{u_0} := \\left\\{ v \\in H^1(\\Omega) \\text{ s. t. } v = u_0 \\text{ on } \\Gamma_D \\right\\}, V_{0} := \\left\\{ v \\in H^1(\\Omega) \\text{ s. t. } v = 0 \\text{ on } \\Gamma_D \\right\\}. To obtain the weak formulation, we multiply the PDE by an arbitrary function v \\in V_0 and integrate over the domain \\Omega leading to -\\int_{\\Omega} \\nabla \\cdot (k \\nabla u) v \\, dx = \\int_\\Omega f v \\, dx\\quad \\forall \\; v \\in V_0. Then, integration by parts the non-conforming term gives \\int_{\\Omega} k \\nabla u \\cdot \\nabla v \\, dx - \\int_{\\partial \\Omega} k \\frac{\\partial u}{\\partial {\\bf n} } v \\, ds = \\int_\\Omega f v \\, dx \\quad \\forall \\; v \\in V_0. Finally by recalling that v = 0 on \\Gamma_D and that k \\frac{\\partial u}{\\partial {\\bf n} } = \\sigma on \\Gamma_N , we find the weak formulation: Find * u \\in V_{u_0} such that* \\int_{\\Omega} k \\nabla u \\cdot \\nabla v \\, dx = \\int_\\Omega f v \\, dx + \\int_{\\Gamma_N} \\sigma v \\, ds \\quad \\forall \\; v \\in V_0. 1. Load modules To start we load the following modules: dolfin: the python/C++ interface to FEniCS math : the python module for mathematical functions numpy : a python package for linear algebra matplotlib : a python package used for plotting the results from __future__ import absolute_import, division, print_function from dolfin import * import math import numpy as np import logging import matplotlib.pyplot as plt %matplotlib inline import nb logging.getLogger('FFC').setLevel(logging.WARNING) logging.getLogger('UFL').setLevel(logging.WARNING) set_log_active(False) 2. Define the mesh and the finite element space We construct a triangulation (mesh) \\mathcal{T}_h of the computational domain \\Omega := [0, 1]^2 with n elements in each direction. On the mesh \\mathcal{T}_h , we then define the finite element space V_h \\subset H^1(\\Omega) consisting of globally continuous piecewise polinomials functions. The degree variable defines the polinomial degree. n = 16 degree = 1 mesh = UnitSquareMesh(n, n) nb.plot(mesh) Vh = FunctionSpace(mesh, 'Lagrange', degree) print(\"dim(Vh) = \", Vh.dim()) dim(Vh) = 289 3. Define boundary labels To partition the boundary of \\Omega in the subdomains \\Gamma_{\\rm top} , \\Gamma_{\\rm bottom} , \\Gamma_{\\rm left} , \\Gamma_{\\rm right} we assign a unique label boundary_parts to each of part of \\partial \\Omega . class TopBoundary(SubDomain): def inside(self, x, on_boundary): return on_boundary and abs(x[1] - 1) < DOLFIN_EPS class BottomBoundary(SubDomain): def inside(self, x, on_boundary): return on_boundary and abs(x[1]) < DOLFIN_EPS class LeftBoundary(SubDomain): def inside(self, x, on_boundary): return on_boundary and abs(x[0]) < DOLFIN_EPS class RightBoundary(SubDomain): def inside(self, x, on_boundary): return on_boundary and abs(x[0] - 1) < DOLFIN_EPS boundary_parts = FacetFunction(\"size_t\", mesh) boundary_parts.set_all(0) Gamma_top = TopBoundary() Gamma_top.mark(boundary_parts, 1) Gamma_bottom = BottomBoundary() Gamma_bottom.mark(boundary_parts, 2) Gamma_left = LeftBoundary() Gamma_left.mark(boundary_parts, 3) Gamma_right = RightBoundary() Gamma_right.mark(boundary_parts, 4) 4. Define the coefficients of the PDE and the boundary conditions We first define the coefficients of the PDE using the Constant and Expression classes. Constant is used to define coefficients that do not depend on the space coordinates, Expression is used to define coefficients that are a known function of the space coordinates x[0] (x-axis direction) and x[1] (y-axis direction). In the finite element method community, Dirichlet boundary conditions are also known as essential boundary conditions since they are imposed directly in the definition of the finite element space. In FEniCS, we use the class DirichletBC to indicate this type of condition. On the other hand, Newman boundary conditions are also known as natural boundary conditions since they are weakly imposed as boundary integrals in the variational formulation (weak form). In FEniCS, we create a new boundary measure ds[i] to integrate over the portion of the boundary marked with label i . u_L = Constant(0.) u_R = Constant(0.) sigma_bottom = Expression('-(pi/2.0)*sin(2*pi*x[0])', degree=5) sigma_top = Constant(0.) f = Expression('(4.0*pi*pi+pi*pi/4.0)*(sin(2*pi*x[0])*sin((pi/2.0)*x[1]))', degree=5) bcs = [DirichletBC(Vh, u_L, boundary_parts, 3), DirichletBC(Vh, u_R, boundary_parts, 4)] ds = Measure(\"ds\", subdomain_data=boundary_parts) 5. Define and solve the variational problem We also define two special types of functions: the TrialFunction u and the TestFunction v . These special types of function are used by FEniCS to generate the finite element vectors and matrices which stem from the weak formulation of the PDE. More specifically, by denoting by \\left[{\\phi_i(x)}\\right]_{i=1}^{{\\rm dim}(V_h)} the finite element basis for the space V_h , a function u_h \\in V_h can be written as u_h = \\sum_{i=1}^{{\\rm dim}(V_h)} {\\rm u}_i \\phi_i(x), where {\\rm u}_i represents the coefficients in the finite element expansion of u_h . We then define the bilinear form a(u_h, v_h) = \\int_\\Omega \\nabla u_h \\cdot \\nabla v_h dx ; the linear form L(v_h) = \\int_\\Omega f v_h dx + + \\int_{\\Gamma_{\\rm top}} \\sigma_{\\rm top} v_h ds \\int_{\\Gamma_{\\rm bottom}} \\sigma_{\\rm bottom} v_h ds . We can then solve the variational problem Find u_h \\in V_h such that a(u_h, v_h) = L(v_h) \\quad \\forall\\, v_h \\in V_h using directly the built-in solve method in FEniCS. NOTE: As an alternative one can also assemble the finite element matrix A and the right hand side b that stems from the discretization of a and L , and then solve the linear system A {\\rm u} = {\\rm b}, where {\\rm u} is the vector collecting the coefficient of the finite element expasion of u_h , the entries of the matrix A are such that A_{ij} = a(\\phi_j, \\phi_i) , the entries of the right hand side b are such that b_i = L(\\phi_i) . u = TrialFunction(Vh) v = TestFunction(Vh) a = inner(nabla_grad(u), nabla_grad(v))*dx L = f*v*dx + sigma_top*v*ds(1) + sigma_bottom*v*ds(2) uh = Function(Vh) #solve(a == L, uh, bcs=bcs) A, b = assemble_system(a,L, bcs=bcs) solve(A, uh.vector(), b, \"cg\") nb.plot(uh) 6. Compute the discretization error For this problem, the exact solution is known. We can therefore compute the following norms of the discretization error (i.e. the of the difference between the finite element solution u_h and the exact solution u_{\\rm ex} ) \\| u_{\\rm ex} - u_h \\|_{L^2{\\Omega}} := \\sqrt{ \\int_{\\Omega} (u_{\\rm ex} - u_h)^2 \\, dx }, and \\| u_{\\rm ex} - u_h \\|_{H^1{\\Omega}} := \\sqrt{ \\int_{\\Omega} (u_{\\rm ex} - u_h)^2 \\, dx + \\int_{\\Omega} |\\nabla u_{\\rm ex} - \\nabla u_h|^2 \\, dx}. u_e = Expression('sin(2*pi*x[0])*sin((pi/2.0)*x[1])', degree=5) grad_u_e = Expression( ('2*pi*cos(2*pi*x[0])*sin((pi/2.0)*x[1])', 'pi/2.0*sin(2*pi*x[0])*cos((pi/2.0)*x[1])'), degree=5) err_L2 = sqrt( assemble( (uh-u_e)**2*dx ) ) err_grad = sqrt( assemble( inner(nabla_grad(uh) - grad_u_e, nabla_grad(uh) - grad_u_e)*dx ) ) err_H1 = sqrt( err_L2**2 + err_grad**2) print(\"|| u_h - u_e ||_L2 = \", err_L2) print(\"|| u_h - u_e ||_H1 = \", err_H1) || u_h - u_e ||_L2 = 0.00880525372208 || u_h - u_e ||_H1 = 0.396718952514 7. Convergence of the finite element method We now verify numerically a well-known convergence result for the finite element method. Let denote with s the polynomial degree of the finite element space, and assume that the solution u_{\\rm ex} is at least in H^{s+1}(\\Omega) . Then we have \\| u_{\\rm ex} - u_h \\|_{H^1} \\leq C h^{s}, \\quad \\| u_{\\rm ex} - u_h \\|_{L^2} \\leq C h^{s+1}. In the code below, the function compute(n, degree) solves the PDE using a mesh with n elements in each direction and finite element spaces of polinomial order degree . The figure below shows the discretization errors in the H^1 and L^2 as a function of the mesh size h ( h = \\frac{1}{n} ) for piecewise linear (P1, s=1 ) and piecewise quadratic (P2, s=2 ) finite elements. We observe that numerical results are consistent with the finite element convergence theory. In particular: for piecewise linear finite element P1 we observe first order convergence in the H^1 -norm and second order convergence in the L^2 -norm; for piecewise quadratic finite element P2 we observe second order convergence in the H^1 -norm and third order convergence in the L^2 -norm. def compute(n, degree): mesh = UnitSquareMesh(n, n) Vh = FunctionSpace(mesh, 'Lagrange', degree) boundary_parts = FacetFunction(\"size_t\", mesh) boundary_parts.set_all(0) Gamma_top = TopBoundary() Gamma_top.mark(boundary_parts, 1) Gamma_bottom = BottomBoundary() Gamma_bottom.mark(boundary_parts, 2) Gamma_left = LeftBoundary() Gamma_left.mark(boundary_parts, 3) Gamma_right = RightBoundary() Gamma_right.mark(boundary_parts, 4) bcs = [DirichletBC(Vh, u_L, boundary_parts, 3), DirichletBC(Vh, u_R, boundary_parts, 4)] ds = Measure(\"ds\", subdomain_data=boundary_parts) u = TrialFunction(Vh) v = TestFunction(Vh) a = inner(nabla_grad(u), nabla_grad(v))*dx L = f*v*dx + sigma_top*v*ds(1) + sigma_bottom*v*ds(2) uh = Function(Vh) solve(a == L, uh, bcs=bcs) err_L2 = sqrt( assemble( (uh-u_e)**2*dx ) ) err_grad = sqrt( assemble( inner(nabla_grad(uh) - grad_u_e, nabla_grad(uh) - grad_u_e)*dx ) ) err_H1 = sqrt( err_L2**2 + err_grad**2) return err_L2, err_H1 nref = 5 n = 8*np.power(2,np.arange(0,nref)) h = 1./n err_L2_P1 = np.zeros(nref) err_H1_P1 = np.zeros(nref) err_L2_P2 = np.zeros(nref) err_H1_P2 = np.zeros(nref) for i in range(nref): err_L2_P1[i], err_H1_P1[i] = compute(n[i], 1) err_L2_P2[i], err_H1_P2[i] = compute(n[i], 2) plt.figure(figsize=(15,5)) plt.subplot(121) plt.loglog(h, err_H1_P1, '-or', label=\"H1 error\") plt.loglog(h, err_L2_P1, '-*b', label=\"L2 error\") plt.loglog(h, h*.5*err_H1_P1[0]/h[0], '--g', label=\"First Order\") plt.loglog(h, np.power(h,2)*.5*np.power( err_L2_P1[0]/h[0], 2), '-.k', label=\"Second Order\") plt.xlabel(\"Mesh size h\") plt.ylabel(\"Error\") plt.title(\"P1 Finite Element\") plt.legend(loc = 'lower right') plt.subplot(122) plt.loglog(h, err_H1_P2, '-or', label=\"H1 error\") plt.loglog(h, err_L2_P2, '-*b', label=\"L2 error\") plt.loglog(h, np.power(h/h[0],2)*.5*err_H1_P2[0], '--g', label=\"Second Order\") plt.loglog(h, np.power(h/h[0],3)*.5*err_L2_P2[0], '-.k', label=\"Third Order\") plt.xlabel(\"Mesh size h\") plt.ylabel(\"Error\") plt.title(\"P2 Finite Element\") plt.legend(loc='lower right') plt.show() Copyright (c) 2016-2018, The University of Texas at Austin & University of California, Merced. All Rights reserved. See file COPYRIGHT for details. This file is part of the hIPPYlib library. For more information and source code availability see https://hippylib.github.io. hIPPYlib is free software; you can redistribute it and/or modify it under the terms of the GNU General Public License (as published by the Free Software Foundation) version 2.0 dated June 1991.","title":"FEniCS101"},{"location":"tutorials_v1.6.0/1_FEniCS101/#fenics101-tutorial","text":"In this tutorial we consider the boundary value problem (BVP) \\begin{eqnarray*} - \\nabla \\cdot (k \\nabla u) = f & \\text{ in } \\Omega,\\\\ u = u_0 & \\text{ on } \\Gamma_D = \\Gamma_{\\rm left} \\bigcup \\Gamma_{\\rm right},\\\\ k \\frac{\\partial u}{\\partial {\\bf{n}}} = \\sigma & \\text{ on } \\Gamma_N = \\Gamma_{\\rm top} \\bigcup \\Gamma_{\\rm bottom}, \\end{eqnarray*} where \\Omega = (0,1) \\times (0,1) , \\Gamma_D and and \\Gamma_N are the union of the left and right, and top and bottom boundaries of \\Omega , respectively. Here \\begin{eqnarray*} k(x,y) = 1 & \\text{ on } \\Omega\\\\ f(x,y) = \\left(4\\pi^2+\\frac{\\pi^2}{4}\\right)\\sin(2 \\pi x) \\sin\\left(\\frac{\\pi}{2} y\\right) & \\text{ on } \\Omega\\\\ u_0(x,y) = 0 & \\text{ on } \\Gamma_D, \\\\ \\sigma(x) = \\left\\{ \\begin{array}{l} -\\frac{\\pi}{2}\\sin(2 \\pi x) \\\\ 0 \\end{array}\\right. & \\begin{array}{l} \\text{ on } \\Gamma_{\\rm bottom},\\\\ \\text{ on } \\Gamma_{\\rm top}.\\end{array} \\end{eqnarray*} The exact solution is u_e(x,y) = \\sin(2\\pi x)\\sin\\left(\\frac{\\pi}{2}y\\right).","title":"FEniCS101 Tutorial"},{"location":"tutorials_v1.6.0/1_FEniCS101/#weak-formulation","text":"Let us define the Hilbert spaces V_{u_0}, V_0 \\in \\Omega as V_{u_0} := \\left\\{ v \\in H^1(\\Omega) \\text{ s. t. } v = u_0 \\text{ on } \\Gamma_D \\right\\}, V_{0} := \\left\\{ v \\in H^1(\\Omega) \\text{ s. t. } v = 0 \\text{ on } \\Gamma_D \\right\\}. To obtain the weak formulation, we multiply the PDE by an arbitrary function v \\in V_0 and integrate over the domain \\Omega leading to -\\int_{\\Omega} \\nabla \\cdot (k \\nabla u) v \\, dx = \\int_\\Omega f v \\, dx\\quad \\forall \\; v \\in V_0. Then, integration by parts the non-conforming term gives \\int_{\\Omega} k \\nabla u \\cdot \\nabla v \\, dx - \\int_{\\partial \\Omega} k \\frac{\\partial u}{\\partial {\\bf n} } v \\, ds = \\int_\\Omega f v \\, dx \\quad \\forall \\; v \\in V_0. Finally by recalling that v = 0 on \\Gamma_D and that k \\frac{\\partial u}{\\partial {\\bf n} } = \\sigma on \\Gamma_N , we find the weak formulation: Find * u \\in V_{u_0} such that* \\int_{\\Omega} k \\nabla u \\cdot \\nabla v \\, dx = \\int_\\Omega f v \\, dx + \\int_{\\Gamma_N} \\sigma v \\, ds \\quad \\forall \\; v \\in V_0.","title":"Weak formulation"},{"location":"tutorials_v1.6.0/1_FEniCS101/#1-load-modules","text":"To start we load the following modules: dolfin: the python/C++ interface to FEniCS math : the python module for mathematical functions numpy : a python package for linear algebra matplotlib : a python package used for plotting the results from __future__ import absolute_import, division, print_function from dolfin import * import math import numpy as np import logging import matplotlib.pyplot as plt %matplotlib inline import nb logging.getLogger('FFC').setLevel(logging.WARNING) logging.getLogger('UFL').setLevel(logging.WARNING) set_log_active(False)","title":"1. Load modules"},{"location":"tutorials_v1.6.0/1_FEniCS101/#2-define-the-mesh-and-the-finite-element-space","text":"We construct a triangulation (mesh) \\mathcal{T}_h of the computational domain \\Omega := [0, 1]^2 with n elements in each direction. On the mesh \\mathcal{T}_h , we then define the finite element space V_h \\subset H^1(\\Omega) consisting of globally continuous piecewise polinomials functions. The degree variable defines the polinomial degree. n = 16 degree = 1 mesh = UnitSquareMesh(n, n) nb.plot(mesh) Vh = FunctionSpace(mesh, 'Lagrange', degree) print(\"dim(Vh) = \", Vh.dim()) dim(Vh) = 289","title":"2. Define the mesh and the finite element space"},{"location":"tutorials_v1.6.0/1_FEniCS101/#3-define-boundary-labels","text":"To partition the boundary of \\Omega in the subdomains \\Gamma_{\\rm top} , \\Gamma_{\\rm bottom} , \\Gamma_{\\rm left} , \\Gamma_{\\rm right} we assign a unique label boundary_parts to each of part of \\partial \\Omega . class TopBoundary(SubDomain): def inside(self, x, on_boundary): return on_boundary and abs(x[1] - 1) < DOLFIN_EPS class BottomBoundary(SubDomain): def inside(self, x, on_boundary): return on_boundary and abs(x[1]) < DOLFIN_EPS class LeftBoundary(SubDomain): def inside(self, x, on_boundary): return on_boundary and abs(x[0]) < DOLFIN_EPS class RightBoundary(SubDomain): def inside(self, x, on_boundary): return on_boundary and abs(x[0] - 1) < DOLFIN_EPS boundary_parts = FacetFunction(\"size_t\", mesh) boundary_parts.set_all(0) Gamma_top = TopBoundary() Gamma_top.mark(boundary_parts, 1) Gamma_bottom = BottomBoundary() Gamma_bottom.mark(boundary_parts, 2) Gamma_left = LeftBoundary() Gamma_left.mark(boundary_parts, 3) Gamma_right = RightBoundary() Gamma_right.mark(boundary_parts, 4)","title":"3. Define boundary labels"},{"location":"tutorials_v1.6.0/1_FEniCS101/#4-define-the-coefficients-of-the-pde-and-the-boundary-conditions","text":"We first define the coefficients of the PDE using the Constant and Expression classes. Constant is used to define coefficients that do not depend on the space coordinates, Expression is used to define coefficients that are a known function of the space coordinates x[0] (x-axis direction) and x[1] (y-axis direction). In the finite element method community, Dirichlet boundary conditions are also known as essential boundary conditions since they are imposed directly in the definition of the finite element space. In FEniCS, we use the class DirichletBC to indicate this type of condition. On the other hand, Newman boundary conditions are also known as natural boundary conditions since they are weakly imposed as boundary integrals in the variational formulation (weak form). In FEniCS, we create a new boundary measure ds[i] to integrate over the portion of the boundary marked with label i . u_L = Constant(0.) u_R = Constant(0.) sigma_bottom = Expression('-(pi/2.0)*sin(2*pi*x[0])', degree=5) sigma_top = Constant(0.) f = Expression('(4.0*pi*pi+pi*pi/4.0)*(sin(2*pi*x[0])*sin((pi/2.0)*x[1]))', degree=5) bcs = [DirichletBC(Vh, u_L, boundary_parts, 3), DirichletBC(Vh, u_R, boundary_parts, 4)] ds = Measure(\"ds\", subdomain_data=boundary_parts)","title":"4. Define the coefficients of the PDE and the boundary conditions"},{"location":"tutorials_v1.6.0/1_FEniCS101/#5-define-and-solve-the-variational-problem","text":"We also define two special types of functions: the TrialFunction u and the TestFunction v . These special types of function are used by FEniCS to generate the finite element vectors and matrices which stem from the weak formulation of the PDE. More specifically, by denoting by \\left[{\\phi_i(x)}\\right]_{i=1}^{{\\rm dim}(V_h)} the finite element basis for the space V_h , a function u_h \\in V_h can be written as u_h = \\sum_{i=1}^{{\\rm dim}(V_h)} {\\rm u}_i \\phi_i(x), where {\\rm u}_i represents the coefficients in the finite element expansion of u_h . We then define the bilinear form a(u_h, v_h) = \\int_\\Omega \\nabla u_h \\cdot \\nabla v_h dx ; the linear form L(v_h) = \\int_\\Omega f v_h dx + + \\int_{\\Gamma_{\\rm top}} \\sigma_{\\rm top} v_h ds \\int_{\\Gamma_{\\rm bottom}} \\sigma_{\\rm bottom} v_h ds . We can then solve the variational problem Find u_h \\in V_h such that a(u_h, v_h) = L(v_h) \\quad \\forall\\, v_h \\in V_h using directly the built-in solve method in FEniCS. NOTE: As an alternative one can also assemble the finite element matrix A and the right hand side b that stems from the discretization of a and L , and then solve the linear system A {\\rm u} = {\\rm b}, where {\\rm u} is the vector collecting the coefficient of the finite element expasion of u_h , the entries of the matrix A are such that A_{ij} = a(\\phi_j, \\phi_i) , the entries of the right hand side b are such that b_i = L(\\phi_i) . u = TrialFunction(Vh) v = TestFunction(Vh) a = inner(nabla_grad(u), nabla_grad(v))*dx L = f*v*dx + sigma_top*v*ds(1) + sigma_bottom*v*ds(2) uh = Function(Vh) #solve(a == L, uh, bcs=bcs) A, b = assemble_system(a,L, bcs=bcs) solve(A, uh.vector(), b, \"cg\") nb.plot(uh)","title":"5. Define and solve the variational problem"},{"location":"tutorials_v1.6.0/1_FEniCS101/#6-compute-the-discretization-error","text":"For this problem, the exact solution is known. We can therefore compute the following norms of the discretization error (i.e. the of the difference between the finite element solution u_h and the exact solution u_{\\rm ex} ) \\| u_{\\rm ex} - u_h \\|_{L^2{\\Omega}} := \\sqrt{ \\int_{\\Omega} (u_{\\rm ex} - u_h)^2 \\, dx }, and \\| u_{\\rm ex} - u_h \\|_{H^1{\\Omega}} := \\sqrt{ \\int_{\\Omega} (u_{\\rm ex} - u_h)^2 \\, dx + \\int_{\\Omega} |\\nabla u_{\\rm ex} - \\nabla u_h|^2 \\, dx}. u_e = Expression('sin(2*pi*x[0])*sin((pi/2.0)*x[1])', degree=5) grad_u_e = Expression( ('2*pi*cos(2*pi*x[0])*sin((pi/2.0)*x[1])', 'pi/2.0*sin(2*pi*x[0])*cos((pi/2.0)*x[1])'), degree=5) err_L2 = sqrt( assemble( (uh-u_e)**2*dx ) ) err_grad = sqrt( assemble( inner(nabla_grad(uh) - grad_u_e, nabla_grad(uh) - grad_u_e)*dx ) ) err_H1 = sqrt( err_L2**2 + err_grad**2) print(\"|| u_h - u_e ||_L2 = \", err_L2) print(\"|| u_h - u_e ||_H1 = \", err_H1) || u_h - u_e ||_L2 = 0.00880525372208 || u_h - u_e ||_H1 = 0.396718952514","title":"6. Compute the discretization error"},{"location":"tutorials_v1.6.0/1_FEniCS101/#7-convergence-of-the-finite-element-method","text":"We now verify numerically a well-known convergence result for the finite element method. Let denote with s the polynomial degree of the finite element space, and assume that the solution u_{\\rm ex} is at least in H^{s+1}(\\Omega) . Then we have \\| u_{\\rm ex} - u_h \\|_{H^1} \\leq C h^{s}, \\quad \\| u_{\\rm ex} - u_h \\|_{L^2} \\leq C h^{s+1}. In the code below, the function compute(n, degree) solves the PDE using a mesh with n elements in each direction and finite element spaces of polinomial order degree . The figure below shows the discretization errors in the H^1 and L^2 as a function of the mesh size h ( h = \\frac{1}{n} ) for piecewise linear (P1, s=1 ) and piecewise quadratic (P2, s=2 ) finite elements. We observe that numerical results are consistent with the finite element convergence theory. In particular: for piecewise linear finite element P1 we observe first order convergence in the H^1 -norm and second order convergence in the L^2 -norm; for piecewise quadratic finite element P2 we observe second order convergence in the H^1 -norm and third order convergence in the L^2 -norm. def compute(n, degree): mesh = UnitSquareMesh(n, n) Vh = FunctionSpace(mesh, 'Lagrange', degree) boundary_parts = FacetFunction(\"size_t\", mesh) boundary_parts.set_all(0) Gamma_top = TopBoundary() Gamma_top.mark(boundary_parts, 1) Gamma_bottom = BottomBoundary() Gamma_bottom.mark(boundary_parts, 2) Gamma_left = LeftBoundary() Gamma_left.mark(boundary_parts, 3) Gamma_right = RightBoundary() Gamma_right.mark(boundary_parts, 4) bcs = [DirichletBC(Vh, u_L, boundary_parts, 3), DirichletBC(Vh, u_R, boundary_parts, 4)] ds = Measure(\"ds\", subdomain_data=boundary_parts) u = TrialFunction(Vh) v = TestFunction(Vh) a = inner(nabla_grad(u), nabla_grad(v))*dx L = f*v*dx + sigma_top*v*ds(1) + sigma_bottom*v*ds(2) uh = Function(Vh) solve(a == L, uh, bcs=bcs) err_L2 = sqrt( assemble( (uh-u_e)**2*dx ) ) err_grad = sqrt( assemble( inner(nabla_grad(uh) - grad_u_e, nabla_grad(uh) - grad_u_e)*dx ) ) err_H1 = sqrt( err_L2**2 + err_grad**2) return err_L2, err_H1 nref = 5 n = 8*np.power(2,np.arange(0,nref)) h = 1./n err_L2_P1 = np.zeros(nref) err_H1_P1 = np.zeros(nref) err_L2_P2 = np.zeros(nref) err_H1_P2 = np.zeros(nref) for i in range(nref): err_L2_P1[i], err_H1_P1[i] = compute(n[i], 1) err_L2_P2[i], err_H1_P2[i] = compute(n[i], 2) plt.figure(figsize=(15,5)) plt.subplot(121) plt.loglog(h, err_H1_P1, '-or', label=\"H1 error\") plt.loglog(h, err_L2_P1, '-*b', label=\"L2 error\") plt.loglog(h, h*.5*err_H1_P1[0]/h[0], '--g', label=\"First Order\") plt.loglog(h, np.power(h,2)*.5*np.power( err_L2_P1[0]/h[0], 2), '-.k', label=\"Second Order\") plt.xlabel(\"Mesh size h\") plt.ylabel(\"Error\") plt.title(\"P1 Finite Element\") plt.legend(loc = 'lower right') plt.subplot(122) plt.loglog(h, err_H1_P2, '-or', label=\"H1 error\") plt.loglog(h, err_L2_P2, '-*b', label=\"L2 error\") plt.loglog(h, np.power(h/h[0],2)*.5*err_H1_P2[0], '--g', label=\"Second Order\") plt.loglog(h, np.power(h/h[0],3)*.5*err_L2_P2[0], '-.k', label=\"Third Order\") plt.xlabel(\"Mesh size h\") plt.ylabel(\"Error\") plt.title(\"P2 Finite Element\") plt.legend(loc='lower right') plt.show() Copyright (c) 2016-2018, The University of Texas at Austin & University of California, Merced. All Rights reserved. See file COPYRIGHT for details. This file is part of the hIPPYlib library. For more information and source code availability see https://hippylib.github.io. hIPPYlib is free software; you can redistribute it and/or modify it under the terms of the GNU General Public License (as published by the Free Software Foundation) version 2.0 dated June 1991.","title":"7. Convergence of the finite element method"},{"location":"tutorials_v1.6.0/2_PoissonDeterministic/","text":"Coefficient field inversion in an elliptic partial differential equation We consider the estimation of a coefficient in an elliptic partial differential equation as a model problem. Depending on the interpretation of the unknowns and the type of measurements, this model problem arises, for instance, in inversion for groundwater flow or heat conductivity. It can also be interpreted as finding a membrane with a certain spatially varying stiffness. Let \\Omega\\subset\\mathbb{R}^n , n\\in\\{1,2,3\\} be an open, bounded domain and consider the following problem: \\min_{a} J(a):=\\frac{1}{2}\\int_\\Omega (u-u_d)^2\\, dx + \\frac{\\gamma}{2}\\int_\\Omega|\\nabla a|^2\\,dx, where u is the solution of \\begin{split} \\quad -\\nabla\\cdot(\\exp(a)\\nabla u) &= f \\text{ in }\\Omega,\\\\ u &= 0 \\text{ on }\\partial\\Omega. \\end{split} Here a\\in U_{ad}:=\\{a\\in H^1(\\Omega) \\bigcap L^{\\infty}(\\Omega)\\} the unknown coefficient field, u_d denotes (possibly noisy) data, f\\in H^{-1}(\\Omega) a given force, and \\gamma\\ge 0 the regularization parameter. The variational (or weak) form of the state equation: Find u\\in H_0^1(\\Omega) such that (\\exp(a)\\nabla u,\\nabla v) - (f,v) = 0, \\text{ for all } v\\in H_0^1(\\Omega), where H_0^1(\\Omega) is the space of functions vanishing on \\partial\\Omega with square integrable derivatives. Here, (\\cdot\\,,\\cdot) denotes the L^2 -inner product, i.e, for scalar functions u,v defined on \\Omega we denote (u,v) := \\int_\\Omega u(x) v(x) \\,dx . Optimality System: The Lagrangian functional \\mathscr{L}:H^1(\\Omega)\\times H_0^1(\\Omega)\\times H_0^1(\\Omega)\\rightarrow \\mathbb{R} , which we use as a tool to derive the optimality system, is given by \\mathscr{L}(a,u,p):= \\frac{1}{2}(u-u_d,u-u_d) + \\frac{\\gamma}{2}(\\nabla a, \\nabla a) + (\\exp(a)\\nabla u,\\nabla p) - (f,p). The Lagrange multiplier theory shows that, at a solution all variations of the Lagrangian functional with respect to all variables must vanish. These variations of \\mathscr{L} with respect to (p,u,a) in directions (\\tilde{u}, \\tilde{p}, \\tilde{a}) are given by \\begin{alignat}{2} \\mathscr{L}_p(a,u,p)(\\tilde{p}) &= (\\exp(a)\\nabla u, \\nabla \\tilde{p}) - (f,\\tilde{p}) &&= 0,\\\\ \\mathscr{L}_u(a,u,p)(\\tilde{u}) &= (\\exp(a)\\nabla p, \\nabla \\tilde{u}) + (u-u_d,\\tilde{u}) && = 0,\\\\ \\mathscr{L}_a(a,u,p)(\\tilde{a}) &= \\gamma(\\nabla a, \\nabla \\tilde{a}) + (\\tilde{a}\\exp(a)\\nabla u, \\nabla p) &&= 0, \\end{alignat} where the variations (\\tilde{u}, \\tilde{p}, \\tilde{a}) are taken from the same spaces as (u,p,a) . The gradient of the cost functional \\mathcal{J}(a) therefore is \\mathcal{G}(a)(\\tilde a) = \\gamma(\\nabla a, \\nabla \\tilde{a}) + (\\tilde{a}\\exp(a)\\nabla u, \\nabla \\tilde{p}). Inexact Newton-CG: Newton's method requires second-order variational derivatives of the Lagrangian . Written in abstract form, it computes an update direction (\\hat a_k, \\hat u_k,\\hat p_k) from the following Newton step for the Lagrangian functional: \\mathscr{L}''(a_k, u_k, p_k)\\left[(\\tilde a, \\tilde u, \\tilde p),(\\hat a_k, \\hat u_k, \\hat p_k)\\right] = -\\mathscr{L}'(a_k,u_k,p_k)(\\tilde a, \\tilde u, \\tilde p), for all variations (\\tilde a, \\tilde u, \\tilde p) , where \\mathscr{L}' and \\mathscr{L}'' denote the first and second variations of the Lagrangian. For the elliptic parameter inversion problem, this Newton step (written in variatonal form) is as follows: Find (\\hat u_k, \\hat a_k,\\hat p_k) as the solution of the linear system \\begin{array}{llll} (\\hat{u}_k, \\tilde u) &+ (\\hat{a}_k \\exp(a_k)\\nabla p_k, \\nabla \\tilde u) &+ (\\exp(a_k) \\nabla \\tilde u, \\nabla \\hat p_k) &= (u_d - u_k, \\tilde u)- (\\exp(a_k) \\nabla p_k, \\nabla \\tilde u)\\\\ (\\tilde a \\exp(a_k) \\nabla \\hat u_k, \\nabla p_k) &+ \\gamma (\\nabla \\hat a_k, \\nabla \\tilde a) + (\\tilde a \\hat a_k \\exp(a_k)\\nabla u, \\nabla p) &+ (\\tilde a \\exp(a_k) \\nabla u_k, \\nabla \\hat p_k) &= - \\gamma (\\nabla a_k, \\nabla\\tilde a) - (\\tilde a \\exp(a_k) \\nabla u_k, \\nabla p_k)\\\\ (\\exp(a_k) \\nabla \\hat u_k, \\nabla \\tilde p) &+ (\\hat a_k \\exp(a_k) \\nabla u_k, \\nabla \\tilde p) & &= - (\\exp(a_k) \\nabla u_k, \\nabla \\tilde p) + (f, \\tilde p), \\end{array} for all (\\tilde u, \\tilde a, \\tilde p) . Discrete Newton system: \\def\\tu{\\tilde u} \\def\\btu{\\bf \\tilde u} \\def\\ta{\\tilde a} \\def\\bta{\\bf \\tilde a} \\def\\tp{\\tilde p} \\def\\btp{\\bf \\tilde p} \\def\\hu{\\hat u} \\def\\bhu{\\bf \\hat u} \\def\\ha{\\hat a} \\def\\bha{\\bf \\hat a} \\def\\hp{\\hat p} \\def\\bhp{\\bf \\hat p} The discretized Newton step: denote the vectors corresponding to the discretization of the functions \\ha_k,\\hu_k, \\hp_k by \\bf \\bha_k, \\bhu_k and \\bhp_k . Then, the discretization of the above system is given by the following symmetric linear system: \\begin{bmatrix} \\bf W_{\\scriptsize\\mbox{uu}} & \\bf W_{\\scriptsize\\mbox{ua}} & \\bf A^T \\\\ \\bf W_{\\scriptsize\\mbox{au}} & \\bf R + \\bf R_{\\scriptsize\\mbox{aa}}& \\bf C^T \\\\ \\bf A & \\bf C & 0 \\end{bmatrix} \\left[ \\begin{array}{c} \\bhu_k \\\\ \\bha_k \\\\ \\bhp_k \\end{array} \\right] = -\\left[ \\begin{array}{ccc} \\bf{g}_u\\\\ \\bf{g}_a\\\\ \\bf{g}_p \\end{array} \\right], where \\bf W_{\\scriptsize \\mbox{uu}} , \\bf W_{\\scriptsize\\mbox{ua}} , \\bf W_{\\scriptsize\\mbox{au}} , and \\bf R are the components of the Hessian matrix of the Lagrangian, \\bf A and \\bf C are the Jacobian of the state equation with respect to the state and the control variables, respectively and \\bf g_u , \\bf g_a , and \\bf g_p are the discrete gradients of the Lagrangian with respect to \\bf u , \\bf a and \\bf p , respectively. Reduced Hessian apply: To eliminate the incremental state and adjoint variables, \\bhu_k and \\bhp_k , from the first and last equations we use \\begin{align} \\bhu_k &= -\\bf A^{-1} \\bf C \\, \\bha_k,\\\\ \\bhp_k &= -\\bf A^{-T} (\\bf W_{\\scriptsize\\mbox{uu}} \\bhu_k + \\bf W_{\\scriptsize\\mbox{ua}}\\,\\bha_k). \\end{align} This results in the following reduced linear system for the Newton step \\bf H \\, \\bha_k = -\\bf{g}_a, with the reduced Hessian \\bf H applied to a vector \\bha given by \\bf H \\bha = \\underbrace{(\\bf R + \\bf R_{\\scriptsize\\mbox{aa}})}_{\\text{Hessian of the regularization}} \\bha + \\underbrace{(\\bf C^{T}\\bf A^{-T} (\\bf W_{\\scriptsize\\mbox{uu}} \\bf A^{-1} \\bf C - \\bf W_{\\scriptsize\\mbox{ua}}) - \\bf W_{\\scriptsize\\mbox{au}} \\bf A^{-1} \\bf C)}_{\\text{Hessian of the data misfit}}\\;\\bha. Goals: By the end of this notebook, you should be able to: solve the forward and adjoint Poisson equations understand the inverse method framework visualise and understand the results modify the problem and code Mathematical tools used: Finite element method Derivation of gradiant and Hessian via the adjoint method inexact Newton-CG Armijo line search List of software used: FEniCS , a parallel finite element element library for the discretization of partial differential equations PETSc , for scalable and efficient linear algebra operations and solvers Matplotlib , a python package used for plotting the results Numpy , a python package for linear algebra Set up Import dependencies from __future__ import absolute_import, division, print_function from dolfin import * import sys import os sys.path.append( os.environ.get('HIPPYLIB_BASE_DIR', \"../\") ) from hippylib import * import numpy as np import logging import matplotlib.pyplot as plt %matplotlib inline import nb logging.getLogger('FFC').setLevel(logging.WARNING) logging.getLogger('UFL').setLevel(logging.WARNING) set_log_active(False) np.random.seed(seed=1) Model set up: As in the introduction, the first thing we need to do is set up the numerical model. In this cell, we set the mesh, the finite element functions u, p, g corresponding to state, adjoint and coefficient/gradient variables, and the corresponding test functions and the parameters for the optimization. # create mesh and define function spaces nx = 64 ny = 64 mesh = UnitSquareMesh(nx, ny) Va = FunctionSpace(mesh, 'Lagrange', 1) Vu = FunctionSpace(mesh, 'Lagrange', 2) # The true and inverted parameter atrue = interpolate(Expression('log(2 + 7*(pow(pow(x[0] - 0.5,2) + pow(x[1] - 0.5,2),0.5) > 0.2))', degree=5),Va) a = interpolate(Expression(\"log(2.0)\", degree=1),Va) # define function for state and adjoint u = Function(Vu) p = Function(Vu) # define Trial and Test Functions u_trial, p_trial, a_trial = TrialFunction(Vu), TrialFunction(Vu), TrialFunction(Va) u_test, p_test, a_test = TestFunction(Vu), TestFunction(Vu), TestFunction(Va) # initialize input functions f = Constant(\"1.0\") u0 = Constant(\"0.0\") # plot plt.figure(figsize=(15,5)) nb.plot(mesh,subplot_loc=121, mytitle=\"Mesh\", show_axis='on') nb.plot(atrue,subplot_loc=122, mytitle=\"True parameter field\") plt.show() # set up dirichlet boundary conditions def boundary(x,on_boundary): return on_boundary bc_state = DirichletBC(Vu, u0, boundary) bc_adj = DirichletBC(Vu, Constant(0.), boundary) Set up synthetic observations: Propose a coefficient field a_{\\text true} shown above The weak form of the pde: Find u\\in H_0^1(\\Omega) such that \\underbrace{(\\exp(a_{\\text true})\\nabla u,\\nabla v)}_{\\; := \\; a_{pde}} - \\underbrace{(f,v)}_{\\; := \\;L_{pde}} = 0, \\text{ for all } v\\in H_0^1(\\Omega) . Perturb the solution: u = u + \\eta , where \\eta \\sim \\mathcal{N}(0, \\sigma) # noise level noise_level = 0.05 # weak form for setting up the synthetic observations a_goal = inner(exp(atrue) * nabla_grad(u_trial), nabla_grad(u_test)) * dx L_goal = f * u_test * dx # solve the forward/state problem to generate synthetic observations goal_A, goal_b = assemble_system(a_goal, L_goal, bc_state) utrue = Function(Vu) solve(goal_A, utrue.vector(), goal_b) ud = Function(Vu) ud.assign(utrue) # perturb state solution and create synthetic measurements ud # ud = u + ||u||/SNR * random.normal MAX = ud.vector().norm(\"linf\") noise = Vector() goal_A.init_vector(noise,1) noise.set_local( noise_level * MAX * np.random.normal(0, 1, Vu.dim())) bc_adj.apply(noise) ud.vector().axpy(1., noise) # plot nb.multi1_plot([utrue, ud], [\"State solution with atrue\", \"Synthetic observations\"]) plt.show() The cost function evaluation: J(a):=\\underbrace{\\frac{1}{2}\\int_\\Omega (u-u_d)^2\\, dx}_{\\text misfit} + \\underbrace{\\frac{\\gamma}{2}\\int_\\Omega|\\nabla a|^2\\,dx}_{\\text reg} In the code below, W and R are symmetric positive definite matrices that stem from finite element discretization of the misfit and regularization component of the cost functional, respectively. # regularization parameter gamma = 1e-8 # weak for for setting up the misfit and regularization compoment of the cost W_equ = inner(u_trial, u_test) * dx R_equ = gamma * inner(nabla_grad(a_trial), nabla_grad(a_test)) * dx W = assemble(W_equ) R = assemble(R_equ) # refine cost function def cost(u, ud, a, W, R): diff = u.vector() - ud.vector() reg = 0.5 * a.vector().inner(R*a.vector() ) misfit = 0.5 * diff.inner(W * diff) return [reg + misfit, misfit, reg] Setting up the state equations, right hand side for the adjoint and the necessary matrices: \\begin{array}{llll} (\\hat{u}_k, \\tilde u) &+ (\\hat{a}_k \\exp(a_k)\\nabla p_k, \\nabla \\tilde u) &+ (\\exp(a_k) \\nabla \\tilde u, \\nabla \\hat p_k) &= (u_d - u_k, \\tilde u)- (\\exp(a_k) \\nabla p_k, \\nabla \\tilde u)\\\\ (\\tilde a \\exp(a_k) \\nabla \\hat u_k, \\nabla p_k) &+ \\gamma (\\nabla \\hat a_k, \\nabla \\tilde a) + (\\tilde a \\hat a_k \\exp(a_k)\\nabla u, \\nabla p) &+ (\\tilde a \\exp(a_k) \\nabla u_k, \\nabla \\hat p_k) &= - \\gamma (\\nabla a_k, \\nabla\\tilde a) - (\\tilde a \\exp(a_k) \\nabla u_k, \\nabla p_k)\\\\ (\\exp(a_k) \\nabla \\hat u_k, \\nabla \\tilde p) &+ (\\hat a_k \\exp(a_k) \\nabla u_k, \\nabla \\tilde p) & &= - (\\exp(a_k) \\nabla u_k, \\nabla \\tilde p) + (f, \\tilde p), \\end{array} # weak form for setting up the state equation a_state = inner(exp(a) * nabla_grad(u_trial), nabla_grad(u_test)) * dx L_state = f * u_test * dx # weak form for setting up the adjoint equation a_adj = inner(exp(a) * nabla_grad(p_trial), nabla_grad(p_test)) * dx L_adj = -inner(u - ud, p_test) * dx # weak form for setting up matrices Wua_equ = inner(exp(a) * a_trial * nabla_grad(p_test), nabla_grad(p)) * dx C_equ = inner(exp(a) * a_trial * nabla_grad(u), nabla_grad(u_test)) * dx Raa_equ = inner(exp(a) * a_trial * a_test * nabla_grad(u), nabla_grad(p)) * dx M_equ = inner(a_trial, a_test) * dx # assemble matrix M M = assemble(M_equ) Initial guess We solve the state equation and compute the cost functional for the initial guess of the parameter a_ini # solve state equation state_A, state_b = assemble_system (a_state, L_state, bc_state) solve (state_A, u.vector(), state_b) # evaluate cost [cost_old, misfit_old, reg_old] = cost(u, ud, a, W, R) # plot plt.figure(figsize=(15,5)) nb.plot(a,subplot_loc=121, mytitle=\"a_ini\", vmin=atrue.vector().min(), vmax=atrue.vector().max()) nb.plot(u,subplot_loc=122, mytitle=\"u(a_ini)\") plt.show() The reduced Hessian apply to a vector v: Here we describe how to apply the reduced Hessian operator to a vector v. For an opportune choice of the regularization, the reduced Hessian operator evaluated in a neighborhood of the solution is positive define, whereas far from the solution the reduced Hessian may be indefinite. On the constrary, the Gauss-Newton approximation of the Hessian is always positive defined. For this reason, it is beneficial to perform a few initial Gauss-Newton steps (5 in this particular example) to accelerate the convergence of the inexact Newton-CG algorithm. The Hessian apply reads: \\begin{align} \\bhu &= -\\bf A^{-1} \\bf C \\bf v\\, & \\text{linearized forward}\\\\ \\bhp &= -\\bf A^{-T} (\\bf W_{\\scriptsize\\mbox{uu}} \\bhu + \\bf W_{\\scriptsize\\mbox{ua}}\\,\\bha) & \\text{adjoint}\\\\ \\bf H \\bf v &= (\\bf R + \\bf R_{\\scriptsize\\mbox{aa}})\\bf v + \\bf C^T \\bhp + \\bf W_{\\scriptsize\\mbox{au}} \\bhu. \\end{align} The Gauss-Newton Hessian apply is obtained by dropping the second derivatives operators \\bf W_{\\scriptsize\\mbox{ua}}\\,\\bha , \\bf R_{\\scriptsize\\mbox{aa}}\\bf v , and \\bf W_{\\scriptsize\\mbox{au}} \\bhu : \\begin{align} \\bhu &= -\\bf A^{-1} \\bf C \\bf v\\, & \\text{linearized forward}\\\\ \\bhp &= -\\bf A^{-T} \\bf W_{\\scriptsize\\mbox{uu}} \\bhu & \\text{adjoint}\\\\ \\bf H_{\\rm GN} \\bf v &= \\bf R \\bf v + \\bf C^T \\bhp. \\end{align} # Class HessianOperator to perform Hessian apply to a vector class HessianOperator(): cgiter = 0 def __init__(self, R, Raa, C, A, adj_A, W, Wua, use_gaussnewton=False): self.R = R self.Raa = Raa self.C = C self.A = A self.adj_A = adj_A self.W = W self.Wua = Wua self.use_gaussnewton = use_gaussnewton # incremental state self.du = Vector() self.A.init_vector(self.du,0) #incremental adjoint self.dp = Vector() self.adj_A.init_vector(self.dp,0) # auxiliary vectors self.CT_dp = Vector() self.C.init_vector(self.CT_dp, 1) self.Wua_du = Vector() self.Wua.init_vector(self.Wua_du, 1) def init_vector(self, v, dim): self.R.init_vector(v,dim) # Hessian performed on x, output as generic vector y def mult(self, v, y): self.cgiter += 1 y.zero() if self.use_gaussnewton: self.mult_GaussNewton(v,y) else: self.mult_Newton(v,y) # define (Gauss-Newton) Hessian apply H * v def mult_GaussNewton(self, v, y): #incremental forward rhs = -(self.C * v) bc_adj.apply(rhs) solve (self.A, self.du, rhs) #incremental adjoint rhs = - (self.W * self.du) bc_adj.apply(rhs) solve (self.adj_A, self.dp, rhs) # Reg/Prior term self.R.mult(v,y) # Misfit term self.C.transpmult(self.dp, self.CT_dp) y.axpy(1, self.CT_dp) # define (Newton) Hessian apply H * v def mult_Newton(self, v, y): #incremental forward rhs = -(self.C * v) bc_adj.apply(rhs) solve (self.A, self.du, rhs) #incremental adjoint rhs = -(self.W * self.du) - self.Wua * v bc_adj.apply(rhs) solve (self.adj_A, self.dp, rhs) #Reg/Prior term self.R.mult(v,y) y.axpy(1., Raa*v) #Misfit term self.C.transpmult(self.dp, self.CT_dp) y.axpy(1., self.CT_dp) self.Wua.transpmult(self.du, self.Wua_du) y.axpy(1., self.Wua_du) The inexact Newton-CG optimization with Armijo line search: We solve the constrained optimization problem using the inexact Newton-CG method with Armijo line search. The stopping criterion is based on a relative reduction of the norm of the gradient (i.e. \\frac{\\|g_{n}\\|}{\\|g_{0}\\|} \\leq \\tau ). First, we compute the gradient by solving the state and adjoint equation for the current parameter a , and then substituing the current state u , parameter a and adjoint p variables in the weak form expression of the gradient: (g, \\tilde{a}) = \\gamma(\\nabla a, \\nabla \\tilde{a}) +(\\tilde{a}\\nabla u, \\nabla p). Then, we compute the Newton direction \\delta a by iteratively solving {\\bf H} {\\delta a} = - {\\bf g} . The Newton system is solved inexactly by early termination of conjugate gradient iterations via Eisenstat\u2013Walker (to prevent oversolving) and Steihaug (to avoid negative curvature) criteria. Finally, the Armijo line search uses backtracking to find \\alpha such that a sufficient reduction in the cost functional is achieved. More specifically, we use backtracking to find \\alpha such that: J( a + \\alpha \\delta a ) \\leq J(a) + \\alpha c_{\\rm armijo} (\\delta a,g). # define parameters for the optimization tol = 1e-8 c = 1e-4 maxiter = 12 plot_on = False # initialize iter counters iter = 1 total_cg_iter = 0 converged = False # initializations g, a_delta = Vector(), Vector() R.init_vector(a_delta,0) R.init_vector(g,0) a_prev = Function(Va) print(\"Nit CGit cost misfit reg sqrt(-G*D) ||grad|| alpha tolcg\") while iter < maxiter and not converged: # assemble matrix C C = assemble(C_equ) # solve the adoint problem adjoint_A, adjoint_RHS = assemble_system(a_adj, L_adj, bc_adj) solve(adjoint_A, p.vector(), adjoint_RHS) # assemble W_ua and R Wua = assemble (Wua_equ) Raa = assemble (Raa_equ) # evaluate the gradient CT_p = Vector() C.init_vector(CT_p,1) C.transpmult(p.vector(), CT_p) MG = CT_p + R * a.vector() solve(M, g, MG) # calculate the norm of the gradient grad2 = g.inner(MG) gradnorm = sqrt(grad2) # set the CG tolerance (use Eisenstat\u2013Walker termination criterion) if iter == 1: gradnorm_ini = gradnorm tolcg = min(0.5, sqrt(gradnorm/gradnorm_ini)) # define the Hessian apply operator (with preconditioner) Hess_Apply = HessianOperator(R, Raa, C, state_A, adjoint_A, W, Wua, use_gaussnewton=(iter<6) ) P = R + gamma * M Psolver = PETScKrylovSolver(\"cg\", amg_method()) Psolver.set_operator(P) solver = CGSolverSteihaug() solver.set_operator(Hess_Apply) solver.set_preconditioner(Psolver) solver.parameters[\"rel_tolerance\"] = tolcg solver.parameters[\"zero_initial_guess\"] = True solver.parameters[\"print_level\"] = -1 # solve the Newton system H a_delta = - MG solver.solve(a_delta, -MG) total_cg_iter += Hess_Apply.cgiter # linesearch alpha = 1 descent = 0 no_backtrack = 0 a_prev.assign(a) while descent == 0 and no_backtrack < 10: a.vector().axpy(alpha, a_delta ) # solve the state/forward problem state_A, state_b = assemble_system(a_state, L_state, bc_state) solve(state_A, u.vector(), state_b) # evaluate cost [cost_new, misfit_new, reg_new] = cost(u, ud, a, W, R) # check if Armijo conditions are satisfied if cost_new < cost_old + alpha * c * MG.inner(a_delta): cost_old = cost_new descent = 1 else: no_backtrack += 1 alpha *= 0.5 a.assign(a_prev) # reset a # calculate sqrt(-G * D) graddir = sqrt(- MG.inner(a_delta) ) sp = \"\" print(\"%2d %2s %2d %3s %8.5e %1s %8.5e %1s %8.5e %1s %8.5e %1s %8.5e %1s %5.2f %1s %5.3e\" % \\ (iter, sp, Hess_Apply.cgiter, sp, cost_new, sp, misfit_new, sp, reg_new, sp, \\ graddir, sp, gradnorm, sp, alpha, sp, tolcg) ) if plot_on: nb.multi1_plot([a,u,p], [\"a\",\"u\",\"p\"], same_colorbar=False) plt.show() # check for convergence if gradnorm < tol and iter > 1: converged = True print(\"Newton's method converged in \",iter,\" iterations\") print(\"Total number of CG iterations: \", total_cg_iter) iter += 1 if not converged: print(\"Newton's method did not converge in \", maxiter, \" iterations\") Nit CGit cost misfit reg sqrt(-G*D) ||grad|| alpha tolcg 1 1 1.12708e-05 1.12708e-05 1.33979e-11 1.56540e-02 3.79427e-04 1.00 5.000e-01 2 1 7.79732e-07 7.79695e-07 3.67737e-11 4.68278e-03 5.35002e-05 1.00 3.755e-01 3 1 3.10620e-07 3.10571e-07 4.91259e-11 9.71633e-04 7.13895e-06 1.00 1.372e-01 4 5 1.92183e-07 1.62405e-07 2.97780e-08 4.51694e-04 1.00276e-06 1.00 5.141e-02 5 1 1.86913e-07 1.57119e-07 2.97941e-08 1.02668e-04 6.12750e-07 1.00 4.019e-02 6 12 1.80408e-07 1.37719e-07 4.26890e-08 1.15975e-04 2.24111e-07 1.00 2.430e-02 7 5 1.80331e-07 1.38935e-07 4.13963e-08 1.23223e-05 4.17399e-08 1.00 1.049e-02 8 15 1.80330e-07 1.39056e-07 4.12734e-08 1.74451e-06 3.43216e-09 1.00 3.008e-03 Newton's method converged in 8 iterations Total number of CG iterations: 41 nb.multi1_plot([atrue, a], [\"atrue\", \"a\"]) nb.multi1_plot([u,p], [\"u\",\"p\"], same_colorbar=False) plt.show() Copyright (c) 2016-2018, The University of Texas at Austin & University of California, Merced. All Rights reserved. See file COPYRIGHT for details. This file is part of the hIPPYlib library. For more information and source code availability see https://hippylib.github.io. hIPPYlib is free software; you can redistribute it and/or modify it under the terms of the GNU General Public License (as published by the Free Software Foundation) version 2.0 dated June 1991.","title":"Poisson Deterministic"},{"location":"tutorials_v1.6.0/2_PoissonDeterministic/#coefficient-field-inversion-in-an-elliptic-partial-differential-equation","text":"We consider the estimation of a coefficient in an elliptic partial differential equation as a model problem. Depending on the interpretation of the unknowns and the type of measurements, this model problem arises, for instance, in inversion for groundwater flow or heat conductivity. It can also be interpreted as finding a membrane with a certain spatially varying stiffness. Let \\Omega\\subset\\mathbb{R}^n , n\\in\\{1,2,3\\} be an open, bounded domain and consider the following problem: \\min_{a} J(a):=\\frac{1}{2}\\int_\\Omega (u-u_d)^2\\, dx + \\frac{\\gamma}{2}\\int_\\Omega|\\nabla a|^2\\,dx, where u is the solution of \\begin{split} \\quad -\\nabla\\cdot(\\exp(a)\\nabla u) &= f \\text{ in }\\Omega,\\\\ u &= 0 \\text{ on }\\partial\\Omega. \\end{split} Here a\\in U_{ad}:=\\{a\\in H^1(\\Omega) \\bigcap L^{\\infty}(\\Omega)\\} the unknown coefficient field, u_d denotes (possibly noisy) data, f\\in H^{-1}(\\Omega) a given force, and \\gamma\\ge 0 the regularization parameter.","title":"Coefficient field inversion in an elliptic partial differential equation"},{"location":"tutorials_v1.6.0/2_PoissonDeterministic/#the-variational-or-weak-form-of-the-state-equation","text":"Find u\\in H_0^1(\\Omega) such that (\\exp(a)\\nabla u,\\nabla v) - (f,v) = 0, \\text{ for all } v\\in H_0^1(\\Omega), where H_0^1(\\Omega) is the space of functions vanishing on \\partial\\Omega with square integrable derivatives. Here, (\\cdot\\,,\\cdot) denotes the L^2 -inner product, i.e, for scalar functions u,v defined on \\Omega we denote (u,v) := \\int_\\Omega u(x) v(x) \\,dx .","title":"The variational (or weak) form of the state equation:"},{"location":"tutorials_v1.6.0/2_PoissonDeterministic/#optimality-system","text":"The Lagrangian functional \\mathscr{L}:H^1(\\Omega)\\times H_0^1(\\Omega)\\times H_0^1(\\Omega)\\rightarrow \\mathbb{R} , which we use as a tool to derive the optimality system, is given by \\mathscr{L}(a,u,p):= \\frac{1}{2}(u-u_d,u-u_d) + \\frac{\\gamma}{2}(\\nabla a, \\nabla a) + (\\exp(a)\\nabla u,\\nabla p) - (f,p). The Lagrange multiplier theory shows that, at a solution all variations of the Lagrangian functional with respect to all variables must vanish. These variations of \\mathscr{L} with respect to (p,u,a) in directions (\\tilde{u}, \\tilde{p}, \\tilde{a}) are given by \\begin{alignat}{2} \\mathscr{L}_p(a,u,p)(\\tilde{p}) &= (\\exp(a)\\nabla u, \\nabla \\tilde{p}) - (f,\\tilde{p}) &&= 0,\\\\ \\mathscr{L}_u(a,u,p)(\\tilde{u}) &= (\\exp(a)\\nabla p, \\nabla \\tilde{u}) + (u-u_d,\\tilde{u}) && = 0,\\\\ \\mathscr{L}_a(a,u,p)(\\tilde{a}) &= \\gamma(\\nabla a, \\nabla \\tilde{a}) + (\\tilde{a}\\exp(a)\\nabla u, \\nabla p) &&= 0, \\end{alignat} where the variations (\\tilde{u}, \\tilde{p}, \\tilde{a}) are taken from the same spaces as (u,p,a) . The gradient of the cost functional \\mathcal{J}(a) therefore is \\mathcal{G}(a)(\\tilde a) = \\gamma(\\nabla a, \\nabla \\tilde{a}) + (\\tilde{a}\\exp(a)\\nabla u, \\nabla \\tilde{p}).","title":"Optimality System:"},{"location":"tutorials_v1.6.0/2_PoissonDeterministic/#inexact-newton-cg","text":"Newton's method requires second-order variational derivatives of the Lagrangian . Written in abstract form, it computes an update direction (\\hat a_k, \\hat u_k,\\hat p_k) from the following Newton step for the Lagrangian functional: \\mathscr{L}''(a_k, u_k, p_k)\\left[(\\tilde a, \\tilde u, \\tilde p),(\\hat a_k, \\hat u_k, \\hat p_k)\\right] = -\\mathscr{L}'(a_k,u_k,p_k)(\\tilde a, \\tilde u, \\tilde p), for all variations (\\tilde a, \\tilde u, \\tilde p) , where \\mathscr{L}' and \\mathscr{L}'' denote the first and second variations of the Lagrangian. For the elliptic parameter inversion problem, this Newton step (written in variatonal form) is as follows: Find (\\hat u_k, \\hat a_k,\\hat p_k) as the solution of the linear system \\begin{array}{llll} (\\hat{u}_k, \\tilde u) &+ (\\hat{a}_k \\exp(a_k)\\nabla p_k, \\nabla \\tilde u) &+ (\\exp(a_k) \\nabla \\tilde u, \\nabla \\hat p_k) &= (u_d - u_k, \\tilde u)- (\\exp(a_k) \\nabla p_k, \\nabla \\tilde u)\\\\ (\\tilde a \\exp(a_k) \\nabla \\hat u_k, \\nabla p_k) &+ \\gamma (\\nabla \\hat a_k, \\nabla \\tilde a) + (\\tilde a \\hat a_k \\exp(a_k)\\nabla u, \\nabla p) &+ (\\tilde a \\exp(a_k) \\nabla u_k, \\nabla \\hat p_k) &= - \\gamma (\\nabla a_k, \\nabla\\tilde a) - (\\tilde a \\exp(a_k) \\nabla u_k, \\nabla p_k)\\\\ (\\exp(a_k) \\nabla \\hat u_k, \\nabla \\tilde p) &+ (\\hat a_k \\exp(a_k) \\nabla u_k, \\nabla \\tilde p) & &= - (\\exp(a_k) \\nabla u_k, \\nabla \\tilde p) + (f, \\tilde p), \\end{array} for all (\\tilde u, \\tilde a, \\tilde p) .","title":"Inexact Newton-CG:"},{"location":"tutorials_v1.6.0/2_PoissonDeterministic/#discrete-newton-system","text":"\\def\\tu{\\tilde u} \\def\\btu{\\bf \\tilde u} \\def\\ta{\\tilde a} \\def\\bta{\\bf \\tilde a} \\def\\tp{\\tilde p} \\def\\btp{\\bf \\tilde p} \\def\\hu{\\hat u} \\def\\bhu{\\bf \\hat u} \\def\\ha{\\hat a} \\def\\bha{\\bf \\hat a} \\def\\hp{\\hat p} \\def\\bhp{\\bf \\hat p} The discretized Newton step: denote the vectors corresponding to the discretization of the functions \\ha_k,\\hu_k, \\hp_k by \\bf \\bha_k, \\bhu_k and \\bhp_k . Then, the discretization of the above system is given by the following symmetric linear system: \\begin{bmatrix} \\bf W_{\\scriptsize\\mbox{uu}} & \\bf W_{\\scriptsize\\mbox{ua}} & \\bf A^T \\\\ \\bf W_{\\scriptsize\\mbox{au}} & \\bf R + \\bf R_{\\scriptsize\\mbox{aa}}& \\bf C^T \\\\ \\bf A & \\bf C & 0 \\end{bmatrix} \\left[ \\begin{array}{c} \\bhu_k \\\\ \\bha_k \\\\ \\bhp_k \\end{array} \\right] = -\\left[ \\begin{array}{ccc} \\bf{g}_u\\\\ \\bf{g}_a\\\\ \\bf{g}_p \\end{array} \\right], where \\bf W_{\\scriptsize \\mbox{uu}} , \\bf W_{\\scriptsize\\mbox{ua}} , \\bf W_{\\scriptsize\\mbox{au}} , and \\bf R are the components of the Hessian matrix of the Lagrangian, \\bf A and \\bf C are the Jacobian of the state equation with respect to the state and the control variables, respectively and \\bf g_u , \\bf g_a , and \\bf g_p are the discrete gradients of the Lagrangian with respect to \\bf u , \\bf a and \\bf p , respectively.","title":"Discrete Newton system:"},{"location":"tutorials_v1.6.0/2_PoissonDeterministic/#reduced-hessian-apply","text":"To eliminate the incremental state and adjoint variables, \\bhu_k and \\bhp_k , from the first and last equations we use \\begin{align} \\bhu_k &= -\\bf A^{-1} \\bf C \\, \\bha_k,\\\\ \\bhp_k &= -\\bf A^{-T} (\\bf W_{\\scriptsize\\mbox{uu}} \\bhu_k + \\bf W_{\\scriptsize\\mbox{ua}}\\,\\bha_k). \\end{align} This results in the following reduced linear system for the Newton step \\bf H \\, \\bha_k = -\\bf{g}_a, with the reduced Hessian \\bf H applied to a vector \\bha given by \\bf H \\bha = \\underbrace{(\\bf R + \\bf R_{\\scriptsize\\mbox{aa}})}_{\\text{Hessian of the regularization}} \\bha + \\underbrace{(\\bf C^{T}\\bf A^{-T} (\\bf W_{\\scriptsize\\mbox{uu}} \\bf A^{-1} \\bf C - \\bf W_{\\scriptsize\\mbox{ua}}) - \\bf W_{\\scriptsize\\mbox{au}} \\bf A^{-1} \\bf C)}_{\\text{Hessian of the data misfit}}\\;\\bha.","title":"Reduced Hessian apply:"},{"location":"tutorials_v1.6.0/2_PoissonDeterministic/#goals","text":"By the end of this notebook, you should be able to: solve the forward and adjoint Poisson equations understand the inverse method framework visualise and understand the results modify the problem and code","title":"Goals:"},{"location":"tutorials_v1.6.0/2_PoissonDeterministic/#mathematical-tools-used","text":"Finite element method Derivation of gradiant and Hessian via the adjoint method inexact Newton-CG Armijo line search","title":"Mathematical tools used:"},{"location":"tutorials_v1.6.0/2_PoissonDeterministic/#list-of-software-used","text":"FEniCS , a parallel finite element element library for the discretization of partial differential equations PETSc , for scalable and efficient linear algebra operations and solvers Matplotlib , a python package used for plotting the results Numpy , a python package for linear algebra","title":"List of software used:"},{"location":"tutorials_v1.6.0/2_PoissonDeterministic/#set-up","text":"","title":"Set up"},{"location":"tutorials_v1.6.0/2_PoissonDeterministic/#import-dependencies","text":"from __future__ import absolute_import, division, print_function from dolfin import * import sys import os sys.path.append( os.environ.get('HIPPYLIB_BASE_DIR', \"../\") ) from hippylib import * import numpy as np import logging import matplotlib.pyplot as plt %matplotlib inline import nb logging.getLogger('FFC').setLevel(logging.WARNING) logging.getLogger('UFL').setLevel(logging.WARNING) set_log_active(False) np.random.seed(seed=1)","title":"Import dependencies"},{"location":"tutorials_v1.6.0/2_PoissonDeterministic/#model-set-up","text":"As in the introduction, the first thing we need to do is set up the numerical model. In this cell, we set the mesh, the finite element functions u, p, g corresponding to state, adjoint and coefficient/gradient variables, and the corresponding test functions and the parameters for the optimization. # create mesh and define function spaces nx = 64 ny = 64 mesh = UnitSquareMesh(nx, ny) Va = FunctionSpace(mesh, 'Lagrange', 1) Vu = FunctionSpace(mesh, 'Lagrange', 2) # The true and inverted parameter atrue = interpolate(Expression('log(2 + 7*(pow(pow(x[0] - 0.5,2) + pow(x[1] - 0.5,2),0.5) > 0.2))', degree=5),Va) a = interpolate(Expression(\"log(2.0)\", degree=1),Va) # define function for state and adjoint u = Function(Vu) p = Function(Vu) # define Trial and Test Functions u_trial, p_trial, a_trial = TrialFunction(Vu), TrialFunction(Vu), TrialFunction(Va) u_test, p_test, a_test = TestFunction(Vu), TestFunction(Vu), TestFunction(Va) # initialize input functions f = Constant(\"1.0\") u0 = Constant(\"0.0\") # plot plt.figure(figsize=(15,5)) nb.plot(mesh,subplot_loc=121, mytitle=\"Mesh\", show_axis='on') nb.plot(atrue,subplot_loc=122, mytitle=\"True parameter field\") plt.show() # set up dirichlet boundary conditions def boundary(x,on_boundary): return on_boundary bc_state = DirichletBC(Vu, u0, boundary) bc_adj = DirichletBC(Vu, Constant(0.), boundary)","title":"Model set up:"},{"location":"tutorials_v1.6.0/2_PoissonDeterministic/#set-up-synthetic-observations","text":"Propose a coefficient field a_{\\text true} shown above The weak form of the pde: Find u\\in H_0^1(\\Omega) such that \\underbrace{(\\exp(a_{\\text true})\\nabla u,\\nabla v)}_{\\; := \\; a_{pde}} - \\underbrace{(f,v)}_{\\; := \\;L_{pde}} = 0, \\text{ for all } v\\in H_0^1(\\Omega) . Perturb the solution: u = u + \\eta , where \\eta \\sim \\mathcal{N}(0, \\sigma) # noise level noise_level = 0.05 # weak form for setting up the synthetic observations a_goal = inner(exp(atrue) * nabla_grad(u_trial), nabla_grad(u_test)) * dx L_goal = f * u_test * dx # solve the forward/state problem to generate synthetic observations goal_A, goal_b = assemble_system(a_goal, L_goal, bc_state) utrue = Function(Vu) solve(goal_A, utrue.vector(), goal_b) ud = Function(Vu) ud.assign(utrue) # perturb state solution and create synthetic measurements ud # ud = u + ||u||/SNR * random.normal MAX = ud.vector().norm(\"linf\") noise = Vector() goal_A.init_vector(noise,1) noise.set_local( noise_level * MAX * np.random.normal(0, 1, Vu.dim())) bc_adj.apply(noise) ud.vector().axpy(1., noise) # plot nb.multi1_plot([utrue, ud], [\"State solution with atrue\", \"Synthetic observations\"]) plt.show()","title":"Set up synthetic observations:"},{"location":"tutorials_v1.6.0/2_PoissonDeterministic/#the-cost-function-evaluation","text":"J(a):=\\underbrace{\\frac{1}{2}\\int_\\Omega (u-u_d)^2\\, dx}_{\\text misfit} + \\underbrace{\\frac{\\gamma}{2}\\int_\\Omega|\\nabla a|^2\\,dx}_{\\text reg} In the code below, W and R are symmetric positive definite matrices that stem from finite element discretization of the misfit and regularization component of the cost functional, respectively. # regularization parameter gamma = 1e-8 # weak for for setting up the misfit and regularization compoment of the cost W_equ = inner(u_trial, u_test) * dx R_equ = gamma * inner(nabla_grad(a_trial), nabla_grad(a_test)) * dx W = assemble(W_equ) R = assemble(R_equ) # refine cost function def cost(u, ud, a, W, R): diff = u.vector() - ud.vector() reg = 0.5 * a.vector().inner(R*a.vector() ) misfit = 0.5 * diff.inner(W * diff) return [reg + misfit, misfit, reg]","title":"The cost function evaluation:"},{"location":"tutorials_v1.6.0/2_PoissonDeterministic/#setting-up-the-state-equations-right-hand-side-for-the-adjoint-and-the-necessary-matrices","text":"\\begin{array}{llll} (\\hat{u}_k, \\tilde u) &+ (\\hat{a}_k \\exp(a_k)\\nabla p_k, \\nabla \\tilde u) &+ (\\exp(a_k) \\nabla \\tilde u, \\nabla \\hat p_k) &= (u_d - u_k, \\tilde u)- (\\exp(a_k) \\nabla p_k, \\nabla \\tilde u)\\\\ (\\tilde a \\exp(a_k) \\nabla \\hat u_k, \\nabla p_k) &+ \\gamma (\\nabla \\hat a_k, \\nabla \\tilde a) + (\\tilde a \\hat a_k \\exp(a_k)\\nabla u, \\nabla p) &+ (\\tilde a \\exp(a_k) \\nabla u_k, \\nabla \\hat p_k) &= - \\gamma (\\nabla a_k, \\nabla\\tilde a) - (\\tilde a \\exp(a_k) \\nabla u_k, \\nabla p_k)\\\\ (\\exp(a_k) \\nabla \\hat u_k, \\nabla \\tilde p) &+ (\\hat a_k \\exp(a_k) \\nabla u_k, \\nabla \\tilde p) & &= - (\\exp(a_k) \\nabla u_k, \\nabla \\tilde p) + (f, \\tilde p), \\end{array} # weak form for setting up the state equation a_state = inner(exp(a) * nabla_grad(u_trial), nabla_grad(u_test)) * dx L_state = f * u_test * dx # weak form for setting up the adjoint equation a_adj = inner(exp(a) * nabla_grad(p_trial), nabla_grad(p_test)) * dx L_adj = -inner(u - ud, p_test) * dx # weak form for setting up matrices Wua_equ = inner(exp(a) * a_trial * nabla_grad(p_test), nabla_grad(p)) * dx C_equ = inner(exp(a) * a_trial * nabla_grad(u), nabla_grad(u_test)) * dx Raa_equ = inner(exp(a) * a_trial * a_test * nabla_grad(u), nabla_grad(p)) * dx M_equ = inner(a_trial, a_test) * dx # assemble matrix M M = assemble(M_equ)","title":"Setting up the state equations, right hand side for the adjoint and the necessary matrices:"},{"location":"tutorials_v1.6.0/2_PoissonDeterministic/#initial-guess","text":"We solve the state equation and compute the cost functional for the initial guess of the parameter a_ini # solve state equation state_A, state_b = assemble_system (a_state, L_state, bc_state) solve (state_A, u.vector(), state_b) # evaluate cost [cost_old, misfit_old, reg_old] = cost(u, ud, a, W, R) # plot plt.figure(figsize=(15,5)) nb.plot(a,subplot_loc=121, mytitle=\"a_ini\", vmin=atrue.vector().min(), vmax=atrue.vector().max()) nb.plot(u,subplot_loc=122, mytitle=\"u(a_ini)\") plt.show()","title":"Initial guess"},{"location":"tutorials_v1.6.0/2_PoissonDeterministic/#the-reduced-hessian-apply-to-a-vector-v","text":"Here we describe how to apply the reduced Hessian operator to a vector v. For an opportune choice of the regularization, the reduced Hessian operator evaluated in a neighborhood of the solution is positive define, whereas far from the solution the reduced Hessian may be indefinite. On the constrary, the Gauss-Newton approximation of the Hessian is always positive defined. For this reason, it is beneficial to perform a few initial Gauss-Newton steps (5 in this particular example) to accelerate the convergence of the inexact Newton-CG algorithm. The Hessian apply reads: \\begin{align} \\bhu &= -\\bf A^{-1} \\bf C \\bf v\\, & \\text{linearized forward}\\\\ \\bhp &= -\\bf A^{-T} (\\bf W_{\\scriptsize\\mbox{uu}} \\bhu + \\bf W_{\\scriptsize\\mbox{ua}}\\,\\bha) & \\text{adjoint}\\\\ \\bf H \\bf v &= (\\bf R + \\bf R_{\\scriptsize\\mbox{aa}})\\bf v + \\bf C^T \\bhp + \\bf W_{\\scriptsize\\mbox{au}} \\bhu. \\end{align} The Gauss-Newton Hessian apply is obtained by dropping the second derivatives operators \\bf W_{\\scriptsize\\mbox{ua}}\\,\\bha , \\bf R_{\\scriptsize\\mbox{aa}}\\bf v , and \\bf W_{\\scriptsize\\mbox{au}} \\bhu : \\begin{align} \\bhu &= -\\bf A^{-1} \\bf C \\bf v\\, & \\text{linearized forward}\\\\ \\bhp &= -\\bf A^{-T} \\bf W_{\\scriptsize\\mbox{uu}} \\bhu & \\text{adjoint}\\\\ \\bf H_{\\rm GN} \\bf v &= \\bf R \\bf v + \\bf C^T \\bhp. \\end{align} # Class HessianOperator to perform Hessian apply to a vector class HessianOperator(): cgiter = 0 def __init__(self, R, Raa, C, A, adj_A, W, Wua, use_gaussnewton=False): self.R = R self.Raa = Raa self.C = C self.A = A self.adj_A = adj_A self.W = W self.Wua = Wua self.use_gaussnewton = use_gaussnewton # incremental state self.du = Vector() self.A.init_vector(self.du,0) #incremental adjoint self.dp = Vector() self.adj_A.init_vector(self.dp,0) # auxiliary vectors self.CT_dp = Vector() self.C.init_vector(self.CT_dp, 1) self.Wua_du = Vector() self.Wua.init_vector(self.Wua_du, 1) def init_vector(self, v, dim): self.R.init_vector(v,dim) # Hessian performed on x, output as generic vector y def mult(self, v, y): self.cgiter += 1 y.zero() if self.use_gaussnewton: self.mult_GaussNewton(v,y) else: self.mult_Newton(v,y) # define (Gauss-Newton) Hessian apply H * v def mult_GaussNewton(self, v, y): #incremental forward rhs = -(self.C * v) bc_adj.apply(rhs) solve (self.A, self.du, rhs) #incremental adjoint rhs = - (self.W * self.du) bc_adj.apply(rhs) solve (self.adj_A, self.dp, rhs) # Reg/Prior term self.R.mult(v,y) # Misfit term self.C.transpmult(self.dp, self.CT_dp) y.axpy(1, self.CT_dp) # define (Newton) Hessian apply H * v def mult_Newton(self, v, y): #incremental forward rhs = -(self.C * v) bc_adj.apply(rhs) solve (self.A, self.du, rhs) #incremental adjoint rhs = -(self.W * self.du) - self.Wua * v bc_adj.apply(rhs) solve (self.adj_A, self.dp, rhs) #Reg/Prior term self.R.mult(v,y) y.axpy(1., Raa*v) #Misfit term self.C.transpmult(self.dp, self.CT_dp) y.axpy(1., self.CT_dp) self.Wua.transpmult(self.du, self.Wua_du) y.axpy(1., self.Wua_du)","title":"The reduced Hessian apply to a vector v:"},{"location":"tutorials_v1.6.0/2_PoissonDeterministic/#the-inexact-newton-cg-optimization-with-armijo-line-search","text":"We solve the constrained optimization problem using the inexact Newton-CG method with Armijo line search. The stopping criterion is based on a relative reduction of the norm of the gradient (i.e. \\frac{\\|g_{n}\\|}{\\|g_{0}\\|} \\leq \\tau ). First, we compute the gradient by solving the state and adjoint equation for the current parameter a , and then substituing the current state u , parameter a and adjoint p variables in the weak form expression of the gradient: (g, \\tilde{a}) = \\gamma(\\nabla a, \\nabla \\tilde{a}) +(\\tilde{a}\\nabla u, \\nabla p). Then, we compute the Newton direction \\delta a by iteratively solving {\\bf H} {\\delta a} = - {\\bf g} . The Newton system is solved inexactly by early termination of conjugate gradient iterations via Eisenstat\u2013Walker (to prevent oversolving) and Steihaug (to avoid negative curvature) criteria. Finally, the Armijo line search uses backtracking to find \\alpha such that a sufficient reduction in the cost functional is achieved. More specifically, we use backtracking to find \\alpha such that: J( a + \\alpha \\delta a ) \\leq J(a) + \\alpha c_{\\rm armijo} (\\delta a,g). # define parameters for the optimization tol = 1e-8 c = 1e-4 maxiter = 12 plot_on = False # initialize iter counters iter = 1 total_cg_iter = 0 converged = False # initializations g, a_delta = Vector(), Vector() R.init_vector(a_delta,0) R.init_vector(g,0) a_prev = Function(Va) print(\"Nit CGit cost misfit reg sqrt(-G*D) ||grad|| alpha tolcg\") while iter < maxiter and not converged: # assemble matrix C C = assemble(C_equ) # solve the adoint problem adjoint_A, adjoint_RHS = assemble_system(a_adj, L_adj, bc_adj) solve(adjoint_A, p.vector(), adjoint_RHS) # assemble W_ua and R Wua = assemble (Wua_equ) Raa = assemble (Raa_equ) # evaluate the gradient CT_p = Vector() C.init_vector(CT_p,1) C.transpmult(p.vector(), CT_p) MG = CT_p + R * a.vector() solve(M, g, MG) # calculate the norm of the gradient grad2 = g.inner(MG) gradnorm = sqrt(grad2) # set the CG tolerance (use Eisenstat\u2013Walker termination criterion) if iter == 1: gradnorm_ini = gradnorm tolcg = min(0.5, sqrt(gradnorm/gradnorm_ini)) # define the Hessian apply operator (with preconditioner) Hess_Apply = HessianOperator(R, Raa, C, state_A, adjoint_A, W, Wua, use_gaussnewton=(iter<6) ) P = R + gamma * M Psolver = PETScKrylovSolver(\"cg\", amg_method()) Psolver.set_operator(P) solver = CGSolverSteihaug() solver.set_operator(Hess_Apply) solver.set_preconditioner(Psolver) solver.parameters[\"rel_tolerance\"] = tolcg solver.parameters[\"zero_initial_guess\"] = True solver.parameters[\"print_level\"] = -1 # solve the Newton system H a_delta = - MG solver.solve(a_delta, -MG) total_cg_iter += Hess_Apply.cgiter # linesearch alpha = 1 descent = 0 no_backtrack = 0 a_prev.assign(a) while descent == 0 and no_backtrack < 10: a.vector().axpy(alpha, a_delta ) # solve the state/forward problem state_A, state_b = assemble_system(a_state, L_state, bc_state) solve(state_A, u.vector(), state_b) # evaluate cost [cost_new, misfit_new, reg_new] = cost(u, ud, a, W, R) # check if Armijo conditions are satisfied if cost_new < cost_old + alpha * c * MG.inner(a_delta): cost_old = cost_new descent = 1 else: no_backtrack += 1 alpha *= 0.5 a.assign(a_prev) # reset a # calculate sqrt(-G * D) graddir = sqrt(- MG.inner(a_delta) ) sp = \"\" print(\"%2d %2s %2d %3s %8.5e %1s %8.5e %1s %8.5e %1s %8.5e %1s %8.5e %1s %5.2f %1s %5.3e\" % \\ (iter, sp, Hess_Apply.cgiter, sp, cost_new, sp, misfit_new, sp, reg_new, sp, \\ graddir, sp, gradnorm, sp, alpha, sp, tolcg) ) if plot_on: nb.multi1_plot([a,u,p], [\"a\",\"u\",\"p\"], same_colorbar=False) plt.show() # check for convergence if gradnorm < tol and iter > 1: converged = True print(\"Newton's method converged in \",iter,\" iterations\") print(\"Total number of CG iterations: \", total_cg_iter) iter += 1 if not converged: print(\"Newton's method did not converge in \", maxiter, \" iterations\") Nit CGit cost misfit reg sqrt(-G*D) ||grad|| alpha tolcg 1 1 1.12708e-05 1.12708e-05 1.33979e-11 1.56540e-02 3.79427e-04 1.00 5.000e-01 2 1 7.79732e-07 7.79695e-07 3.67737e-11 4.68278e-03 5.35002e-05 1.00 3.755e-01 3 1 3.10620e-07 3.10571e-07 4.91259e-11 9.71633e-04 7.13895e-06 1.00 1.372e-01 4 5 1.92183e-07 1.62405e-07 2.97780e-08 4.51694e-04 1.00276e-06 1.00 5.141e-02 5 1 1.86913e-07 1.57119e-07 2.97941e-08 1.02668e-04 6.12750e-07 1.00 4.019e-02 6 12 1.80408e-07 1.37719e-07 4.26890e-08 1.15975e-04 2.24111e-07 1.00 2.430e-02 7 5 1.80331e-07 1.38935e-07 4.13963e-08 1.23223e-05 4.17399e-08 1.00 1.049e-02 8 15 1.80330e-07 1.39056e-07 4.12734e-08 1.74451e-06 3.43216e-09 1.00 3.008e-03 Newton's method converged in 8 iterations Total number of CG iterations: 41 nb.multi1_plot([atrue, a], [\"atrue\", \"a\"]) nb.multi1_plot([u,p], [\"u\",\"p\"], same_colorbar=False) plt.show() Copyright (c) 2016-2018, The University of Texas at Austin & University of California, Merced. All Rights reserved. See file COPYRIGHT for details. This file is part of the hIPPYlib library. For more information and source code availability see https://hippylib.github.io. hIPPYlib is free software; you can redistribute it and/or modify it under the terms of the GNU General Public License (as published by the Free Software Foundation) version 2.0 dated June 1991.","title":"The inexact Newton-CG optimization with Armijo line search:"},{"location":"tutorials_v1.6.0/3_SubsurfaceBayesian/","text":"\\def\\data{\\bf d_\\rm{obs}} \\def\\vec{\\bf} \\def\\m{\\bf m} \\def\\map{\\bf m_{\\text{MAP}}} \\def\\postcov{\\bf \\Gamma_{\\text{post}}} \\def\\prcov{\\bf \\Gamma_{\\text{prior}}} \\def\\matrix{\\bf} \\def\\Hmisfit{\\bf H_{\\text{misfit}}} \\def\\HT{\\tilde{\\bf H}_{\\text{misfit}}} \\def\\diag{diag} \\def\\Vr{\\matrix V_r} \\def\\Wr{\\matrix W_r} \\def\\Ir{\\matrix I_r} \\def\\Dr{\\matrix D_r} \\def\\H{\\matrix H} Example: Bayesian quantification of parameter uncertainty: Estimating the (Gaussian) posterior pdf of the coefficient parameter field in an elliptic PDE In this example we tackle the problem of quantifying the uncertainty in the solution of an inverse problem governed by an elliptic PDE via the Bayesian inference framework. Hence, we state the inverse problem as a problem of statistical inference over the space of uncertain parameters, which are to be inferred from data and a physical model. The resulting solution to the statistical inverse problem is a posterior distribution that assigns to any candidate set of parameter fields our belief (expressed as a probability) that a member of this candidate set is the ``true'' parameter field that gave rise to the observed data. For simplicity, in what follows we give finite-dimensional expressions (i.e., after discretization of the parameter space) for the Bayesian formulation of the inverse problem. Bayes' Theorem: The posterior probability distribution combines the prior pdf \\pi_{\\text{prior}}(\\m) over the parameter space, which encodes any knowledge or assumptions about the parameter space that we may wish to impose before the data are considered, with a likelihood pdf \\pi_{\\text{like}}(\\vec{d}_{\\text{obs}} \\; | \\; \\m) , which explicitly represents the probability that a given set of parameters \\m might give rise to the observed data \\vec{d}_{\\text{obs}} \\in \\mathbb{R}^m , namely: + + + + + + +
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Tutorial

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These tutorials are the best place to learn about the basic features and the algorithms in hIPPYlib.

+
    +
  1. FEniCS101 notebook illustrates the use of FEniCS for the solution of a linear boundary value problem.
  2. +
  3. Poisson Deterministic notebook illustrates how to compute gradient/Hessian information and solve a non-linear parameter inversion for the Poisson equation in a deterministic setting.
  4. +
  5. Subsurface Bayesian notebook illustrates how to solve a non-linear parameter inversion for the Poisson equation in a Bayesian setting.
  6. +
  7. Advection-Diffusion Bayesian notebook illustrates how to solve a time-dependent linear inverse problem in a Bayesian setting.
  8. +
  9. Hessian Spectrum notebook illustrates the spectral property of the Hessian operator for a linear source inversion problem.
  10. +
+

The interactive ipython notebooks are located in the tutorial folder of the hIPPYlib release.

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To run the notebooks follow these instructions.

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  1. Open a FEniCS terminal and type
  2. +
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$ cd tutorial
+$ jupyter notebook
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  1. A new tab will open in your web-brower showing the notebooks.
  2. +
  3. Click on the notebook you would like to use.
  4. +
  5. To run all the code in the notebook simply click on Cell --> Run All.
  6. +
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For more information on installing ipython and using notebooks see here.

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+ +
+
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Copyright © 2016-2023 The University of Texas at Austin, University of California Merced.
Copyright © 2018-2020 The University of Texas at Austin, University of California Merced, Washington University in St. Louis.
This material is based on work partially supported by the National Science Foundation under Grants No ACI-1550593, ACI-1550547.

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Documentation built with MkDocs.

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+ + + + + + + + + + + diff --git a/tutorial_v2.3.0/index.html b/tutorial_v2.3.0/index.html new file mode 100644 index 0000000..80f7b22 --- /dev/null +++ b/tutorial_v2.3.0/index.html @@ -0,0 +1,316 @@ + + + + + + + + + + + README - hIPPYlib + + + + + + + + + + + +
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Tutorial hIPPYlib 2.3.0

+

These tutorials are the best place to learn about the basic features and the algorithms in hIPPYlib.

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For the complete API reference click here.

+
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    +
  1. FEniCS101 notebook illustrates the use of FEniCS for the solution of a linear boundary value problem.
  2. +
  3. Poisson Deterministic notebook illustrates how to compute gradient/Hessian information and solve a non-linear parameter inversion for the Poisson equation in a deterministic setting.
  4. +
  5. Subsurface Bayesian notebook illustrates how to solve a non-linear parameter inversion for the Poisson equation in a Bayesian setting
  6. +
  7. Advection-Diffusion Bayesian notebook illustrates how to solve a time-dependent linear inverse problem in a Bayesian setting.
  8. +
  9. Hessian Spectrum notebook illustrates the spectral property of the Hessian operator for a linear source inversion problem.
  10. +
+
+

Tutorials 3 and 4 contain a line-by-line explanation of the model problems presented in hIPPYlib: An Extensible Software Framework for Large-Scale Inverse Problems Governed by PDEs; Part I: Deterministic Inversion and Linearized Bayesian Inference.

+
+

Interactive tutorials

+

hIPPYlib 2.3.0 is available as source code or prebuilt docker image.

+

The interactive ipython notebooks are located in the tutorial folder of the hIPPYlib release.

+

To run the notebooks follow these instructions.

+
    +
  1. Open a FEniCS terminal and type
  2. +
+
$ cd tutorial
+$ jupyter notebook
+
+
    +
  1. A new tab will open in your web-brower showing the notebooks.
  2. +
  3. Click on the notebook you would like to use.
  4. +
  5. To run all the code in the notebook simply click on Cell --> Run All.
  6. +
+

For more information on installing ipython and using notebooks see here.

+

Additional resources

+

For additional resources and tutorials please see the teaching material for the +2018 Gene Golub SIAM Summer School on Inverse Problems: Systematic Integration of Data with Models under Uncertainty available here.

+
+
+ +
+
+

Copyright © 2016-2023 The University of Texas at Austin, University of California Merced.
Copyright © 2018-2020 The University of Texas at Austin, University of California Merced, Washington University in St. Louis.
This material is based on work partially supported by the National Science Foundation under Grants No ACI-1550593, ACI-1550547.

+

Documentation built with MkDocs.

+
+ + + + + + + + + + + diff --git a/tutorial_v3.0.0/index.html b/tutorial_v3.0.0/index.html new file mode 100644 index 0000000..69103c1 --- /dev/null +++ b/tutorial_v3.0.0/index.html @@ -0,0 +1,315 @@ + + + + + + + + + + + README - hIPPYlib + + + + + + + + + + + +
+
+
+
+ +

Tutorial - Version 3.0.0

+

These tutorials are the best place to learn about the basic features and the algorithms in hIPPYlib.

+
+

For the complete API reference click here.

+
+
    +
  1. FEniCS101 notebook illustrates the use of FEniCS for the solution of a linear boundary value problem.
  2. +
  3. Poisson Deterministic notebook illustrates how to compute gradient/Hessian information and solve a non-linear parameter inversion for the Poisson equation in a deterministic setting.
  4. +
  5. Subsurface Bayesian notebook illustrates how to solve a non-linear parameter inversion for the Poisson equation in a Bayesian setting.
  6. +
  7. Advection-Diffusion Bayesian notebook illustrates how to solve a time-dependent linear inverse problem in a Bayesian setting.
  8. +
  9. Hessian Spectrum notebook illustrates the spectral property of the Hessian operator for a linear source inversion problem.
  10. +
+

Interactive tutorials

+

The interactive ipython notebooks are located in the tutorial folder of the hIPPYlib repository.

+

To run the notebooks follow these instructions.

+
    +
  1. Open a FEniCS terminal and type
  2. +
+
$ cd tutorial
+$ jupyter notebook
+
+
    +
  1. A new tab will open in your web-brower showing the notebooks.
  2. +
  3. Click on the notebook you would like to use.
  4. +
  5. To run all the code in the notebook simply click on Cell --> Run All.
  6. +
+

For more information on installing ipython and using notebooks see here.

+

Additional resources

+

For additional resources and tutorials please see the teaching material for the +2018 Gene Golub SIAM Summer School on Inverse Problems: Systematic Integration of Data with Models under Uncertainty available here.

+
+

These tutorials require hIPPYlib versions 2.1.1 - 2.3.0.

+
+
+
+ +
+
+

Copyright © 2016-2023 The University of Texas at Austin, University of California Merced.
Copyright © 2018-2020 The University of Texas at Austin, University of California Merced, Washington University in St. Louis.
This material is based on work partially supported by the National Science Foundation under Grants No ACI-1550593, ACI-1550547.

+

Documentation built with MkDocs.

+
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+
+ +
+ +

FEniCS101 Tutorial

+

In this tutorial we consider the boundary value problem (BVP)

+

+ +

+

where , and and are the union of +the left and right, and top and bottom boundaries of , +respectively.

+

Here + +

+

The exact solution is + +

+

Weak formulation

+

Let us define the Hilbert spaces as + + +

+

To obtain the weak formulation, we multiply the PDE by an arbitrary function and integrate over the domain leading to

+

+ +

+

Then, integration by parts the non-conforming term gives

+

+ +

+

Finally by recalling that on and that on , we find the weak formulation:

+

Find * such that* + +

+

1. Load modules

+

To start we load the following modules:

+
    +
  • +

    dolfin: the python/C++ interface to FEniCS

    +
  • +
  • +

    math: the python module for mathematical functions

    +
  • +
  • +

    numpy: a python package for linear algebra

    +
  • +
  • +

    matplotlib: a python package used for plotting the results

    +
  • +
+
from __future__ import absolute_import, division, print_function
+
+from dolfin import *
+
+import math
+import numpy as np
+import logging
+
+import matplotlib.pyplot as plt
+%matplotlib inline
+import nb
+
+logging.getLogger('FFC').setLevel(logging.WARNING)
+logging.getLogger('UFL').setLevel(logging.WARNING)
+set_log_active(False)
+
+

2. Define the mesh and the finite element space

+

We construct a triangulation (mesh) of the computational domain with n elements in each direction.

+

On the mesh , we then define the finite element space consisting of globally continuous piecewise polinomials functions. The degree variable defines the polinomial degree.

+
n = 16
+degree = 1
+mesh = UnitSquareMesh(n, n)
+nb.plot(mesh)
+
+Vh  = FunctionSpace(mesh, 'Lagrange', degree)
+print("dim(Vh) = ", Vh.dim())
+
+
dim(Vh) =  289
+
+

png

+

3. Define boundary labels

+

To partition the boundary of in the subdomains , , , we assign a unique label boundary_parts to each of part of .

+
class TopBoundary(SubDomain):
+    def inside(self, x, on_boundary):
+        return on_boundary and abs(x[1] - 1) < DOLFIN_EPS
+
+class BottomBoundary(SubDomain):
+    def inside(self, x, on_boundary):
+        return on_boundary and abs(x[1]) < DOLFIN_EPS
+
+class LeftBoundary(SubDomain):
+    def inside(self, x, on_boundary):
+        return on_boundary and abs(x[0]) < DOLFIN_EPS
+
+class RightBoundary(SubDomain):
+    def inside(self, x, on_boundary):
+        return on_boundary and abs(x[0] - 1) < DOLFIN_EPS
+
+boundary_parts = FacetFunction("size_t", mesh)
+boundary_parts.set_all(0)
+
+Gamma_top = TopBoundary()
+Gamma_top.mark(boundary_parts, 1)
+Gamma_bottom = BottomBoundary()
+Gamma_bottom.mark(boundary_parts, 2)
+Gamma_left = LeftBoundary()
+Gamma_left.mark(boundary_parts, 3)
+Gamma_right = RightBoundary()
+Gamma_right.mark(boundary_parts, 4)
+
+

4. Define the coefficients of the PDE and the boundary conditions

+

We first define the coefficients of the PDE using the Constant and Expression classes. Constant is used to define coefficients that do not depend on the space coordinates, Expression is used to define coefficients that are a known function of the space coordinates x[0] (x-axis direction) and x[1] (y-axis direction).

+

In the finite element method community, Dirichlet boundary conditions are also known as essential boundary conditions since they are imposed directly in the definition of the finite element space. In FEniCS, we use the class DirichletBC to indicate this type of condition.

+

On the other hand, Newman boundary conditions are also known as natural boundary conditions since they are weakly imposed as boundary integrals in the variational formulation (weak form). In FEniCS, we create a new boundary measure ds[i] to integrate over the portion of the boundary marked with label i.

+
u_L = Constant(0.)
+u_R = Constant(0.)
+
+sigma_bottom = Expression('-(pi/2.0)*sin(2*pi*x[0])', degree=5)
+sigma_top    = Constant(0.)
+
+f = Expression('(4.0*pi*pi+pi*pi/4.0)*(sin(2*pi*x[0])*sin((pi/2.0)*x[1]))', degree=5)
+
+bcs = [DirichletBC(Vh, u_L, boundary_parts, 3),
+       DirichletBC(Vh, u_R, boundary_parts, 4)]
+
+ds = Measure("ds", subdomain_data=boundary_parts)
+
+

5. Define and solve the variational problem

+

We also define two special types of functions: the TrialFunction u and the TestFunction v. These special types of function are used by FEniCS to generate the finite element vectors and matrices which stem from the weak formulation of the PDE.

+

More specifically, by denoting by the finite element basis for the space , a function can be written as + +where represents the coefficients in the finite element expansion of .

+

We then define

+
    +
  • +

    the bilinear form ;

    +
  • +
  • +

    the linear form .

    +
  • +
+

We can then solve the variational problem

+

Find + such that + +

+

using directly the built-in solve method in FEniCS.

+

NOTE: As an alternative one can also assemble the finite element matrix and the right hand side that stems from the discretization of and , and then solve the linear system + +where

+
    +
  • +

    + is the vector collecting the coefficient of the finite element expasion of ,

    +
  • +
  • +

    the entries of the matrix A are such that ,

    +
  • +
  • +

    the entries of the right hand side b are such that .

    +
  • +
+
u = TrialFunction(Vh)
+v = TestFunction(Vh)
+a = inner(nabla_grad(u), nabla_grad(v))*dx
+L = f*v*dx + sigma_top*v*ds(1) + sigma_bottom*v*ds(2)
+
+uh = Function(Vh)
+
+#solve(a == L, uh, bcs=bcs)
+A, b = assemble_system(a,L, bcs=bcs)
+solve(A, uh.vector(), b, "cg")
+
+nb.plot(uh)
+
+

png

+

6. Compute the discretization error

+

For this problem, the exact solution is known. +We can therefore compute the following norms of the discretization error (i.e. the of the difference between the finite element solution and the exact solution ) + +and + +

+
u_e = Expression('sin(2*pi*x[0])*sin((pi/2.0)*x[1])', degree=5)
+grad_u_e = Expression( ('2*pi*cos(2*pi*x[0])*sin((pi/2.0)*x[1])', 'pi/2.0*sin(2*pi*x[0])*cos((pi/2.0)*x[1])'), degree=5)
+
+err_L2 = sqrt( assemble( (uh-u_e)**2*dx ) )
+err_grad = sqrt( assemble( inner(nabla_grad(uh) - grad_u_e, nabla_grad(uh) - grad_u_e)*dx ) )
+err_H1 = sqrt( err_L2**2 + err_grad**2)
+
+print("|| u_h - u_e ||_L2 = ", err_L2)
+print("|| u_h - u_e ||_H1 = ", err_H1)
+
+
|| u_h - u_e ||_L2 =  0.00880525372208
+|| u_h - u_e ||_H1 =  0.396718952514
+
+

7. Convergence of the finite element method

+

We now verify numerically a well-known convergence result for the finite element method.

+

Let denote with the polynomial degree of the finite element space, and assume that the solution is at least in . Then we have + +

+

In the code below, the function compute(n, degree) solves the PDE using a mesh with n elements in each direction and finite element spaces of polinomial order degree.

+

The figure below shows the discretization errors in the and as a function of the mesh size () for piecewise linear (P1, ) and piecewise quadratic (P2, ) finite elements. We observe that numerical results are consistent with the finite element convergence theory. In particular:

+
    +
  • +

    for piecewise linear finite element P1 we observe first order convergence in the -norm and second order convergence in the -norm;

    +
  • +
  • +

    for piecewise quadratic finite element P2 we observe second order convergence in the -norm and third order convergence in the -norm.

    +
  • +
+
def compute(n, degree):
+    mesh = UnitSquareMesh(n, n)
+    Vh  = FunctionSpace(mesh, 'Lagrange', degree)
+    boundary_parts = FacetFunction("size_t", mesh)
+    boundary_parts.set_all(0)
+
+    Gamma_top = TopBoundary()
+    Gamma_top.mark(boundary_parts, 1)
+    Gamma_bottom = BottomBoundary()
+    Gamma_bottom.mark(boundary_parts, 2)
+    Gamma_left = LeftBoundary()
+    Gamma_left.mark(boundary_parts, 3)
+    Gamma_right = RightBoundary()
+    Gamma_right.mark(boundary_parts, 4)
+
+    bcs = [DirichletBC(Vh, u_L, boundary_parts, 3), DirichletBC(Vh, u_R, boundary_parts, 4)]
+    ds = Measure("ds", subdomain_data=boundary_parts)
+
+    u = TrialFunction(Vh)
+    v = TestFunction(Vh)
+    a = inner(nabla_grad(u), nabla_grad(v))*dx
+    L = f*v*dx + sigma_top*v*ds(1) + sigma_bottom*v*ds(2)
+    uh = Function(Vh)
+    solve(a == L, uh, bcs=bcs)
+    err_L2 = sqrt( assemble( (uh-u_e)**2*dx ) )
+    err_grad = sqrt( assemble( inner(nabla_grad(uh) - grad_u_e, nabla_grad(uh) - grad_u_e)*dx ) )
+    err_H1 = sqrt( err_L2**2 + err_grad**2)
+
+    return err_L2, err_H1
+
+nref = 5
+n = 8*np.power(2,np.arange(0,nref))
+h = 1./n
+
+err_L2_P1 = np.zeros(nref)
+err_H1_P1 = np.zeros(nref)
+err_L2_P2 = np.zeros(nref)
+err_H1_P2 = np.zeros(nref)
+
+for i in range(nref):
+    err_L2_P1[i], err_H1_P1[i] = compute(n[i], 1)
+    err_L2_P2[i], err_H1_P2[i] = compute(n[i], 2)
+
+plt.figure(figsize=(15,5))
+
+plt.subplot(121)
+plt.loglog(h, err_H1_P1, '-or', label="H1 error")
+plt.loglog(h, err_L2_P1, '-*b', label="L2 error")
+plt.loglog(h, h*.5*err_H1_P1[0]/h[0], '--g', label="First Order")
+plt.loglog(h, np.power(h,2)*.5*np.power( err_L2_P1[0]/h[0], 2), '-.k', label="Second Order")
+plt.xlabel("Mesh size h")
+plt.ylabel("Error")
+plt.title("P1 Finite Element")
+plt.legend(loc = 'lower right')
+
+
+plt.subplot(122)
+plt.loglog(h, err_H1_P2, '-or', label="H1 error")
+plt.loglog(h, err_L2_P2, '-*b', label="L2 error")
+plt.loglog(h, np.power(h/h[0],2)*.5*err_H1_P2[0], '--g', label="Second Order")
+plt.loglog(h, np.power(h/h[0],3)*.5*err_L2_P2[0], '-.k', label="Third Order")
+plt.xlabel("Mesh size h")
+plt.ylabel("Error")
+plt.title("P2 Finite Element")
+plt.legend(loc='lower right')
+
+plt.show()
+
+

png

+

Copyright (c) 2016-2018, The University of Texas at Austin & University of California, Merced. +All Rights reserved. +See file COPYRIGHT for details.

+

This file is part of the hIPPYlib library. For more information and source code +availability see https://hippylib.github.io.

+

hIPPYlib is free software; you can redistribute it and/or modify it under the terms of the GNU General Public License (as published by the Free Software Foundation) version 2.0 dated June 1991.

+
+
+ +
+
+

Copyright © 2016-2023 The University of Texas at Austin, University of California Merced.
Copyright © 2018-2020 The University of Texas at Austin, University of California Merced, Washington University in St. Louis.
This material is based on work partially supported by the National Science Foundation under Grants No ACI-1550593, ACI-1550547.

+

Documentation built with MkDocs.

+
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+ +
+
+ +
+ +

Coefficient field inversion in an elliptic partial differential equation

+

We consider the estimation of a coefficient in an elliptic partial +differential equation as a model problem. Depending on the +interpretation of the unknowns and the type of measurements, this +model problem arises, for instance, in inversion for groundwater flow +or heat conductivity. It can also be interpreted as finding a +membrane with a certain spatially varying stiffness. Let +, be an open, bounded +domain and consider the following problem:

+

+ +

+

where is the solution of

+

+ +

+

Here the unknown coefficient field, denotes (possibly noisy) data, a given force, and the regularization parameter.

+

The variational (or weak) form of the state equation:

+

Find such that +where is the space of functions vanishing on with square integrable derivatives. Here, denotes the -inner product, i.e, for scalar functions defined on we denote .

+

Optimality System:

+

The Lagrangian functional , which we use as a tool to derive the optimality system, is given by

+

+ +

+

The Lagrange multiplier theory shows that, at a solution all variations of the Lagrangian functional with respect to all variables must vanish. These variations of with respect to in directions are given by

+

+ +

+

where the variations are taken from the same spaces as .

+

The gradient of the cost functional therefore is

+

+ +

+

Inexact Newton-CG:

+

Newton's method requires second-order variational derivatives of the Lagrangian . Written in abstract form, it computes an update direction from the following Newton step for the Lagrangian functional:

+

+ +

+

for all variations , where and denote the first and +second variations of the Lagrangian. For the elliptic parameter inversion problem, this Newton step (written in variatonal form) is as follows: Find as the solution of the linear system

+

+ +

+

for all .

+

Discrete Newton system:

+

+ +The discretized Newton step: denote the vectors corresponding to the discretization of the functions by and . Then, the discretization of the above system is given by the following symmetric linear system:

+

+ +

+

where , , , and are the components of the Hessian matrix of the Lagrangian, and are the Jacobian of the state equation with respect to the state and the control variables, respectively and , , and are the discrete gradients of the Lagrangian with respect to , and , respectively.

+

Reduced Hessian apply:

+

To eliminate the incremental state and adjoint variables, and , from the first and last equations we use

+

+ +

+

This results in the following reduced linear system for the Newton step

+

+ +

+

with the reduced Hessian applied to a vector given by

+

+ +

+

Goals:

+

By the end of this notebook, you should be able to:

+
    +
  • solve the forward and adjoint Poisson equations
  • +
  • understand the inverse method framework
  • +
  • visualise and understand the results
  • +
  • modify the problem and code
  • +
+

Mathematical tools used:

+
    +
  • Finite element method
  • +
  • Derivation of gradiant and Hessian via the adjoint method
  • +
  • inexact Newton-CG
  • +
  • Armijo line search
  • +
+

List of software used:

+
    +
  • FEniCS, a parallel finite element element library for the discretization of partial differential equations
  • +
  • PETSc, for scalable and efficient linear algebra operations and solvers
  • +
  • Matplotlib, a python package used for plotting the results
  • +
  • Numpy, a python package for linear algebra
  • +
+

Set up

+

Import dependencies

+
from __future__ import absolute_import, division, print_function
+
+from dolfin import *
+
+import sys
+import os
+sys.path.append( os.environ.get('HIPPYLIB_BASE_DIR', "../") )
+from hippylib import *
+
+import numpy as np
+import logging
+
+import matplotlib.pyplot as plt
+%matplotlib inline
+import nb
+
+
+logging.getLogger('FFC').setLevel(logging.WARNING)
+logging.getLogger('UFL').setLevel(logging.WARNING)
+set_log_active(False)
+
+np.random.seed(seed=1)
+
+

Model set up:

+

As in the introduction, the first thing we need to do is set up the numerical model. In this cell, we set the mesh, the finite element functions corresponding to state, adjoint and coefficient/gradient variables, and the corresponding test functions and the parameters for the optimization.

+
# create mesh and define function spaces
+nx = 64
+ny = 64
+mesh = UnitSquareMesh(nx, ny)
+Va = FunctionSpace(mesh, 'Lagrange', 1)
+Vu = FunctionSpace(mesh, 'Lagrange', 2)
+
+# The true and inverted parameter
+atrue = interpolate(Expression('log(2 + 7*(pow(pow(x[0] - 0.5,2) + pow(x[1] - 0.5,2),0.5) > 0.2))', degree=5),Va)
+a = interpolate(Expression("log(2.0)", degree=1),Va)
+
+# define function for state and adjoint
+u = Function(Vu)
+p = Function(Vu)
+
+# define Trial and Test Functions
+u_trial, p_trial, a_trial = TrialFunction(Vu), TrialFunction(Vu), TrialFunction(Va)
+u_test, p_test, a_test = TestFunction(Vu), TestFunction(Vu), TestFunction(Va)
+
+# initialize input functions
+f = Constant("1.0")
+u0 = Constant("0.0")
+
+# plot
+plt.figure(figsize=(15,5))
+nb.plot(mesh,subplot_loc=121, mytitle="Mesh", show_axis='on')
+nb.plot(atrue,subplot_loc=122, mytitle="True parameter field")
+plt.show()
+
+

png

+
# set up dirichlet boundary conditions
+def boundary(x,on_boundary):
+    return on_boundary
+
+bc_state = DirichletBC(Vu, u0, boundary)
+bc_adj = DirichletBC(Vu, Constant(0.), boundary)
+
+

Set up synthetic observations:

+
    +
  • Propose a coefficient field shown above
  • +
  • +

    The weak form of the pde: + Find such that .

    +
  • +
  • +

    Perturb the solution: , where +

    +
  • +
+
# noise level
+noise_level = 0.05
+
+# weak form for setting up the synthetic observations
+a_goal = inner(exp(atrue) * nabla_grad(u_trial), nabla_grad(u_test)) * dx
+L_goal = f * u_test * dx
+
+# solve the forward/state problem to generate synthetic observations
+goal_A, goal_b = assemble_system(a_goal, L_goal, bc_state)
+
+utrue = Function(Vu)
+solve(goal_A, utrue.vector(), goal_b)
+
+ud = Function(Vu)
+ud.assign(utrue)
+
+# perturb state solution and create synthetic measurements ud
+# ud = u + ||u||/SNR * random.normal
+MAX = ud.vector().norm("linf")
+noise = Vector()
+goal_A.init_vector(noise,1)
+noise.set_local( noise_level * MAX * np.random.normal(0, 1, Vu.dim()))
+bc_adj.apply(noise)
+
+ud.vector().axpy(1., noise)
+
+# plot
+nb.multi1_plot([utrue, ud], ["State solution with atrue", "Synthetic observations"])
+plt.show()
+
+

png

+

The cost function evaluation:

+

+ +

+

In the code below, and are symmetric positive definite matrices that stem from finite element discretization of the misfit and regularization component of the cost functional, respectively.

+
# regularization parameter
+gamma = 1e-8
+
+# weak for for setting up the misfit and regularization compoment of the cost
+W_equ   = inner(u_trial, u_test) * dx
+R_equ   = gamma * inner(nabla_grad(a_trial), nabla_grad(a_test)) * dx
+
+W = assemble(W_equ)
+R = assemble(R_equ)
+
+# refine cost function
+def cost(u, ud, a, W, R):
+    diff = u.vector() - ud.vector()
+    reg = 0.5 * a.vector().inner(R*a.vector() ) 
+    misfit = 0.5 * diff.inner(W * diff)
+    return [reg + misfit, misfit, reg]
+
+

Setting up the state equations, right hand side for the adjoint and the necessary matrices:

+

+ +

+
# weak form for setting up the state equation
+a_state = inner(exp(a) * nabla_grad(u_trial), nabla_grad(u_test)) * dx
+L_state = f * u_test * dx
+
+# weak form for setting up the adjoint equation
+a_adj = inner(exp(a) * nabla_grad(p_trial), nabla_grad(p_test)) * dx
+L_adj = -inner(u - ud, p_test) * dx
+
+# weak form for setting up matrices
+Wua_equ = inner(exp(a) * a_trial * nabla_grad(p_test), nabla_grad(p)) * dx
+C_equ   = inner(exp(a) * a_trial * nabla_grad(u), nabla_grad(u_test)) * dx
+Raa_equ = inner(exp(a) * a_trial * a_test *  nabla_grad(u),  nabla_grad(p)) * dx
+
+M_equ   = inner(a_trial, a_test) * dx
+
+# assemble matrix M
+M = assemble(M_equ)
+
+

Initial guess

+

We solve the state equation and compute the cost functional for the initial guess of the parameter a_ini

+
# solve state equation
+state_A, state_b = assemble_system (a_state, L_state, bc_state)
+solve (state_A, u.vector(), state_b)
+
+# evaluate cost
+[cost_old, misfit_old, reg_old] = cost(u, ud, a, W, R)
+
+# plot
+plt.figure(figsize=(15,5))
+nb.plot(a,subplot_loc=121, mytitle="a_ini", vmin=atrue.vector().min(), vmax=atrue.vector().max())
+nb.plot(u,subplot_loc=122, mytitle="u(a_ini)")
+plt.show()
+
+

png

+

The reduced Hessian apply to a vector v:

+

Here we describe how to apply the reduced Hessian operator to a vector v. For an opportune choice of the regularization, the reduced Hessian operator evaluated in a neighborhood of the solution is positive define, whereas far from the solution the reduced Hessian may be indefinite. On the constrary, the Gauss-Newton approximation of the Hessian is always positive defined.

+

For this reason, it is beneficial to perform a few initial Gauss-Newton steps (5 in this particular example) to accelerate the convergence of the inexact Newton-CG algorithm.

+

The Hessian apply reads: + +

+

The Gauss-Newton Hessian apply is obtained by dropping the second derivatives operators , , and : + +

+
# Class HessianOperator to perform Hessian apply to a vector
+class HessianOperator():
+    cgiter = 0
+    def __init__(self, R, Raa, C, A, adj_A, W, Wua, use_gaussnewton=False):
+        self.R = R
+        self.Raa = Raa
+        self.C = C
+        self.A = A
+        self.adj_A = adj_A
+        self.W = W
+        self.Wua = Wua
+        self.use_gaussnewton = use_gaussnewton
+
+        # incremental state
+        self.du = Vector()
+        self.A.init_vector(self.du,0)
+
+        #incremental adjoint
+        self.dp = Vector()
+        self.adj_A.init_vector(self.dp,0)
+
+        # auxiliary vectors
+        self.CT_dp = Vector()
+        self.C.init_vector(self.CT_dp, 1)
+        self.Wua_du = Vector()
+        self.Wua.init_vector(self.Wua_du, 1)
+
+    def init_vector(self, v, dim):
+        self.R.init_vector(v,dim)
+
+    # Hessian performed on x, output as generic vector y
+    def mult(self, v, y):
+        self.cgiter += 1
+        y.zero()
+        if self.use_gaussnewton:
+            self.mult_GaussNewton(v,y)
+        else:
+            self.mult_Newton(v,y)
+
+    # define (Gauss-Newton) Hessian apply H * v
+    def mult_GaussNewton(self, v, y):
+
+        #incremental forward
+        rhs = -(self.C * v)
+        bc_adj.apply(rhs)
+        solve (self.A, self.du, rhs)
+
+        #incremental adjoint
+        rhs = - (self.W * self.du)
+        bc_adj.apply(rhs)
+        solve (self.adj_A, self.dp, rhs)
+
+        # Reg/Prior term
+        self.R.mult(v,y)
+
+        # Misfit term
+        self.C.transpmult(self.dp, self.CT_dp)
+        y.axpy(1, self.CT_dp)
+
+    # define (Newton) Hessian apply H * v
+    def mult_Newton(self, v, y):
+
+        #incremental forward
+        rhs = -(self.C * v)
+        bc_adj.apply(rhs)
+        solve (self.A, self.du, rhs)
+
+        #incremental adjoint
+        rhs = -(self.W * self.du) -  self.Wua * v
+        bc_adj.apply(rhs)
+        solve (self.adj_A, self.dp, rhs)
+
+        #Reg/Prior term
+        self.R.mult(v,y)
+        y.axpy(1., Raa*v)
+
+        #Misfit term
+        self.C.transpmult(self.dp, self.CT_dp)
+        y.axpy(1., self.CT_dp)
+        self.Wua.transpmult(self.du, self.Wua_du)
+        y.axpy(1., self.Wua_du)
+
+ +

We solve the constrained optimization problem using the inexact Newton-CG method with Armijo line search.

+

The stopping criterion is based on a relative reduction of the norm of the gradient (i.e. ).

+

First, we compute the gradient by solving the state and adjoint equation for the current parameter , and then substituing the current state , parameter and adjoint variables in the weak form expression of the gradient: + +

+

Then, we compute the Newton direction by iteratively solving . +The Newton system is solved inexactly by early termination of conjugate gradient iterations via Eisenstat–Walker (to prevent oversolving) and Steihaug (to avoid negative curvature) criteria.

+

Finally, the Armijo line search uses backtracking to find such that a sufficient reduction in the cost functional is achieved. +More specifically, we use backtracking to find such that: + +

+
# define parameters for the optimization
+tol = 1e-8
+c = 1e-4
+maxiter = 12
+plot_on = False
+
+# initialize iter counters
+iter = 1
+total_cg_iter = 0
+converged = False
+
+# initializations
+g, a_delta = Vector(), Vector()
+R.init_vector(a_delta,0)
+R.init_vector(g,0)
+
+a_prev = Function(Va)
+
+print("Nit   CGit   cost          misfit        reg           sqrt(-G*D)    ||grad||       alpha  tolcg")
+
+while iter <  maxiter and not converged:
+
+    # assemble matrix C
+    C =  assemble(C_equ)
+
+    # solve the adoint problem
+    adjoint_A, adjoint_RHS = assemble_system(a_adj, L_adj, bc_adj)
+    solve(adjoint_A, p.vector(), adjoint_RHS)
+
+    # assemble W_ua and R
+    Wua = assemble (Wua_equ)
+    Raa = assemble (Raa_equ)
+
+    # evaluate the  gradient
+    CT_p = Vector()
+    C.init_vector(CT_p,1)
+    C.transpmult(p.vector(), CT_p)
+    MG = CT_p + R * a.vector()
+    solve(M, g, MG)
+
+    # calculate the norm of the gradient
+    grad2 = g.inner(MG)
+    gradnorm = sqrt(grad2)
+
+    # set the CG tolerance (use Eisenstat–Walker termination criterion)
+    if iter == 1:
+        gradnorm_ini = gradnorm
+    tolcg = min(0.5, sqrt(gradnorm/gradnorm_ini))
+
+    # define the Hessian apply operator (with preconditioner)
+    Hess_Apply = HessianOperator(R, Raa, C, state_A, adjoint_A, W, Wua, use_gaussnewton=(iter<6) )
+    P = R + gamma * M
+    Psolver = PETScKrylovSolver("cg", amg_method())
+    Psolver.set_operator(P)
+
+    solver = CGSolverSteihaug()
+    solver.set_operator(Hess_Apply)
+    solver.set_preconditioner(Psolver)
+    solver.parameters["rel_tolerance"] = tolcg
+    solver.parameters["zero_initial_guess"] = True
+    solver.parameters["print_level"] = -1
+
+    # solve the Newton system H a_delta = - MG
+    solver.solve(a_delta, -MG)
+    total_cg_iter += Hess_Apply.cgiter
+
+    # linesearch
+    alpha = 1
+    descent = 0
+    no_backtrack = 0
+    a_prev.assign(a)
+    while descent == 0 and no_backtrack < 10:
+        a.vector().axpy(alpha, a_delta )
+
+        # solve the state/forward problem
+        state_A, state_b = assemble_system(a_state, L_state, bc_state)
+        solve(state_A, u.vector(), state_b)
+
+        # evaluate cost
+        [cost_new, misfit_new, reg_new] = cost(u, ud, a, W, R)
+
+        # check if Armijo conditions are satisfied
+        if cost_new < cost_old + alpha * c * MG.inner(a_delta):
+            cost_old = cost_new
+            descent = 1
+        else:
+            no_backtrack += 1
+            alpha *= 0.5
+            a.assign(a_prev)  # reset a
+
+    # calculate sqrt(-G * D)
+    graddir = sqrt(- MG.inner(a_delta) )
+
+    sp = ""
+    print("%2d %2s %2d %3s %8.5e %1s %8.5e %1s %8.5e %1s %8.5e %1s %8.5e %1s %5.2f %1s %5.3e" % \
+        (iter, sp, Hess_Apply.cgiter, sp, cost_new, sp, misfit_new, sp, reg_new, sp, \
+         graddir, sp, gradnorm, sp, alpha, sp, tolcg) )
+
+    if plot_on:
+        nb.multi1_plot([a,u,p], ["a","u","p"], same_colorbar=False)
+        plt.show()
+
+    # check for convergence
+    if gradnorm < tol and iter > 1:
+        converged = True
+        print("Newton's method converged in ",iter,"  iterations")
+        print("Total number of CG iterations: ", total_cg_iter)
+
+    iter += 1
+
+if not converged:
+    print("Newton's method did not converge in ", maxiter, " iterations")
+
+
Nit   CGit   cost          misfit        reg           sqrt(-G*D)    ||grad||       alpha  tolcg
+ 1     1     1.12708e-05   1.12708e-05   1.33979e-11   1.56540e-02   3.79427e-04    1.00   5.000e-01
+ 2     1     7.79732e-07   7.79695e-07   3.67737e-11   4.68278e-03   5.35002e-05    1.00   3.755e-01
+ 3     1     3.10620e-07   3.10571e-07   4.91259e-11   9.71633e-04   7.13895e-06    1.00   1.372e-01
+ 4     5     1.92183e-07   1.62405e-07   2.97780e-08   4.51694e-04   1.00276e-06    1.00   5.141e-02
+ 5     1     1.86913e-07   1.57119e-07   2.97941e-08   1.02668e-04   6.12750e-07    1.00   4.019e-02
+ 6    12     1.80408e-07   1.37719e-07   4.26890e-08   1.15975e-04   2.24111e-07    1.00   2.430e-02
+ 7     5     1.80331e-07   1.38935e-07   4.13963e-08   1.23223e-05   4.17399e-08    1.00   1.049e-02
+ 8    15     1.80330e-07   1.39056e-07   4.12734e-08   1.74451e-06   3.43216e-09    1.00   3.008e-03
+Newton's method converged in  8   iterations
+Total number of CG iterations:  41
+
+
nb.multi1_plot([atrue, a], ["atrue", "a"])
+nb.multi1_plot([u,p], ["u","p"], same_colorbar=False)
+plt.show()
+
+

png

+

png

+

Copyright (c) 2016-2018, The University of Texas at Austin & University of California, Merced. +All Rights reserved. +See file COPYRIGHT for details.

+

This file is part of the hIPPYlib library. For more information and source code +availability see https://hippylib.github.io.

+

hIPPYlib is free software; you can redistribute it and/or modify it under the terms of the GNU General Public License (as published by the Free Software Foundation) version 2.0 dated June 1991.

+
+
+ +
+
+

Copyright © 2016-2023 The University of Texas at Austin, University of California Merced.
Copyright © 2018-2020 The University of Texas at Austin, University of California Merced, Washington University in St. Louis.
This material is based on work partially supported by the National Science Foundation under Grants No ACI-1550593, ACI-1550547.

+

Documentation built with MkDocs.

+
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    +
    + +
    + +

    + +

    +

    Example: Bayesian quantification of parameter uncertainty:

    +

    Estimating the (Gaussian) posterior pdf of the coefficient parameter field in an elliptic PDE

    +

    In this example we tackle the problem of quantifying the +uncertainty in the solution of an inverse problem governed by an +elliptic PDE via the Bayesian inference framework. +Hence, we state the inverse problem as a +problem of statistical inference over the space of uncertain +parameters, which are to be inferred from data and a physical +model. The resulting solution to the statistical inverse problem +is a posterior distribution that assigns to any candidate set of +parameter fields our belief (expressed as a probability) that a +member of this candidate set is the ``true'' parameter field that +gave rise to the observed data.

    +

    For simplicity, in what follows we give finite-dimensional expressions (i.e., after +discretization of the parameter space) for the Bayesian +formulation of the inverse problem.

    +

    Bayes' Theorem:

    +

    The posterior probability distribution combines the prior pdf + over the parameter space, which encodes +any knowledge or assumptions about the parameter space that we may +wish to impose before the data are considered, with a likelihood pdf +, which explicitly +represents the probability that a given set of parameters +might give rise to the observed data , namely:

    +

    + + +

    +

    Note that infinite-dimensional analog of Bayes' formula requires the use Radon-Nikodym derivatives instead of probability density functions.

    +

    Gaussian prior and noise:

    +

    The prior:

    +

    We consider a Gaussian prior with mean and covariance . The covariance is given by the discretization of the inverse of differential operator , where , control the correlation length and the variance of the prior operator. This choice of prior ensures that it is a trace-class operator, guaranteeing bounded pointwise variance and a well-posed infinite-dimensional Bayesian inverse problem

    +

    The likelihood:

    +

    + +

    +

    + +

    +

    Here is the parameter-to-observable map that takes a parameter vector and maps +it to the space observation vector .

    +

    The posterior:

    +

    + +

    +

    The Gaussian approximation of the posterior: +

    +

    The mean of this posterior distribution, , is the +parameter vector maximizing the posterior, and +is known as the maximum a posteriori (MAP) point. It can be found +by minimizing the negative log of the posterior, which amounts to +solving a deterministic inverse problem) with appropriately weighted norms,

    +

    + +

    +

    The posterior covariance matrix is then given by the inverse of +the Hessian matrix of at , namely

    +

    + +

    +

    The prior-preconditioned Hessian of the data misfit:

    +

    + +

    +

    The generalized eigenvalue problem:

    +

    + +

    +

    where +contains the generalized eigenvalues and the columns of the generalized eigenvectors such that +.

    +

    Randomized eigensolvers to construct the approximate spectral decomposition:

    +

    When the generalized eigenvalues decay rapidly, we can +extract a low-rank approximation of by retaining only the +largest eigenvalues and corresponding eigenvectors,

    +

    + +

    +

    Here, contains only the +generalized eigenvectors of that correspond to the largest eigenvalues, +which are assembled into the diagonal matrix .

    +

    The approximate posterior covariance::

    +

    Using the Sherman–Morrison–Woodbury formula, we write + +

    +

    where . The last term in this expression captures the +error due to truncation in terms of the discarded eigenvalues; this +provides a criterion for truncating the spectrum, namely that is +chosen such that is small relative to 1.

    +

    Therefore we can approximate the posterior covariance as

    +

    + +

    +

    Drawing samples from a Gaussian distribution with covariance +

    +

    Let be a sample for the prior distribution, i.e. , then, using the low rank approximation of the posterior covariance, we compute a sample as + +

    +

    This tutorial shows:

    +
      +
    • description of the inverse problem (the forward problem, the prior, and the misfit functional)
    • +
    • convergence of the inexact Newton-CG algorithm
    • +
    • low-rank-based approximation of the posterior covariance (built on a low-rank +approximation of the Hessian of the data misfit)
    • +
    • how to construct the low-rank approximation of the Hessian of the data misfit
    • +
    • how to apply the inverse and square-root inverse Hessian to a vector efficiently
    • +
    • samples from the Gaussian approximation of the posterior
    • +
    +

    Goals:

    +

    By the end of this notebook, you should be able to:

    +
      +
    • understand the Bayesian inverse framework
    • +
    • visualise and understand the results
    • +
    • modify the problem and code
    • +
    +

    Mathematical tools used:

    +
      +
    • Finite element method
    • +
    • Derivation of gradiant and Hessian via the adjoint method
    • +
    • inexact Newton-CG
    • +
    • Armijo line search
    • +
    • Bayes' formula
    • +
    • randomized eigensolvers
    • +
    +

    List of software used:

    +
      +
    • FEniCS, a parallel finite element element library for the discretization of partial differential equations
    • +
    • PETSc, for scalable and efficient linear algebra operations and solvers
    • +
    • Matplotlib, A great python package that I used for plotting many of the results
    • +
    • Numpy, A python package for linear algebra. While extensive, this is mostly used to compute means and sums in this notebook.
    • +
    +

    1. Load modules

    +
    from __future__ import absolute_import, division, print_function
    +
    +import dolfin as dl
    +import math
    +import numpy as np
    +import matplotlib.pyplot as plt
    +%matplotlib inline
    +import sys
    +import os
    +sys.path.append( os.environ.get('HIPPYLIB_BASE_DIR', "../") )
    +from hippylib import *
    +
    +import nb
    +
    +import logging
    +logging.getLogger('FFC').setLevel(logging.WARNING)
    +logging.getLogger('UFL').setLevel(logging.WARNING)
    +dl.set_log_active(False)
    +
    +np.random.seed(seed=1)
    +
    +

    2. Generate the true parameter

    +

    This function generates a random field with a prescribed anysotropic covariance function.

    +
    def true_model(Vh, gamma, delta, anis_diff):
    +    prior = BiLaplacianPrior(Vh, gamma, delta, anis_diff )
    +    noise = dl.Vector()
    +    prior.init_vector(noise,"noise")
    +    noise_size = get_local_size(noise)
    +    noise.set_local( np.random.randn( noise_size ) )
    +    atrue = dl.Vector()
    +    prior.init_vector(atrue, 0)
    +    prior.sample(noise,atrue)
    +    return atrue
    +
    +

    3. Set up the mesh and finite element spaces

    +

    We compute a two dimensional mesh of a unit square with nx by ny elements. +We define a P2 finite element space for the state and adjoint variable and P1 for the parameter.

    +
    ndim = 2
    +nx = 64
    +ny = 64
    +mesh = dl.UnitSquareMesh(nx, ny)
    +Vh2 = dl.FunctionSpace(mesh, 'Lagrange', 2)
    +Vh1 = dl.FunctionSpace(mesh, 'Lagrange', 1)
    +Vh = [Vh2, Vh1, Vh2]
    +print("Number of dofs: STATE={0}, PARAMETER={1}, ADJOINT={2}".format(
    +    Vh[STATE].dim(), Vh[PARAMETER].dim(), Vh[ADJOINT].dim()) )
    +
    +
    Number of dofs: STATE=16641, PARAMETER=4225, ADJOINT=16641
    +
    +

    4. Set up the forward problem

    +

    To set up the forward problem we use the PDEVariationalProblem class, which requires the following inputs +- the finite element spaces for the state, parameter, and adjoint variables Vh +- the pde in weak form pde_varf +- the boundary conditions bc for the forward problem and bc0 for the adjoint and incremental problems.

    +

    The PDEVariationalProblem class offer the following functionality: +- solving the forward/adjoint and incremental problems +- evaluate first and second partial derivative of the forward problem with respect to the state, parameter, and adojnt variables.

    +
    def u_boundary(x, on_boundary):
    +    return on_boundary and ( x[1] < dl.DOLFIN_EPS or x[1] > 1.0 - dl.DOLFIN_EPS)
    +
    +u_bdr = dl.Expression("x[1]", degree=1)
    +u_bdr0 = dl.Constant(0.0)
    +bc = dl.DirichletBC(Vh[STATE], u_bdr, u_boundary)
    +bc0 = dl.DirichletBC(Vh[STATE], u_bdr0, u_boundary)
    +
    +f = dl.Constant(0.0)
    +
    +def pde_varf(u,a,p):
    +    return dl.exp(a)*dl.inner(dl.nabla_grad(u), dl.nabla_grad(p))*dl.dx - f*p*dl.dx
    +
    +pde = PDEVariationalProblem(Vh, pde_varf, bc, bc0, is_fwd_linear=True)
    +
    +

    4. Set up the prior

    +

    To obtain the synthetic true paramter we generate a realization of a Gaussian random field with zero average and covariance matrix , where is a differential operator of the form + +Here is an s.p.d. anisotropic tensor of the form + +

    +

    For the prior model, we assume that we can measure the log-permeability coefficient at locations, and we denote with , , such measures. +We also introduce the mollifier functions + +and we let + +where is a penalization costant (10 for this problem) and .

    +

    We then compute , the mean of the prior measure, as a regularized +least-squares fit of these point observations by solving + +

    +

    Finally the prior distribution is , with .

    +
    gamma = .1
    +delta = .5
    +
    +anis_diff = dl.Expression(code_AnisTensor2D, degree=1)
    +anis_diff.theta0 = 2.
    +anis_diff.theta1 = .5
    +anis_diff.alpha = math.pi/4
    +atrue = true_model(Vh[PARAMETER], gamma, delta,anis_diff)
    +
    +locations = np.array([[0.1, 0.1], [0.1, 0.9], [.5,.5], [.9, .1], [.9, .9]])
    +pen = 1e1
    +prior = MollifiedBiLaplacianPrior(Vh[PARAMETER], gamma, delta, locations, atrue, anis_diff, pen)
    +
    +print("Prior regularization: (delta_x - gamma*Laplacian)^order: delta={0}, gamma={1}, order={2}".format(
    +    delta, gamma,2) )   
    +
    +objs = [dl.Function(Vh[PARAMETER],atrue), dl.Function(Vh[PARAMETER],prior.mean)]
    +mytitles = ["True Parameter", "Prior mean"]
    +nb.multi1_plot(objs, mytitles)
    +plt.show()
    +
    +model = Model(pde,prior, misfit)
    +
    +
    Prior regularization: (delta_x - gamma*Laplacian)^order: delta=0.5, gamma=0.1, order=2
    +
    +

    png

    +

    5. Set up the misfit functional and generate synthetic observations

    +

    To setup the observation operator, we generate ntargets random locations where to evaluate the value of the state.

    +

    To generate the synthetic observation, we first solve the forward problem using the true parameter . Synthetic observations are obtained by perturbing the state variable at the observation points with a random gaussian noise. +rel_noise is the signal to noise ratio.

    +
    ntargets = 300
    +rel_noise = 0.01
    +
    +
    +targets = np.random.uniform(0.1,0.9, [ntargets, ndim] )
    +print("Number of observation points: {0}".format(ntargets))
    +misfit = PointwiseStateObservation(Vh[STATE], targets)
    +
    +utrue = pde.generate_state()
    +x = [utrue, atrue, None]
    +pde.solveFwd(x[STATE], x, 1e-9)
    +misfit.B.mult(x[STATE], misfit.d)
    +MAX = misfit.d.norm("linf")
    +noise_std_dev = rel_noise * MAX
    +randn_perturb(misfit.d, noise_std_dev)
    +misfit.noise_variance = noise_std_dev*noise_std_dev
    +
    +vmax = max( utrue.max(), misfit.d.max() )
    +vmin = min( utrue.min(), misfit.d.min() )
    +
    +plt.figure(figsize=(15,5))
    +nb.plot(dl.Function(Vh[STATE], utrue), mytitle="True State", subplot_loc=121, vmin=vmin, vmax=vmax)
    +nb.plot_pts(targets, misfit.d, mytitle="Observations", subplot_loc=122, vmin=vmin, vmax=vmax)
    +plt.show()
    +
    +
    Number of observation points: 300
    +
    +

    png

    +

    6. Set up the model and test gradient and Hessian

    +

    The model is defined by three component: +- the PDEVariationalProblem pde which provides methods for the solution of the forward problem, adjoint problem, and incremental forward and adjoint problems. +- the Prior prior which provides methods to apply the regularization (precision) operator to a vector or to apply the prior covariance operator (i.e. to solve linear system with the regularization operator) +- the Misfit misfit which provides methods to compute the cost functional and its partial derivatives with respect to the state and parameter variables.

    +

    To test gradient and the Hessian of the model we use forward finite differences.

    +
    model = Model(pde, prior, misfit)
    +
    +a0 = dl.interpolate(dl.Expression("sin(x[0])", degree=5), Vh[PARAMETER])
    +modelVerify(model, a0.vector(), 1e-12)
    +
    +
    (yy, H xx) - (xx, H yy) =  7.57910091653e-15
    +
    +

    png

    +

    7. Compute the MAP point

    +

    We used the globalized Newtown-CG method to compute the MAP point.

    +
    a0 = prior.mean.copy()
    +solver = ReducedSpaceNewtonCG(model)
    +solver.parameters["rel_tolerance"] = 1e-9
    +solver.parameters["abs_tolerance"] = 1e-12
    +solver.parameters["max_iter"]      = 25
    +solver.parameters["inner_rel_tolerance"] = 1e-15
    +solver.parameters["c_armijo"] = 1e-4
    +solver.parameters["GN_iter"] = 5
    +
    +x = solver.solve(a0)
    +
    +if solver.converged:
    +    print("\nConverged in ", solver.it, " iterations.")
    +else:
    +    print("\nNot Converged")
    +
    +print("Termination reason: ", solver.termination_reasons[solver.reason])
    +print("Final gradient norm: ", solver.final_grad_norm)
    +print("Final cost: ", solver.final_cost)
    +
    +plt.figure(figsize=(15,5))
    +nb.plot(dl.Function(Vh[STATE], x[STATE]), subplot_loc=121,mytitle="State")
    +nb.plot(dl.Function(Vh[PARAMETER], x[PARAMETER]), subplot_loc=122,mytitle="Parameter")
    +plt.show()
    +
    +
    It  cg_it cost            misfit          reg             (g,da)          ||g||L2        alpha          tolcg         
    +  1   1    1.206547e+03    1.206233e+03    3.147385e-01   -1.570552e+04   1.042537e+05   1.000000e+00   5.000000e-01
    +  2   3    3.459350e+02    3.446814e+02    1.253641e+00   -1.846772e+03   1.431429e+04   1.000000e+00   3.705435e-01
    +  3   1    2.746765e+02    2.733672e+02    1.309257e+00   -1.424615e+02   1.003736e+04   1.000000e+00   3.102873e-01
    +  4   7    1.691763e+02    1.647605e+02    4.415777e+00   -2.128491e+02   3.871034e+03   1.000000e+00   1.926938e-01
    +  5   6    1.573210e+02    1.522926e+02    5.028367e+00   -2.345647e+01   1.821115e+03   1.000000e+00   1.321670e-01
    +  6  14    1.424926e+02    1.297495e+02    1.274310e+01   -2.988285e+01   1.157426e+03   1.000000e+00   1.053661e-01
    +  7   2    1.421601e+02    1.294134e+02    1.274672e+01   -6.643211e-01   7.325602e+02   1.000000e+00   8.382545e-02
    +  8  22    1.407914e+02    1.253200e+02    1.547141e+01   -2.734442e+00   4.409802e+02   1.000000e+00   6.503749e-02
    +  9  14    1.407826e+02    1.253781e+02    1.540453e+01   -1.744746e-02   5.057992e+01   1.000000e+00   2.202639e-02
    + 10  29    1.407817e+02    1.253307e+02    1.545107e+01   -1.819642e-03   1.502434e+01   1.000000e+00   1.200472e-02
    + 11  38    1.407817e+02    1.253304e+02    1.545139e+01   -2.502663e-07   1.390539e-01   1.000000e+00   1.154904e-03
    + 12  54    1.407817e+02    1.253303e+02    1.545139e+01   -1.380451e-12   2.625467e-04   1.000000e+00   5.018311e-05
    +
    +Converged in  12  iterations.
    +Termination reason:  Norm of the gradient less than tolerance
    +Final gradient norm:  5.41022518768e-09
    +Final cost:  140.781736843
    +
    +

    png

    +

    8. Compute the low rank Gaussian approximation of the posterior

    +

    We used the double pass algorithm to compute a low-rank decomposition of the Hessian Misfit. +In particular, we solve

    +

    + +

    +

    The Figure shows the largest k generalized eigenvectors of the Hessian misfit. +The effective rank of the Hessian misfit is the number of eigenvalues above the red line (y=1). +The effective rank is independent of the mesh size.

    +
    model.setPointForHessianEvaluations(x)
    +Hmisfit = ReducedHessian(model, solver.parameters["inner_rel_tolerance"], gauss_newton_approx=False, misfit_only=True)
    +k = 50
    +p = 20
    +print("Single/Double Pass Algorithm. Requested eigenvectors: {0}; Oversampling {1}.".format(k,p))
    +Omega = np.random.randn(get_local_size(x[PARAMETER]), k+p)
    +d, U = doublePassG(Hmisfit, prior.R, prior.Rsolver, Omega, k)
    +
    +posterior = GaussianLRPosterior(prior, d, U)
    +posterior.mean = x[PARAMETER]
    +
    +plt.plot(range(0,k), d, 'b*', range(0,k+1), np.ones(k+1), '-r')
    +plt.yscale('log')
    +plt.xlabel('number')
    +plt.ylabel('eigenvalue')
    +
    +nb.plot_eigenvectors(Vh[PARAMETER], U, mytitle="Eigenvector", which=[0,1,2,5,10,15])
    +
    +
    Single/Double Pass Algorithm. Requested eigenvectors: 50; Oversampling 20.
    +
    +

    png

    +

    png

    +

    9. Prior and posterior pointwise variance fields

    +
    compute_trace = True
    +if compute_trace:
    +    post_tr, prior_tr, corr_tr = posterior.trace(method="Randomized", r=200)
    +    print("Posterior trace {0:5e}; Prior trace {1:5e}; Correction trace {2:5e}".format(post_tr, prior_tr, corr_tr))
    +post_pw_variance, pr_pw_variance, corr_pw_variance = posterior.pointwise_variance(method="Randomized", r=200)
    +
    +objs = [dl.Function(Vh[PARAMETER], pr_pw_variance),
    +        dl.Function(Vh[PARAMETER], post_pw_variance)]
    +mytitles = ["Prior variance", "Posterior variance"]
    +nb.multi1_plot(objs, mytitles, logscale=True)
    +plt.show()
    +
    +
    Posterior trace 1.058859e-01; Prior trace 3.949847e-01; Correction trace 2.890987e-01
    +
    +

    png

    +

    10. Generate samples from Prior and Posterior

    +
    nsamples = 5
    +noise = dl.Vector()
    +posterior.init_vector(noise,"noise")
    +noise_size = get_local_size(noise)
    +s_prior = dl.Function(Vh[PARAMETER], name="sample_prior")
    +s_post = dl.Function(Vh[PARAMETER], name="sample_post")
    +
    +pr_max   =  2.5*math.sqrt( pr_pw_variance.max() ) + prior.mean.max()
    +pr_min   = -2.5*math.sqrt( pr_pw_variance.max() ) + prior.mean.min()
    +ps_max   =  2.5*math.sqrt( post_pw_variance.max() ) + posterior.mean.max()
    +ps_min   = -2.5*math.sqrt( post_pw_variance.max() ) + posterior.mean.min()
    +
    +for i in range(nsamples):
    +    noise.set_local( np.random.randn( noise_size ) )
    +    posterior.sample(noise, s_prior.vector(), s_post.vector())
    +    plt.figure(figsize=(15,5))
    +    nb.plot(s_prior, subplot_loc=121, mytitle="Prior sample",     vmin=pr_min, vmax=pr_max)
    +    nb.plot(s_post,  subplot_loc=122, mytitle="Posterior sample", vmin=ps_min, vmax=ps_max)
    +    plt.show()
    +
    +

    png

    +

    png

    +

    png

    +

    png

    +

    png

    +

    Copyright (c) 2016-2018, The University of Texas at Austin & University of California, Merced. +All Rights reserved. +See file COPYRIGHT for details.

    +

    This file is part of the hIPPYlib library. For more information and source code +availability see https://hippylib.github.io.

    +

    hIPPYlib is free software; you can redistribute it and/or modify it under the terms of the GNU General Public License (as published by the Free Software Foundation) version 2.0 dated June 1991.

    +
    +
    + +
    +
    +

    Copyright © 2016-2023 The University of Texas at Austin, University of California Merced.
    Copyright © 2018-2020 The University of Texas at Austin, University of California Merced, Washington University in St. Louis.
    This material is based on work partially supported by the National Science Foundation under Grants No ACI-1550593, ACI-1550547.

    +

    Documentation built with MkDocs.

    +
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    +
    + +
    + +

    + +

    +

    Example: Bayesian initial condition inversion in an advection-diffusion problem

    +

    In this example we tackle the problem of quantifying the uncertainty in the solution of an inverse problem governed by a parabolic PDE via the Bayesian inference framework. The underlying PDE is a time-dependent advection-diffusion equation in which we seek to infer an unknown initial condition from spatio-temporal point measurements.

    +

    The Bayesian inverse problem:

    +

    Following the Bayesian framework, we utilize +a Gaussian prior measure , +with where is an elliptic differential operator as +described in the PoissonBayesian example, and use an additive +Gaussian noise model. Therefore, the solution of the Bayesian inverse +problem is the posterior measure, with + and .

    +
      +
    • The posterior mean is characterized as the minimizer of
    • +
    +

    + +

    +

    which can also be interpreted as the regularized functional to be +minimized in deterministic inversion. The observation operator extracts the values of the forward solution on a set of +locations at +times .

    +
      +
    • The posterior covariance is the inverse of the Hessian of , i.e.,
    • +
    +

    + +

    +

    The forward problem:

    +

    The PDE in the parameter-to-observable map models diffusive transport +in a domain ():

    +

    + +

    +

    Here, is the diffusion coefficient and is the final +time. The velocity field + is computed by solving the following steady-state +Navier-Stokes equation with the side walls driving the flow:

    +

    + +

    +

    Here, is pressure, is the Reynolds number. The Dirichlet boundary data + is given by + on the left wall of the domain, + on the right wall, and everywhere else.

    +

    The adjoint problem:

    +

    + +

    +

    1. Load modules

    +
    from __future__ import absolute_import, division, print_function
    +
    +import dolfin as dl
    +import math
    +import numpy as np
    +import matplotlib.pyplot as plt
    +%matplotlib inline
    +import sys
    +import os
    +sys.path.append( os.environ.get('HIPPYLIB_BASE_DIR', "../") )
    +from hippylib import *
    +sys.path.append( os.environ.get('HIPPYLIB_BASE_DIR', "..") + "/applications/ad_diff/" )
    +from model_ad_diff import TimeDependentAD
    +
    +import nb
    +
    +import logging
    +logging.getLogger('FFC').setLevel(logging.WARNING)
    +logging.getLogger('UFL').setLevel(logging.WARNING)
    +dl.set_log_active(False)
    +
    +np.random.seed(1)
    +
    +

    2. Construct the velocity field

    +
    def v_boundary(x,on_boundary):
    +    return on_boundary
    +
    +def q_boundary(x,on_boundary):
    +    return x[0] < dl.DOLFIN_EPS and x[1] < dl.DOLFIN_EPS
    +
    +def computeVelocityField(mesh):
    +    Xh = dl.VectorFunctionSpace(mesh,'Lagrange', 2)
    +    Wh = dl.FunctionSpace(mesh, 'Lagrange', 1)
    +    if dlversion() <= (1,6,0):
    +        XW = dl.MixedFunctionSpace([Xh, Wh])
    +    else:
    +        mixed_element = dl.MixedElement([Xh.ufl_element(), Wh.ufl_element()])
    +        XW = dl.FunctionSpace(mesh, mixed_element)
    +
    +    Re = 1e2
    +
    +    g = dl.Expression(('0.0','(x[0] < 1e-14) - (x[0] > 1 - 1e-14)'), degree=1)
    +    bc1 = dl.DirichletBC(XW.sub(0), g, v_boundary)
    +    bc2 = dl.DirichletBC(XW.sub(1), dl.Constant(0), q_boundary, 'pointwise')
    +    bcs = [bc1, bc2]
    +
    +    vq = dl.Function(XW)
    +    (v,q) = dl.split(vq)
    +    (v_test, q_test) = dl.TestFunctions (XW)
    +
    +    def strain(v):
    +        return dl.sym(dl.nabla_grad(v))
    +
    +    F = ( (2./Re)*dl.inner(strain(v),strain(v_test))+ dl.inner (dl.nabla_grad(v)*v, v_test)
    +           - (q * dl.div(v_test)) + ( dl.div(v) * q_test) ) * dl.dx
    +
    +    dl.solve(F == 0, vq, bcs, solver_parameters={"newton_solver":
    +                                         {"relative_tolerance":1e-4, "maximum_iterations":100}})
    +
    +    plt.figure(figsize=(15,5))
    +    vh = dl.project(v,Xh)
    +    qh = dl.project(q,Wh)
    +    nb.plot(nb.coarsen_v(vh), subplot_loc=121,mytitle="Velocity")
    +    nb.plot(qh, subplot_loc=122,mytitle="Pressure")
    +    plt.show()
    +
    +    return v
    +
    +

    3. Set up the mesh and finite element spaces

    +
    mesh = dl.refine( dl.Mesh("ad_20.xml") )
    +wind_velocity = computeVelocityField(mesh)
    +Vh = dl.FunctionSpace(mesh, "Lagrange", 1)
    +print("Number of dofs: {0}".format( Vh.dim() ) )
    +
    +

    png

    +
    Number of dofs: 2023
    +
    +

    4. Set up model (prior, true/proposed initial condition)

    +
    #gamma = 1
    +#delta = 10
    +#prior = LaplacianPrior(Vh, gamma, delta)
    +
    +gamma = 1
    +delta = 8
    +prior = BiLaplacianPrior(Vh, gamma, delta)
    +
    +prior.mean = dl.interpolate(dl.Constant(0.5), Vh).vector()
    +true_initial_condition = dl.interpolate(dl.Expression('min(0.5,exp(-100*(pow(x[0]-0.35,2) +  pow(x[1]-0.7,2))))', degree=5), Vh).vector()
    +problem = TimeDependentAD(mesh, [Vh,Vh,Vh], 0., 4., 1., .2, wind_velocity, True, prior)
    +
    +objs = [dl.Function(Vh,true_initial_condition),
    +        dl.Function(Vh,prior.mean)]
    +mytitles = ["True Initial Condition", "Prior mean"]
    +nb.multi1_plot(objs, mytitles)
    +plt.show()
    +
    +

    png

    +

    5. Generate the synthetic observations

    +
    rel_noise = 0.001
    +utrue = problem.generate_vector(STATE)
    +x = [utrue, true_initial_condition, None]
    +problem.solveFwd(x[STATE], x, 1e-9)
    +MAX = utrue.norm("linf", "linf")
    +noise_std_dev = rel_noise * MAX
    +problem.ud.copy(utrue)
    +problem.ud.randn_perturb(noise_std_dev)
    +problem.noise_variance = noise_std_dev*noise_std_dev
    +
    +nb.show_solution(Vh, true_initial_condition, utrue, "Solution")
    +
    +

    png

    +

    6. Test the gradient and the Hessian of the cost (negative log posterior)

    +
    a0 = true_initial_condition.copy()
    +modelVerify(problem, a0, 1e-12, is_quadratic=True)
    +
    +
    (yy, H xx) - (xx, H yy) =  -4.75341826113e-14
    +
    +

    png

    +

    7. Evaluate the gradient

    +
    [u,a,p] = problem.generate_vector()
    +problem.solveFwd(u, [u,a,p], 1e-12)
    +problem.solveAdj(p, [u,a,p], 1e-12)
    +mg = problem.generate_vector(PARAMETER)
    +grad_norm = problem.evalGradientParameter([u,a,p], mg)
    +
    +print("(g,g) = ", grad_norm)
    +
    +
    (g,g) =  1.66716039169e+12
    +
    +

    8. The Gaussian approximation of the posterior

    +
    H = ReducedHessian(problem, 1e-12, gauss_newton_approx=False, misfit_only=True) 
    +
    +k = 80
    +p = 20
    +print("Single Pass Algorithm. Requested eigenvectors: {0}; Oversampling {1}.".format(k,p))
    +Omega = np.random.randn(get_local_size(a), k+p)
    +d, U = singlePassG(H, prior.R, prior.Rsolver, Omega, k)
    +
    +
    +posterior = GaussianLRPosterior( prior, d, U )
    +
    +plt.plot(range(0,k), d, 'b*', range(0,k+1), np.ones(k+1), '-r')
    +plt.yscale('log')
    +plt.xlabel('number')
    +plt.ylabel('eigenvalue')
    +
    +nb.plot_eigenvectors(Vh, U, mytitle="Eigenvector", which=[0,1,2,5,10,20,30,45,60])
    +
    +
    Single Pass Algorithm. Requested eigenvectors: 80; Oversampling 20.
    +
    +

    png

    +

    png

    +

    9. Compute the MAP point

    +
    H.misfit_only = False
    +
    +solver = CGSolverSteihaug()
    +solver.set_operator(H)
    +solver.set_preconditioner( posterior.Hlr )
    +solver.parameters["print_level"] = 1
    +solver.parameters["rel_tolerance"] = 1e-6
    +solver.solve(a, -mg)
    +problem.solveFwd(u, [u,a,p], 1e-12)
    +
    +total_cost, reg_cost, misfit_cost = problem.cost([u,a,p])
    +print("Total cost {0:5g}; Reg Cost {1:5g}; Misfit {2:5g}".format(total_cost, reg_cost, misfit_cost))
    +
    +posterior.mean = a
    +
    +plt.figure(figsize=(7.5,5))
    +nb.plot(dl.Function(Vh, a), mytitle="Initial Condition")
    +plt.show()
    +
    +nb.show_solution(Vh, a, u, "Solution")
    +
    +
     Iterartion :  0  (B r, r) =  30140.7469425
    + Iteration :  1  (B r, r) =  0.0653733954192
    + Iteration :  2  (B r, r) =  6.28002367536e-06
    + Iteration :  3  (B r, r) =  9.57007003125e-10
    +Relative/Absolute residual less than tol
    +Converged in  3  iterations with final norm  3.09355297858e-05
    +Total cost 84.2612; Reg Cost 68.8823; Misfit 15.3789
    +
    +

    png

    +

    png

    +

    10. Prior and posterior pointwise variance fields

    +
    compute_trace = True
    +if compute_trace:
    +    post_tr, prior_tr, corr_tr = posterior.trace(method="Randomized", r=200)
    +    print("Posterior trace {0:5g}; Prior trace {1:5g}; Correction trace {2:5g}".format(post_tr, prior_tr, corr_tr))
    +post_pw_variance, pr_pw_variance, corr_pw_variance = posterior.pointwise_variance(method="Randomized", r=300)
    +
    +objs = [dl.Function(Vh, pr_pw_variance),
    +        dl.Function(Vh, post_pw_variance)]
    +mytitles = ["Prior Variance", "Posterior Variance"]
    +nb.multi1_plot(objs, mytitles, logscale=True)
    +plt.show()
    +
    +
    Posterior trace 0.000602854; Prior trace 0.0285673; Correction trace 0.0279644
    +
    +

    png

    +

    11. Draw samples from the prior and posterior distributions

    +
    nsamples = 5
    +noise = dl.Vector()
    +posterior.init_vector(noise,"noise")
    +noise_size = get_local_size(noise)
    +s_prior = dl.Function(Vh, name="sample_prior")
    +s_post = dl.Function(Vh, name="sample_post")
    +
    +pr_max =  2.5*math.sqrt( pr_pw_variance.max() ) + prior.mean.max()
    +pr_min = -2.5*math.sqrt( pr_pw_variance.min() ) + prior.mean.min()
    +ps_max =  2.5*math.sqrt( post_pw_variance.max() ) + posterior.mean.max()
    +ps_min = -2.5*math.sqrt( post_pw_variance.max() ) + posterior.mean.min()
    +
    +for i in range(nsamples):
    +    noise.set_local( np.random.randn( noise_size ) )
    +    posterior.sample(noise, s_prior.vector(), s_post.vector())
    +    plt.figure(figsize=(15,5))
    +    nb.plot(s_prior, subplot_loc=121,mytitle="Prior sample", vmin=pr_min, vmax=pr_max)
    +    nb.plot(s_post, subplot_loc=122,mytitle="Posterior sample", vmin=ps_min, vmax=ps_max)
    +    plt.show()
    +
    +

    png

    +

    png

    +

    png

    +

    png

    +

    png

    +

    Copyright (c) 2016-2018, The University of Texas at Austin & University of California, Merced. +All Rights reserved. +See file COPYRIGHT for details.

    +

    This file is part of the hIPPYlib library. For more information and source code +availability see https://hippylib.github.io.

    +

    hIPPYlib is free software; you can redistribute it and/or modify it under the terms of the GNU General Public License (as published by the Free Software Foundation) version 2.0 dated June 1991.

    +
    +
    + +
    +
    +

    Copyright © 2016-2023 The University of Texas at Austin, University of California Merced.
    Copyright © 2018-2020 The University of Texas at Austin, University of California Merced, Washington University in St. Louis.
    This material is based on work partially supported by the National Science Foundation under Grants No ACI-1550593, ACI-1550547.

    +

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    +
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z>Oo$Oy;EB``bQgoZ|&x2B>Dh1(-r*Jf6S#L>D z(ijujV8B6P3G^@ZR)5iXd=OUo^&JJvs^s{i`|_Zbd)CU*N`^(5SS7TuZBbpJP^%k_ z<5P)%5w&7F+r<52TJT}Plq`NM!Lum)^A;(IM6yE(S48i{nG#|2l0tE9HP9 zb@~TR>%1}W91OsA7*GIiM&|cjT!h$yJ;w5w@5_gx(QiMNcUgW3rj#dQ`}4eV literal 0 HcmV?d00001 diff --git a/tutorials_v1.6.0/5_HessianSpectrum/index.html b/tutorials_v1.6.0/5_HessianSpectrum/index.html new file mode 100644 index 0000000..48caae7 --- /dev/null +++ b/tutorials_v1.6.0/5_HessianSpectrum/index.html @@ -0,0 +1,598 @@ + + + + + + + + + + + Hessian Spectrum - hIPPYlib + + + + + + + + + +

    + +
    +
    + +
    + +

    Spectrum of the Reduced Hessian

    +

    The linear source inversion problem

    +

    We consider the following linear source inversion problem. +Find the state and the source (parameter) that solves + +

    +

    Here:

    +
      +
    • +

      + is a finite dimensional vector that denotes noisy observations of the state in locations . More specifically, , where are i.i.d. .

      +
    • +
    • +

      + is the linear operator that evaluates the state at the observation locations .

      +
    • +
    • +

      + and are the parameters of the regularization penalizing the and norm of , respectively.

      +
    • +
    • +

      +, , are given coefficients representing the diffusivity coefficient, the advective velocity and the reaction term, respectively.

      +
    • +
    • +

      +, represents the subdomain of where we impose Dirichlet or Neumann boundary conditions, respectively.

      +
    • +
    +

    1. Load modules

    +
    from __future__ import absolute_import, division, print_function
    +
    +import dolfin as dl
    +import numpy as np
    +import matplotlib.pyplot as plt
    +%matplotlib inline
    +import nb
    +
    +import sys
    +import os
    +sys.path.append( os.environ.get('HIPPYLIB_BASE_DIR', "../") )
    +from hippylib import *
    +
    +import logging
    +logging.getLogger('FFC').setLevel(logging.WARNING)
    +logging.getLogger('UFL').setLevel(logging.WARNING)
    +dl.set_log_active(False)
    +
    +

    2. The linear source inversion problem

    +
    def pde_varf(u,a,p):
    +    return k*dl.inner(dl.nabla_grad(u), dl.nabla_grad(p))*dl.dx \
    +           + dl.inner(dl.nabla_grad(u), v*p)*dl.dx \
    +           + c*u*p*dl.dx \
    +           - a*p*dl.dx
    +
    +def u_boundary(x, on_boundary):
    +    return on_boundary and x[1] < dl.DOLFIN_EPS
    +
    +def solve(nx,ny, targets, rel_noise, gamma, delta, verbose=True):
    +    np.random.seed(seed=2)
    +    mesh = dl.UnitSquareMesh(nx, ny)
    +    Vh1 = dl.FunctionSpace(mesh, 'Lagrange', 1)
    +
    +    Vh = [Vh1, Vh1, Vh1]
    +    if verbose:
    +        print("Number of dofs: STATE={0}, PARAMETER={1}, ADJOINT={2}".format(
    +            Vh[STATE].dim(), Vh[PARAMETER].dim(), Vh[ADJOINT].dim()) )
    +
    +
    +    u_bdr = dl.Constant(0.0)
    +    u_bdr0 = dl.Constant(0.0)
    +    bc = dl.DirichletBC(Vh[STATE], u_bdr, u_boundary)
    +    bc0 = dl.DirichletBC(Vh[STATE], u_bdr0, u_boundary)
    +
    +    atrue = dl.interpolate( dl.Expression('min(0.5,exp(-100*(pow(x[0]-0.35,2) +  pow(x[1]-0.7,2))))',degree=5), Vh[PARAMETER]).vector()
    +    a0 = dl.interpolate(dl.Constant(0.0), Vh[PARAMETER]).vector()
    +
    +    pde = PDEVariationalProblem(Vh, pde_varf, bc, bc0, is_fwd_linear=True)
    +
    +    if verbose:
    +        print("Number of observation points: {0}".format(targets.shape[0]))
    +
    +    misfit = PointwiseStateObservation(Vh[STATE], targets)
    +
    +    reg = LaplacianPrior(Vh[PARAMETER], gamma, delta)
    +
    +    #Generate synthetic observations
    +    utrue = pde.generate_state()
    +    x = [utrue, atrue, None]
    +    pde.solveFwd(x[STATE], x, 1e-9)
    +    misfit.B.mult(x[STATE], misfit.d)
    +    MAX = misfit.d.norm("linf")
    +    noise_std_dev = rel_noise * MAX
    +    randn_perturb(misfit.d, noise_std_dev)
    +    misfit.noise_variance = noise_std_dev*noise_std_dev
    +
    +    if verbose:
    +        plt.figure(figsize=(18,4))
    +        nb.plot(dl.Function(Vh[PARAMETER], atrue), mytitle = "True source", subplot_loc=131)
    +        nb.plot(dl.Function(Vh[STATE], utrue), mytitle="True state", subplot_loc=132)
    +        nb.plot_pts(targets, misfit.d,mytitle="Observations", subplot_loc=133)
    +        plt.show()
    +
    +    model = Model(pde, reg, misfit)
    +    u = model.generate_vector(STATE)
    +    a = a0.copy()
    +    p = model.generate_vector(ADJOINT)
    +    x = [u,a,p]
    +    mg = model.generate_vector(PARAMETER)
    +    model.solveFwd(u, x)
    +    model.solveAdj(p, x)
    +    model.evalGradientParameter(x, mg)
    +    model.setPointForHessianEvaluations(x)
    +
    +    H = ReducedHessian(model, 1e-12)
    +
    +    solver = CGSolverSteihaug()
    +    solver.set_operator(H)
    +    solver.set_preconditioner( reg.Rsolver )
    +    solver.parameters["print_level"] = -1
    +    solver.parameters["rel_tolerance"] = 1e-9
    +    solver.solve(a, -mg)
    +
    +    if solver.converged:
    +        if verbose:
    +            print("CG converged in ", solver.iter, " iterations.")
    +    else:
    +        print("CG did not converged.")
    +        raise
    +
    +    model.solveFwd(u, x, 1e-12)
    +
    +    total_cost, reg_cost, misfit_cost = model.cost(x)
    +
    +    if verbose:
    +        plt.figure(figsize=(18,4))
    +        nb.plot(dl.Function(Vh[PARAMETER], a), mytitle = "Reconstructed source", subplot_loc=131)
    +        nb.plot(dl.Function(Vh[STATE], u), mytitle="Reconstructed state", subplot_loc=132)
    +        nb.plot_pts(targets, misfit.B*u - misfit.d, mytitle="Misfit", subplot_loc=133)
    +        plt.show()
    +
    +    H.misfit_only = True
    +    k_evec = 80
    +    p_evec = 5
    +    if verbose:
    +        print("Double Pass Algorithm. Requested eigenvectors: {0}; Oversampling {1}.".format(k_evec,p_evec))
    +    Omega = np.random.randn(get_local_size(a), k_evec+p_evec)
    +    d, U = doublePassG(H, reg.R, reg.Rsolver, Omega, k_evec)
    +
    +    if verbose:
    +        plt.figure()
    +        nb.plot_eigenvalues(d, mytitle="Generalized Eigenvalues")
    +        nb.plot_eigenvectors(Vh[PARAMETER], U, mytitle="Eigenvectors", which=[0,1,2,5,10,15])
    +        plt.show()
    +
    +    return d, U, Vh[PARAMETER], solver.iter
    +
    +
    +

    3. Solution of the source inversion problem

    +
    ndim = 2
    +nx = 32
    +ny = 32
    +
    +ntargets = 300
    +np.random.seed(seed=1)
    +targets = np.random.uniform(0.1,0.9, [ntargets, ndim] )
    +rel_noise = 0.01
    +
    +gamma = 70.
    +delta = 1e-1
    +
    +k = dl.Constant(1.0)
    +v = dl.Constant((0.0, 0.0))
    +c = dl.Constant(0.)
    +
    +d, U, Va, nit = solve(nx,ny, targets, rel_noise, gamma, delta)
    +
    +
    Number of dofs: STATE=1089, PARAMETER=1089, ADJOINT=1089
    +Number of observation points: 300
    +
    +

    png

    +
    CG converged in  74  iterations.
    +
    +

    png

    +
    Double Pass Algorithm. Requested eigenvectors: 80; Oversampling 5.
    +
    +

    png

    +

    png

    +

    4. Mesh independence of the spectrum of the preconditioned Hessian

    +
    gamma = 70.
    +delta = 1e-1
    +
    +k = dl.Constant(1.0)
    +v = dl.Constant((0.0, 0.0))
    +c = dl.Constant(0.)
    +
    +n = [16,32,64]
    +d1, U1, Va1, niter1 = solve(n[0],n[0], targets, rel_noise, gamma, delta,verbose=False)
    +d2, U2, Va2, niter2 = solve(n[1],n[1], targets, rel_noise, gamma, delta,verbose=False)
    +d3, U3, Va3, niter3 = solve(n[2],n[2], targets, rel_noise, gamma, delta,verbose=False)
    +
    +print("Number of Iterations: ", niter1, niter2, niter3)
    +plt.figure(figsize=(18,4))
    +nb.plot_eigenvalues(d1, mytitle="Eigenvalues Mesh {0} by {1}".format(n[0],n[0]), subplot_loc=131)
    +nb.plot_eigenvalues(d2, mytitle="Eigenvalues Mesh {0} by {1}".format(n[1],n[1]), subplot_loc=132)
    +nb.plot_eigenvalues(d3, mytitle="Eigenvalues Mesh {0} by {1}".format(n[2],n[2]), subplot_loc=133)
    +
    +nb.plot_eigenvectors(Va1, U1, mytitle="Mesh {0} by {1} Eigen".format(n[0],n[0]), which=[0,1,5])
    +nb.plot_eigenvectors(Va2, U2, mytitle="Mesh {0} by {1} Eigen".format(n[1],n[1]), which=[0,1,5])
    +nb.plot_eigenvectors(Va3, U3, mytitle="Mesh {0} by {1} Eigen".format(n[2],n[2]), which=[0,1,5])
    +
    +plt.show()
    +
    +
    Number of Iterations:  62 74 76
    +
    +

    png

    +

    png

    +

    png

    +

    png

    +

    5. Dependence on the noise level

    +

    We solve the problem for different noise levels. The higher the noise level the more important becomes the effect of the regularization.

    +
    gamma = 70.
    +delta = 1e-1
    +
    +k = dl.Constant(1.0)
    +v = dl.Constant((0.0, 0.0))
    +c = dl.Constant(0.)
    +
    +rel_noise = [1e-3,1e-2,1e-1]
    +d1, U1, Va1, niter1 = solve(nx,ny, targets, rel_noise[0], gamma, delta,verbose=False)
    +d2, U2, Va2, niter2 = solve(nx,ny, targets, rel_noise[1], gamma, delta,verbose=False)
    +d3, U3, Va3, niter3 = solve(nx,ny, targets, rel_noise[2], gamma, delta,verbose=False)
    +
    +print("Number of Iterations: ", niter1, niter2, niter3)
    +plt.figure(figsize=(18,4))
    +nb.plot_eigenvalues(d1, mytitle="Eigenvalues rel_noise {0:g}".format(rel_noise[0]), subplot_loc=131)
    +nb.plot_eigenvalues(d2, mytitle="Eigenvalues rel_noise {0:g}".format(rel_noise[1]), subplot_loc=132)
    +nb.plot_eigenvalues(d3, mytitle="Eigenvalues rel_noise {0:g}".format(rel_noise[2]), subplot_loc=133)
    +
    +nb.plot_eigenvectors(Va1, U1, mytitle="rel_noise {0:g} Eigen".format(rel_noise[0]), which=[0,1,5])
    +nb.plot_eigenvectors(Va2, U2, mytitle="rel_noise {0:g} Eigen".format(rel_noise[1]), which=[0,1,5])
    +nb.plot_eigenvectors(Va3, U3, mytitle="rel_noise {0:g} Eigen".format(rel_noise[2]), which=[0,1,5])
    +
    +plt.show()
    +
    +
    Number of Iterations:  161 74 21
    +
    +

    png

    +

    png

    +

    png

    +

    png

    +

    6. Dependence on the PDE coefficients

    +

    Assume a constant reaction term , and we consider different values for the diffusivity coefficient .

    +

    The smaller the value of the slower the decay in the spectrum.

    +
    rel_noise = 0.01
    +
    +k = dl.Constant(1.0)
    +v = dl.Constant((0.0, 0.0))
    +c = dl.Constant(1.0)
    +
    +d1, U1, Va1, niter1 = solve(nx,ny, targets, rel_noise, gamma, delta,verbose=False)
    +k = dl.Constant(0.1)
    +d2, U2, Va2, niter2 = solve(nx,ny, targets, rel_noise, gamma, delta,verbose=False)
    +k = dl.Constant(0.01)
    +d3, U3, Va3, niter3 = solve(nx,ny, targets, rel_noise, gamma, delta,verbose=False)
    +
    +print("Number of Iterations: ", niter1, niter2, niter3)
    +plt.figure(figsize=(18,4))
    +nb.plot_eigenvalues(d1, mytitle="Eigenvalues k=1.0", subplot_loc=131)
    +nb.plot_eigenvalues(d2, mytitle="Eigenvalues k=0.1", subplot_loc=132)
    +nb.plot_eigenvalues(d3, mytitle="Eigenvalues k=0.01", subplot_loc=133)
    +
    +nb.plot_eigenvectors(Va1, U1, mytitle="k=1. Eigen", which=[0,1,5])
    +nb.plot_eigenvectors(Va2, U2, mytitle="k=0.1 Eigen", which=[0,1,5])
    +nb.plot_eigenvectors(Va3, U3, mytitle="k=0.01 Eigen", which=[0,1,5])
    +
    +plt.show()
    +
    +
    Number of Iterations:  88 147 256
    +
    +

    png

    +

    png

    +

    png

    +

    png

    +

    Copyright (c) 2016-2018, The University of Texas at Austin & University of California, Merced. +All Rights reserved. +See file COPYRIGHT for details.

    +

    This file is part of the hIPPYlib library. For more information and source code +availability see https://hippylib.github.io.

    +

    hIPPYlib is free software; you can redistribute it and/or modify it under the terms of the GNU General Public License (as published by the Free Software Foundation) version 2.0 dated June 1991.

    +
    +
    + +
    +
    +

    Copyright © 2016-2023 The University of Texas at Austin, University of California Merced.
    Copyright © 2018-2020 The University of Texas at Austin, University of California Merced, Washington University in St. Louis.
    This material is based on work partially supported by the National Science Foundation under Grants No ACI-1550593, ACI-1550547.

    +

    Documentation built with MkDocs.

    +
    + + + + + + + + + + + diff --git a/tutorials_v2.3.0/1_FEniCS101/1_FEniCS101/index.html b/tutorials_v2.3.0/1_FEniCS101/1_FEniCS101/index.html new file mode 100644 index 0000000..1ec641b --- /dev/null +++ b/tutorials_v2.3.0/1_FEniCS101/1_FEniCS101/index.html @@ -0,0 +1,605 @@ + + + + + + + + + + + FEniCS101 - hIPPYlib + + + + + + + + + + + +
    +
    + +
    + +

    FEniCS101 Tutorial

    +

    In this tutorial we consider the boundary value problem (BVP)

    +

    + +

    +

    where , and are the union of +the left and right, and top and bottom boundaries of , +respectively.

    +

    Here + +

    +

    The exact solution is + +

    +

    Weak formulation

    +

    Let us define the Hilbert spaces as + + +

    +

    To obtain the weak formulation, we multiply the PDE by an arbitrary function and integrate over the domain leading to

    +

    + +

    +

    Then, integration by parts the non-conforming term gives

    +

    + +

    +

    Finally by recalling that on and that on , we find the weak formulation:

    +

    Find * such that* + +

    +

    1. Load modules

    +

    To start we load the following modules:

    +
      +
    • +

      dolfin: the python/C++ interface to FEniCS

      +
    • +
    • +

      math: the python module for mathematical functions

      +
    • +
    • +

      numpy: a python package for linear algebra

      +
    • +
    • +

      matplotlib: a python package used for plotting the results

      +
    • +
    +
    from __future__ import absolute_import, division, print_function
    +
    +from dolfin import *
    +
    +import math
    +import numpy as np
    +import logging
    +
    +import matplotlib.pyplot as plt
    +%matplotlib inline
    +
    +import sys
    +import os
    +sys.path.append( os.environ.get('HIPPYLIB_BASE_DIR', "../") )
    +from hippylib import nb
    +
    +logging.getLogger('FFC').setLevel(logging.WARNING)
    +logging.getLogger('UFL').setLevel(logging.WARNING)
    +set_log_active(False)
    +
    +

    2. Define the mesh and the finite element space

    +

    We construct a triangulation (mesh) of the computational domain with n elements in each direction.

    +

    On the mesh , we then define the finite element space consisting of globally continuous piecewise polynomials. The degree variable defines the polynomial degree.

    +
    n = 16
    +degree = 1
    +mesh = UnitSquareMesh(n, n)
    +nb.plot(mesh)
    +
    +Vh  = FunctionSpace(mesh, 'Lagrange', degree)
    +print( "dim(Vh) = ", Vh.dim() )
    +
    +
    dim(Vh) =  289
    +
    +

    png

    +

    3. Define boundary labels

    +

    To partition the boundary of in the subdomains , , , we assign a unique label boundary_parts to each of part of .

    +
    class TopBoundary(SubDomain):
    +    def inside(self, x, on_boundary):
    +        return on_boundary and abs(x[1] - 1) < DOLFIN_EPS
    +
    +class BottomBoundary(SubDomain):
    +    def inside(self, x, on_boundary):
    +        return on_boundary and abs(x[1]) < DOLFIN_EPS
    +
    +class LeftBoundary(SubDomain):
    +    def inside(self, x, on_boundary):
    +        return on_boundary and abs(x[0]) < DOLFIN_EPS
    +
    +class RightBoundary(SubDomain):
    +    def inside(self, x, on_boundary):
    +        return on_boundary and abs(x[0] - 1) < DOLFIN_EPS
    +
    +boundary_parts = FacetFunction("size_t", mesh)
    +boundary_parts.set_all(0)
    +
    +Gamma_top = TopBoundary()
    +Gamma_top.mark(boundary_parts, 1)
    +Gamma_bottom = BottomBoundary()
    +Gamma_bottom.mark(boundary_parts, 2)
    +Gamma_left = LeftBoundary()
    +Gamma_left.mark(boundary_parts, 3)
    +Gamma_right = RightBoundary()
    +Gamma_right.mark(boundary_parts, 4)
    +
    +

    4. Define the coefficients of the PDE and the boundary conditions

    +

    We first define the coefficients of the PDE using the Constant and Expression classes. Constant is used to define coefficients that do not depend on the space coordinates, Expression is used to define coefficients that are a known function of the space coordinates x[0] (x-axis direction) and x[1] (y-axis direction).

    +

    In the finite element method community, Dirichlet boundary conditions are also known as essential boundary conditions since they are imposed directly in the definition of the finite element space. In FEniCS, we use the class DirichletBC to indicate this type of condition.

    +

    On the other hand, Newman boundary conditions are also known as natural boundary conditions since they are weakly imposed as boundary integrals in the variational formulation (weak form). In FEniCS, we create a new boundary measure ds[i] to integrate over the portion of the boundary marked with label i.

    +
    u_L = Constant(0.)
    +u_R = Constant(0.)
    +
    +sigma_bottom = Expression('-(pi/2.0)*sin(2*pi*x[0])', degree=5)
    +sigma_top    = Constant(0.)
    +
    +f = Expression('(4.0*pi*pi+pi*pi/4.0)*(sin(2*pi*x[0])*sin((pi/2.0)*x[1]))', degree=5)
    +
    +bcs = [DirichletBC(Vh, u_L, boundary_parts, 3),
    +       DirichletBC(Vh, u_R, boundary_parts, 4)]
    +
    +ds = Measure("ds", subdomain_data=boundary_parts)
    +
    +

    5. Define and solve the variational problem

    +

    We also define two special types of functions: the TrialFunction u and the TestFunction v. These special types of function are used by FEniCS to generate the finite element vectors and matrices which stem from the weak formulation of the PDE.

    +

    More specifically, by denoting by the finite element basis for the space , a function can be written as + +where represents the coefficients in the finite element expansion of .

    +

    We then define

    +
      +
    • +

      the bilinear form ;

      +
    • +
    • +

      the linear form .

      +
    • +
    +

    We can then solve the variational problem

    +

    Find + such that + +

    +

    using directly the built-in solve method in FEniCS.

    +

    NOTE: As an alternative one can also assemble the finite element matrix and the right hand side that stems from the discretization of and , and then solve the linear system + +where

    +
      +
    • +

      + is the vector collecting the coefficients of the finite element expasion of ,

      +
    • +
    • +

      the entries of the matrix A are such that ,

      +
    • +
    • +

      the entries of the right hand side b are such that .

      +
    • +
    +
    u = TrialFunction(Vh)
    +v = TestFunction(Vh)
    +a = inner(nabla_grad(u), nabla_grad(v))*dx
    +L = f*v*dx + sigma_top*v*ds(1) + sigma_bottom*v*ds(2)
    +
    +uh = Function(Vh)
    +
    +#solve(a == L, uh, bcs=bcs)
    +A, b = assemble_system(a,L, bcs=bcs)
    +solve(A, uh.vector(), b, "cg")
    +
    +nb.plot(uh)
    +
    +
    <matplotlib.collections.TriMesh at 0x11e2f0748>
    +
    +

    png

    +

    6. Compute the discretization error

    +

    For this problem, the exact solution is known. +We can therefore compute the following norms of the discretization error (i.e. the difference between the finite element solution and the exact solution ) + +and + +

    +
    u_e = Expression('sin(2*pi*x[0])*sin((pi/2.0)*x[1])', degree=5)
    +grad_u_e = Expression( ('2*pi*cos(2*pi*x[0])*sin((pi/2.0)*x[1])', 'pi/2.0*sin(2*pi*x[0])*cos((pi/2.0)*x[1])'), degree=5)
    +
    +err_L2 = sqrt( assemble( (uh-u_e)**2*dx ) )
    +err_grad = sqrt( assemble( inner(nabla_grad(uh) - grad_u_e, nabla_grad(uh) - grad_u_e)*dx ) )
    +err_H1 = sqrt( err_L2**2 + err_grad**2)
    +
    +print ("|| u_h - u_e ||_L2 = ", err_L2)
    +print ("|| u_h - u_e ||_H1 = ", err_H1)
    +
    +
    || u_h - u_e ||_L2 =  0.008805253722075506
    +|| u_h - u_e ||_H1 =  0.39671895251414124
    +
    +

    7. Convergence of the finite element method

    +

    We now verify numerically a well-known convergence result for the finite element method.

    +

    Let denote the polynomial degree of the finite element space, and assume that the solution is at least in . Then we have + +

    +

    In the code below, the function compute(n, degree) solves the PDE using a mesh with n elements in each direction and finite element spaces of polynomial order degree.

    +

    The figure below shows the discretization errors in the and as a function of the mesh size () for piecewise linear (P1, ) and piecewise quadratic (P2, ) finite elements. We observe that numerical results are consistent with the finite element convergence theory. In particular:

    +
      +
    • +

      for piecewise linear finite element P1 we observe first order convergence in the -norm and second order convergence in the -norm;

      +
    • +
    • +

      for piecewise quadratic finite element P2 we observe second order convergence in the -norm and third order convergence in the -norm.

      +
    • +
    +
    def compute(n, degree):
    +    mesh = UnitSquareMesh(n, n)
    +    Vh  = FunctionSpace(mesh, 'Lagrange', degree)
    +    boundary_parts = FacetFunction("size_t", mesh)
    +    boundary_parts.set_all(0)
    +
    +    Gamma_top = TopBoundary()
    +    Gamma_top.mark(boundary_parts, 1)
    +    Gamma_bottom = BottomBoundary()
    +    Gamma_bottom.mark(boundary_parts, 2)
    +    Gamma_left = LeftBoundary()
    +    Gamma_left.mark(boundary_parts, 3)
    +    Gamma_right = RightBoundary()
    +    Gamma_right.mark(boundary_parts, 4)
    +
    +    bcs = [DirichletBC(Vh, u_L, boundary_parts, 3), DirichletBC(Vh, u_R, boundary_parts, 4)]
    +    ds = Measure("ds", subdomain_data=boundary_parts)
    +
    +    u = TrialFunction(Vh)
    +    v = TestFunction(Vh)
    +    a = inner(nabla_grad(u), nabla_grad(v))*dx
    +    L = f*v*dx + sigma_top*v*ds(1) + sigma_bottom*v*ds(2)
    +    uh = Function(Vh)
    +    solve(a == L, uh, bcs=bcs)
    +    err_L2 = sqrt( assemble( (uh-u_e)**2*dx ) )
    +    err_grad = sqrt( assemble( inner(nabla_grad(uh) - grad_u_e, nabla_grad(uh) - grad_u_e)*dx ) )
    +    err_H1 = sqrt( err_L2**2 + err_grad**2)
    +
    +    return err_L2, err_H1
    +
    +nref = 5
    +n = 8*np.power(2,np.arange(0,nref))
    +h = 1./n
    +
    +err_L2_P1 = np.zeros(nref)
    +err_H1_P1 = np.zeros(nref)
    +err_L2_P2 = np.zeros(nref)
    +err_H1_P2 = np.zeros(nref)
    +
    +for i in range(nref):
    +    err_L2_P1[i], err_H1_P1[i] = compute(n[i], 1)
    +    err_L2_P2[i], err_H1_P2[i] = compute(n[i], 2)
    +
    +plt.figure(figsize=(15,5))
    +
    +plt.subplot(121)
    +plt.loglog(h, err_H1_P1, '-or', label = "H1 error")
    +plt.loglog(h, err_L2_P1, '-*b', label = "L2 error")
    +plt.loglog(h, h*.5*err_H1_P1[0]/h[0], '--g', label = "First Order")
    +plt.loglog(h, np.power(h,2)*.5*np.power( err_L2_P1[0]/h[0], 2), '-.k', label = "Second Order")
    +plt.xlabel("Mesh size h")
    +plt.ylabel("Error")
    +plt.title("P1 Finite Element")
    +plt.legend(loc='lower right')
    +
    +
    +plt.subplot(122)
    +plt.loglog(h, err_H1_P2, '-or', label = "H1 error")
    +plt.loglog(h, err_L2_P2, '-*b', label = "L2 error")
    +plt.loglog(h, np.power(h/h[0],2)*.5*err_H1_P2[0], '--g', label = "Second Order")
    +plt.loglog(h, np.power(h/h[0],3)*.5*err_L2_P2[0], '-.k', label = "Third Order")
    +plt.xlabel("Mesh size h")
    +plt.ylabel("Error")
    +plt.title("P2 Finite Element")
    +plt.legend(loc='lower right')
    +
    +plt.show()
    +
    +

    png

    +

    Copyright (c) 2016-2018, The University of Texas at Austin & University of California, Merced.
    +All Rights reserved.
    +See file COPYRIGHT for details.

    +

    This file is part of the hIPPYlib library. For more information and source code +availability see https://hippylib.github.io.

    +

    hIPPYlib is free software; you can redistribute it and/or modify it under the terms of the GNU General Public License (as published by the Free Software Foundation) version 2.0 dated June 1991.

    +
    +
    + +
    +
    +

    Copyright © 2016-2023 The University of Texas at Austin, University of California Merced.
    Copyright © 2018-2020 The University of Texas at Austin, University of California Merced, Washington University in St. Louis.
    This material is based on work partially supported by the National Science Foundation under Grants No ACI-1550593, ACI-1550547.

    +

    Documentation built with MkDocs.

    +
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z$g2;^YHuI5wOtG>jVqTDqneqY(On|?2z zb=9=5TD%)bF9E`O`zdL{t)&Ms03G8>wfg98}K0NJl8n25v88b6(r+H z;>eKJFgWA1)Y@`c_#SKGkXoETlxR;ii+RFquJEvNd4n|gWXwHBK{*a + + + + + + + + + + Poisson Deterministic - hIPPYlib + + + + + + + + + +

    + +
    +
    + +
    + +

    Coefficient field inversion in an elliptic partial differential equation

    +

    We consider the estimation of a coefficient in an elliptic partial +differential equation as a model problem. Depending on the +interpretation of the unknowns and the type of measurements, this +model problem arises, for instance, in inversion for groundwater flow +or heat conductivity. It can also be interpreted as finding a +membrane with a certain spatially varying stiffness. Let +, be an open, bounded +domain and consider the following problem:

    +

    + +

    +

    where is the solution of

    +

    + +

    +

    Here the unknown coefficient field, denotes (possibly noisy) data, a given force, and the regularization parameter.

    +

    The variational (or weak) form of the state equation:

    +

    Find such that

    +

    + +

    +

    where is the space of functions vanishing on with square integrable derivatives. Here, denotes the -inner product, i.e, for scalar functions we denote

    +

    + +

    +

    Gradient evaluation:

    +

    The Lagrangian functional is given by

    +

    + +

    +

    Then the gradient of the cost functional with respect to the parameter is

    +

    + +

    +

    where is the solution of the forward problem,

    +

    + +

    +

    and is the solution of the adjoint problem,

    +

    + +

    +

    Hessian action:

    +

    To evaluate the action of the Hessian in a given direction , we consider variations of the meta-Lagrangian functional

    +

    + +

    +

    Then the action of the Hessian in a given direction is

    +

    + +

    +

    where

    +
      +
    • +

      + and are the solution of the forward and adjoint problem, respectively;

      +
    • +
    • +

      + is the solution of the incremental forward problem,

      +
    • +
    +

    + +

    +
      +
    • and is the solution of the incremental adjoint problem, + +
    • +
    +

    Inexact Newton-CG:

    +

    Written in abstract form, the Newton Method computes an update direction by solving the linear system

    +

    + +

    +

    where the evaluation of the gradient involve the solution and of the forward and adjoint problem (respectively) for . +Similarly, the Hessian action requires to additional solve the incremental forward and adjoint problems.

    +

    Discrete Newton system:

    +

    + + + +

    +

    Let us denote the vectors corresponding to the discretization of the functions by and of the functions by .

    +

    Then, the discretization of the above system is given by the following symmetric linear system:

    +

    + +

    +

    The gradient is computed using the following three steps

    +
      +
    • Given we solve the forward problem
    • +
    +

    + +

    +

    where stems from the discretization , and stands for the discretization of the right hand side .

    +
      +
    • Given and solve the adjoint problem
    • +
    +

    + +

    +

    where stems from the discretization of , is the mass matrix corresponding to the inner product in the state space, and stems from the data.

    +
      +
    • Define the gradient
    • +
    +

    + +

    +

    where is the matrix stemming from discretization of the regularization operator , and stems from discretization of the term .

    +

    Similarly the action of the Hessian in a direction (by using the CG algorithm we only need the action of to solve the Newton step) is given by

    +
      +
    • Solve the incremental forward problem
    • +
    +

    + +

    +

    where stems from discretization of .

    +
      +
    • Solve the incremental adjoint problem
    • +
    +

    + +

    +

    where stems for the discretization of .

    +
      +
    • Define the Hessian action
    • +
    +

    + +

    +

    Goals:

    +

    By the end of this notebook, you should be able to:

    +
      +
    • solve the forward and adjoint Poisson equations
    • +
    • understand the inverse method framework
    • +
    • visualise and understand the results
    • +
    • modify the problem and code
    • +
    +

    Mathematical tools used:

    +
      +
    • Finite element method
    • +
    • Derivation of gradiant and Hessian via the adjoint method
    • +
    • Inexact Newton-CG
    • +
    • Armijo line search
    • +
    +

    List of software used:

    +
      +
    • FEniCS, a parallel finite element element library for the discretization of partial differential equations
    • +
    • PETSc, for scalable and efficient linear algebra operations and solvers
    • +
    • Matplotlib, a python package used for plotting the results
    • +
    +

    Set up

    +

    Import dependencies

    +
    from __future__ import absolute_import, division, print_function
    +
    +from dolfin import *
    +
    +import sys
    +import os
    +sys.path.append( os.environ.get('HIPPYLIB_BASE_DIR', "../") )
    +from hippylib import *
    +
    +import logging
    +
    +import matplotlib.pyplot as plt
    +%matplotlib inline
    +
    +logging.getLogger('FFC').setLevel(logging.WARNING)
    +logging.getLogger('UFL').setLevel(logging.WARNING)
    +set_log_active(False)
    +
    +

    Model set up:

    +

    As in the introduction, the first thing we need to do is set up the numerical model. In this cell, we set the mesh, the finite element functions corresponding to state, adjoint and coefficient/gradient variables, and the corresponding test functions and the parameters for the optimization.

    +
    # create mesh and define function spaces
    +nx = 64
    +ny = 64
    +mesh = UnitSquareMesh(nx, ny)
    +Vm = FunctionSpace(mesh, 'Lagrange', 1)
    +Vu = FunctionSpace(mesh, 'Lagrange', 2)
    +
    +# The true and inverted parameter
    +mtrue = interpolate(Expression('log(2 + 7*(pow(pow(x[0] - 0.5,2) + pow(x[1] - 0.5,2),0.5) > 0.2))', degree=5),Vm)
    +m = interpolate(Expression("log(2.0)", degree=1),Vm)
    +
    +# define function for state and adjoint
    +u = Function(Vu)
    +p = Function(Vu)
    +
    +# define Trial and Test Functions
    +u_trial, p_trial, m_trial = TrialFunction(Vu), TrialFunction(Vu), TrialFunction(Vm)
    +u_test, p_test, m_test = TestFunction(Vu), TestFunction(Vu), TestFunction(Vm)
    +
    +# initialize input functions
    +f = Constant("1.0")
    +u0 = Constant("0.0")
    +
    +# plot
    +plt.figure(figsize=(15,5))
    +nb.plot(mesh,subplot_loc=121, mytitle="Mesh", show_axis='on')
    +nb.plot(mtrue,subplot_loc=122, mytitle="True parameter field")
    +plt.show()
    +
    +

    png

    +
    # set up dirichlet boundary conditions
    +def boundary(x,on_boundary):
    +    return on_boundary
    +
    +bc_state = DirichletBC(Vu, u0, boundary)
    +bc_adj = DirichletBC(Vu, Constant(0.), boundary)
    +
    +

    Set up synthetic observations:

    +
      +
    • Propose a coefficient field shown above
    • +
    • +

      The weak form of the pde: + Find such that .

      +
    • +
    • +

      Perturb the solution: , where +

      +
    • +
    +
    # noise level
    +noise_level = 0.05
    +
    +# weak form for setting up the synthetic observations
    +a_goal = inner(exp(mtrue) * nabla_grad(u_trial), nabla_grad(u_test)) * dx
    +L_goal = f * u_test * dx
    +
    +# solve the forward/state problem to generate synthetic observations
    +goal_A, goal_b = assemble_system(a_goal, L_goal, bc_state)
    +
    +utrue = Function(Vu)
    +solve(goal_A, utrue.vector(), goal_b)
    +
    +ud = Function(Vu)
    +ud.assign(utrue)
    +
    +# perturb state solution and create synthetic measurements ud
    +# ud = u + ||u||/SNR * random.normal
    +MAX = ud.vector().norm("linf")
    +noise = Vector()
    +goal_A.init_vector(noise,1)
    +parRandom.normal(noise_level * MAX, noise)
    +bc_adj.apply(noise)
    +
    +ud.vector().axpy(1., noise)
    +
    +# plot
    +nb.multi1_plot([utrue, ud], ["State solution with mtrue", "Synthetic observations"])
    +plt.show()
    +
    +

    png

    +

    The cost function evaluation:

    +

    + +

    +

    In the code below, and are symmetric positive definite matrices that stem from finite element discretization of the misfit and regularization component of the cost functional, respectively.

    +
    # regularization parameter
    +gamma = 1e-8
    +
    +# weak for for setting up the misfit and regularization compoment of the cost
    +W_equ   = inner(u_trial, u_test) * dx
    +R_equ   = gamma * inner(nabla_grad(m_trial), nabla_grad(m_test)) * dx
    +
    +W = assemble(W_equ)
    +R = assemble(R_equ)
    +
    +# refine cost function
    +def cost(u, ud, m, W, R):
    +    diff = u.vector() - ud.vector()
    +    reg = 0.5 * m.vector().inner(R*m.vector() ) 
    +    misfit = 0.5 * diff.inner(W * diff)
    +    return [reg + misfit, misfit, reg]
    +
    +

    Setting up the state equations, right hand side for the adjoint and the necessary matrices:

    +
    # weak form for setting up the state equation
    +a_state = inner(exp(m) * nabla_grad(u_trial), nabla_grad(u_test)) * dx
    +L_state = f * u_test * dx
    +
    +# weak form for setting up the adjoint equation
    +a_adj = inner(exp(m) * nabla_grad(p_trial), nabla_grad(p_test)) * dx
    +L_adj = -inner(u - ud, p_test) * dx
    +
    +# weak form for setting up matrices
    +Wum_equ = inner(exp(m) * m_trial * nabla_grad(p_test), nabla_grad(p)) * dx
    +C_equ   = inner(exp(m) * m_trial * nabla_grad(u), nabla_grad(u_test)) * dx
    +Wmm_equ = inner(exp(m) * m_trial * m_test *  nabla_grad(u),  nabla_grad(p)) * dx
    +
    +M_equ   = inner(m_trial, m_test) * dx
    +
    +# assemble matrix M
    +M = assemble(M_equ)
    +
    +

    Initial guess

    +

    We solve the state equation and compute the cost functional for the initial guess of the parameter m_ini

    +
    # solve state equation
    +state_A, state_b = assemble_system (a_state, L_state, bc_state)
    +solve (state_A, u.vector(), state_b)
    +
    +# evaluate cost
    +[cost_old, misfit_old, reg_old] = cost(u, ud, m, W, R)
    +
    +# plot
    +plt.figure(figsize=(15,5))
    +nb.plot(m,subplot_loc=121, mytitle="m_ini", vmin=mtrue.vector().min(), vmax=mtrue.vector().max())
    +nb.plot(u,subplot_loc=122, mytitle="u(m_ini)")
    +plt.show()
    +
    +

    png

    +

    The reduced Hessian apply to a vector :

    +

    Here we describe how to apply the reduced Hessian operator to a vector . For an opportune choice of the regularization, the reduced Hessian operator evaluated in a neighborhood of the solution is positive define, whereas far from the solution the reduced Hessian may be indefinite. On the constrary, the Gauss-Newton approximation of the Hessian is always positive defined.

    +

    For this reason, it is beneficial to perform a few initial Gauss-Newton steps (5 in this particular example) to accelerate the convergence of the inexact Newton-CG algorithm.

    +

    The Hessian apply reads: + +

    +

    The Gauss-Newton Hessian apply is obtained by dropping the second derivatives operators , , and : + +

    +
    # Class HessianOperator to perform Hessian apply to a vector
    +class HessianOperator():
    +    cgiter = 0
    +    def __init__(self, R, Wmm, C, A, adj_A, W, Wum, gauss_newton_approx=False):
    +        self.R = R
    +        self.Wmm = Wmm
    +        self.C = C
    +        self.A = A
    +        self.adj_A = adj_A
    +        self.W = W
    +        self.Wum = Wum
    +        self.gauss_newton_approx = gauss_newton_approx
    +
    +        # incremental state
    +        self.du = Vector()
    +        self.A.init_vector(self.du,0)
    +
    +        # incremental adjoint
    +        self.dp = Vector()
    +        self.adj_A.init_vector(self.dp,0)
    +
    +        # auxiliary vectors
    +        self.CT_dp = Vector()
    +        self.C.init_vector(self.CT_dp, 1)
    +        self.Wum_du = Vector()
    +        self.Wum.init_vector(self.Wum_du, 1)
    +
    +    def init_vector(self, v, dim):
    +        self.R.init_vector(v,dim)
    +
    +    # Hessian performed on v, output as generic vector y
    +    def mult(self, v, y):
    +        self.cgiter += 1
    +        y.zero()
    +        if self.gauss_newton_approx:
    +            self.mult_GaussNewton(v,y)
    +        else:
    +            self.mult_Newton(v,y)
    +
    +    # define (Gauss-Newton) Hessian apply H * v
    +    def mult_GaussNewton(self, v, y):
    +
    +        # incremental forward
    +        rhs = -(self.C * v)
    +        bc_adj.apply(rhs)
    +        solve (self.A, self.du, rhs)
    +
    +        # incremental adjoint
    +        rhs = - (self.W * self.du)
    +        bc_adj.apply(rhs)
    +        solve (self.adj_A, self.dp, rhs)
    +
    +        # Reg/Prior term
    +        self.R.mult(v,y)
    +
    +        # Misfit term
    +        self.C.transpmult(self.dp, self.CT_dp)
    +        y.axpy(1, self.CT_dp)
    +
    +    # define (Newton) Hessian apply H * v
    +    def mult_Newton(self, v, y):
    +
    +        # incremental forward
    +        rhs = -(self.C * v)
    +        bc_adj.apply(rhs)
    +        solve (self.A, self.du, rhs)
    +
    +        # incremental adjoint
    +        rhs = -(self.W * self.du) -  self.Wum * v
    +        bc_adj.apply(rhs)
    +        solve (self.adj_A, self.dp, rhs)
    +
    +        # Reg/Prior term
    +        self.R.mult(v,y)
    +        y.axpy(1., self.Wmm*v)
    +
    +        # Misfit term
    +        self.C.transpmult(self.dp, self.CT_dp)
    +        y.axpy(1., self.CT_dp)
    +        self.Wum.transpmult(self.du, self.Wum_du)
    +        y.axpy(1., self.Wum_du)
    +
    + +

    We solve the constrained optimization problem using the inexact Newton-CG method with Armijo line search.

    +

    The stopping criterion is based on a relative reduction of the norm of the gradient (i.e. ).

    +

    First, we compute the gradient by solving the state and adjoint equation for the current parameter , and then substituing the current state , parameter and adjoint variables in the weak form expression of the gradient: + +

    +

    Then, we compute the Newton direction by iteratively solving . +The Newton system is solved inexactly by early termination of conjugate gradient iterations via Eisenstat–Walker (to prevent oversolving) and Steihaug (to avoid negative curvature) criteria.

    +

    Finally, the Armijo line search uses backtracking to find such that a sufficient reduction in the cost functional is achieved. +More specifically, we use backtracking to find such that: + +

    +
    # define parameters for the optimization
    +tol = 1e-8
    +c = 1e-4
    +maxiter = 12
    +plot_on = False
    +
    +# initialize iter counters
    +iter = 1
    +total_cg_iter = 0
    +converged = False
    +
    +# initializations
    +g, m_delta = Vector(), Vector()
    +R.init_vector(m_delta,0)
    +R.init_vector(g,0)
    +
    +m_prev = Function(Vm)
    +
    +print ("Nit   CGit   cost          misfit        reg           sqrt(-G*D)    ||grad||       alpha  tolcg")
    +
    +while iter <  maxiter and not converged:
    +
    +    # assemble matrix C
    +    C =  assemble(C_equ)
    +
    +    # solve the adoint problem
    +    adjoint_A, adjoint_RHS = assemble_system(a_adj, L_adj, bc_adj)
    +    solve(adjoint_A, p.vector(), adjoint_RHS)
    +
    +    # assemble W_ua and R
    +    Wum = assemble (Wum_equ)
    +    Wmm = assemble (Wmm_equ)
    +
    +    # evaluate the  gradient
    +    CT_p = Vector()
    +    C.init_vector(CT_p,1)
    +    C.transpmult(p.vector(), CT_p)
    +    MG = CT_p + R * m.vector()
    +    solve(M, g, MG)
    +
    +    # calculate the norm of the gradient
    +    grad2 = g.inner(MG)
    +    gradnorm = sqrt(grad2)
    +
    +    # set the CG tolerance (use Eisenstat–Walker termination criterion)
    +    if iter == 1:
    +        gradnorm_ini = gradnorm
    +    tolcg = min(0.5, sqrt(gradnorm/gradnorm_ini))
    +
    +    # define the Hessian apply operator (with preconditioner)
    +    Hess_Apply = HessianOperator(R, Wmm, C, state_A, adjoint_A, W, Wum, gauss_newton_approx=(iter<6) )
    +    P = R + gamma * M
    +    Psolver = PETScKrylovSolver("cg", amg_method())
    +    Psolver.set_operator(P)
    +
    +    solver = CGSolverSteihaug()
    +    solver.set_operator(Hess_Apply)
    +    solver.set_preconditioner(Psolver)
    +    solver.parameters["rel_tolerance"] = tolcg
    +    solver.parameters["zero_initial_guess"] = True
    +    solver.parameters["print_level"] = -1
    +
    +    # solve the Newton system H a_delta = - MG
    +    solver.solve(m_delta, -MG)
    +    total_cg_iter += Hess_Apply.cgiter
    +
    +    # linesearch
    +    alpha = 1
    +    descent = 0
    +    no_backtrack = 0
    +    m_prev.assign(m)
    +    while descent == 0 and no_backtrack < 10:
    +        m.vector().axpy(alpha, m_delta )
    +
    +        # solve the state/forward problem
    +        state_A, state_b = assemble_system(a_state, L_state, bc_state)
    +        solve(state_A, u.vector(), state_b)
    +
    +        # evaluate cost
    +        [cost_new, misfit_new, reg_new] = cost(u, ud, m, W, R)
    +
    +        # check if Armijo conditions are satisfied
    +        if cost_new < cost_old + alpha * c * MG.inner(m_delta):
    +            cost_old = cost_new
    +            descent = 1
    +        else:
    +            no_backtrack += 1
    +            alpha *= 0.5
    +            m.assign(m_prev)  # reset a
    +
    +    # calculate sqrt(-G * D)
    +    graddir = sqrt(- MG.inner(m_delta) )
    +
    +    sp = ""
    +    print( "%2d %2s %2d %3s %8.5e %1s %8.5e %1s %8.5e %1s %8.5e %1s %8.5e %1s %5.2f %1s %5.3e" % \
    +        (iter, sp, Hess_Apply.cgiter, sp, cost_new, sp, misfit_new, sp, reg_new, sp, \
    +         graddir, sp, gradnorm, sp, alpha, sp, tolcg) )
    +
    +    if plot_on:
    +        nb.multi1_plot([m,u,p], ["m","u","p"], same_colorbar=False)
    +        plt.show()
    +
    +    # check for convergence
    +    if gradnorm < tol and iter > 1:
    +        converged = True
    +        print( "Newton's method converged in ",iter,"  iterations")
    +        print( "Total number of CG iterations: ", total_cg_iter)
    +
    +    iter += 1
    +
    +if not converged:
    +    print( "Newton's method did not converge in ", maxiter, " iterations")
    +
    +
    Nit   CGit   cost          misfit        reg           sqrt(-G*D)    ||grad||       alpha  tolcg
    + 1     1     1.12916e-05   1.12916e-05   1.34131e-11   1.56616e-02   3.79614e-04    1.00   5.000e-01
    + 2     1     7.83203e-07   7.83166e-07   3.68374e-11   4.68686e-03   5.35268e-05    1.00   3.755e-01
    + 3     1     3.12289e-07   3.12240e-07   4.92387e-11   9.73515e-04   7.14567e-06    1.00   1.372e-01
    + 4     6     1.91792e-07   1.61389e-07   3.04037e-08   4.54694e-04   1.00593e-06    1.00   5.148e-02
    + 5     1     1.86420e-07   1.56000e-07   3.04197e-08   1.03668e-04   6.15515e-07    1.00   4.027e-02
    + 6    11     1.80340e-07   1.36887e-07   4.34527e-08   1.12151e-04   2.14951e-07    1.00   2.380e-02
    + 7     5     1.80268e-07   1.38103e-07   4.21646e-08   1.19478e-05   3.96243e-08    1.00   1.022e-02
    + 8    15     1.80266e-07   1.38241e-07   4.20247e-08   1.70777e-06   3.37236e-09    1.00   2.981e-03
    +Newton's method converged in  8   iterations
    +Total number of CG iterations:  41
    +
    +
    nb.multi1_plot([mtrue, m], ["mtrue", "m"])
    +nb.multi1_plot([u,p], ["u","p"], same_colorbar=False)
    +plt.show()
    +
    +

    png

    +

    png

    +

    Copyright (c) 2016-2018, The University of Texas at Austin & University of California, Merced.
    +All Rights reserved.
    +See file COPYRIGHT for details.

    +

    This file is part of the hIPPYlib library. For more information and source code +availability see https://hippylib.github.io.

    +

    hIPPYlib is free software; you can redistribute it and/or modify it under the terms of the GNU General Public License (as published by the Free Software Foundation) version 2.0 dated June 1991.

    +
    +
    + +
    +
    +

    Copyright © 2016-2023 The University of Texas at Austin, University of California Merced.
    Copyright © 2018-2020 The University of Texas at Austin, University of California Merced, Washington University in St. Louis.
    This material is based on work partially supported by the National Science Foundation under Grants No ACI-1550593, ACI-1550547.

    +

    Documentation built with MkDocs.

    +
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    + +
    +
    + +
    + +

    + +

    +

    Bayesian quantification of parameter uncertainty:

    +

    Estimating the Gaussian approximation of posterior pdf of the coefficient parameter field in an elliptic PDE

    +

    In this example we tackle the problem of quantifying the +uncertainty in the solution of an inverse problem governed by an +elliptic PDE via the Bayesian inference framework. +Hence, we state the inverse problem as a +problem of statistical inference over the space of uncertain +parameters, which are to be inferred from data and a physical +model. The resulting solution to the statistical inverse problem +is a posterior distribution that assigns to any candidate set of +parameter fields our belief (expressed as a probability) that a +member of this candidate set is the ``true'' parameter field that +gave rise to the observed data.

    +

    For simplicity, in what follows we give finite-dimensional expressions (i.e., after +discretization of the parameter space) for the Bayesian +formulation of the inverse problem.

    +

    Bayes' Theorem:

    +

    The posterior probability distribution combines the prior pdf + over the parameter space, which encodes +any knowledge or assumptions about the parameter space that we may +wish to impose before the data are considered, with a likelihood pdf +, which explicitly +represents the probability that a given set of parameters +might give rise to the observed data , namely:

    +

    + +

    +

    Note that infinite-dimensional analog of Bayes' formula requires the use Radon-Nikodym derivatives instead of probability density functions.

    +

    Gaussian prior and noise:

    +

    The prior:

    +

    We consider a Gaussian prior with mean and covariance . The covariance is given by the discretization of the inverse of differential operator , where , control the correlation length and the variance of the prior operator. This choice of prior ensures that it is a trace-class operator, guaranteeing bounded pointwise variance and a well-posed infinite-dimensional Bayesian inverse problem.

    +

    The likelihood:

    +

    + +

    +

    + +

    +

    Here is the parameter-to-observable map that takes a parameter vector and maps +it to the space observation vector .

    +

    The posterior:

    +

    + +

    +

    The Gaussian approximation of the posterior: +

    +

    The mean of this posterior distribution, , is the +parameter vector maximizing the posterior, and +is known as the maximum a posteriori (MAP) point. It can be found +by minimizing the negative log of the posterior, which amounts to +solving a deterministic inverse problem with appropriately weighted norms,

    +

    + +

    +

    The posterior covariance matrix is then given by the inverse of +the Hessian matrix of at , namely

    +

    + +

    +

    The generalized eigenvalue problem:

    +

    + +

    +

    where +contains the generalized eigenvalues and the columns of the generalized eigenvectors such that +.

    +

    Randomized eigensolvers to construct the approximate spectral decomposition:

    +

    When the generalized eigenvalues decay rapidly, we can +extract a low-rank approximation of by retaining only the +largest eigenvalues and corresponding eigenvectors,

    +

    + +

    +

    Here, contains only the +generalized eigenvectors of that correspond to the largest eigenvalues, +which are assembled into the diagonal matrix .

    +

    The approximate posterior covariance:

    +

    Using the Sherman–Morrison–Woodbury formula, we write

    +

    + +

    +

    where . The last term in this expression captures the +error due to truncation in terms of the discarded eigenvalues; this +provides a criterion for truncating the spectrum, namely that is +chosen such that is small relative to 1.

    +

    Therefore we can approximate the posterior covariance as

    +

    + +

    +

    Drawing samples from a Gaussian distribution with covariance +

    +

    Let be a sample for the prior distribution, i.e. , then, using the low rank approximation of the posterior covariance, we compute a sample as

    +

    + +

    +

    This tutorial shows:

    +
      +
    • Description of the inverse problem (the forward problem, the prior, and the misfit functional)
    • +
    • Convergence of the inexact Newton-CG algorithm
    • +
    • Low-rank-based approximation of the posterior covariance (built on a low-rank +approximation of the Hessian of the data misfit)
    • +
    • How to construct the low-rank approximation of the Hessian of the data misfit
    • +
    • How to apply the inverse and square-root inverse Hessian to a vector efficiently
    • +
    • Samples from the Gaussian approximation of the posterior
    • +
    +

    Goals:

    +

    By the end of this notebook, you should be able to:

    +
      +
    • Understand the Bayesian inverse framework
    • +
    • Visualise and understand the results
    • +
    • Modify the problem and code
    • +
    +

    Mathematical tools used:

    +
      +
    • Finite element method
    • +
    • Derivation of gradient and Hessian via the adjoint method
    • +
    • inexact Newton-CG
    • +
    • Armijo line search
    • +
    • Bayes' formula
    • +
    • randomized eigensolvers
    • +
    +

    List of software used:

    +
      +
    • FEniCS, a parallel finite element element library for the discretization of partial differential equations
    • +
    • PETSc, for scalable and efficient linear algebra operations and solvers
    • +
    • Matplotlib, A great python package that I used for plotting many of the results
    • +
    • Numpy, A python package for linear algebra. While extensive, this is mostly used to compute means and sums in this notebook.
    • +
    +

    1. Load modules

    +
    from __future__ import absolute_import, division, print_function
    +
    +import dolfin as dl
    +import math
    +import numpy as np
    +import matplotlib.pyplot as plt
    +%matplotlib inline
    +import sys
    +import os
    +sys.path.append( os.environ.get('HIPPYLIB_BASE_DIR', "../") )
    +from hippylib import *
    +
    +import logging
    +logging.getLogger('FFC').setLevel(logging.WARNING)
    +logging.getLogger('UFL').setLevel(logging.WARNING)
    +dl.set_log_active(False)
    +
    +np.random.seed(seed=1)
    +
    +

    2. Generate the true parameter

    +

    This function generates a random field with a prescribed anysotropic covariance function.

    +
    def true_model(prior):
    +    noise = dl.Vector()
    +    prior.init_vector(noise,"noise")
    +    parRandom.normal(1., noise)
    +    mtrue = dl.Vector()
    +    prior.init_vector(mtrue, 0)
    +    prior.sample(noise,mtrue)
    +    return mtrue
    +
    +

    3. Set up the mesh and finite element spaces

    +

    We compute a two dimensional mesh of a unit square with nx by ny elements. +We define a P2 finite element space for the state and adjoint variable and P1 for the parameter.

    +
    ndim = 2
    +nx = 64
    +ny = 64
    +mesh = dl.UnitSquareMesh(nx, ny)
    +Vh2 = dl.FunctionSpace(mesh, 'Lagrange', 2)
    +Vh1 = dl.FunctionSpace(mesh, 'Lagrange', 1)
    +Vh = [Vh2, Vh1, Vh2]
    +print( "Number of dofs: STATE={0}, PARAMETER={1}, ADJOINT={2}".format(
    +    Vh[STATE].dim(), Vh[PARAMETER].dim(), Vh[ADJOINT].dim()) )
    +
    +
    Number of dofs: STATE=16641, PARAMETER=4225, ADJOINT=16641
    +
    +

    4. Set up the forward problem

    +

    To set up the forward problem we use the PDEVariationalProblem class, which requires the following inputs +- the finite element spaces for the state, parameter, and adjoint variables Vh +- the pde in weak form pde_varf +- the boundary conditions bc for the forward problem and bc0 for the adjoint and incremental problems.

    +

    The PDEVariationalProblem class offer the following functionality: +- solving the forward/adjoint and incremental problems +- evaluate first and second partial derivative of the forward problem with respect to the state, parameter, and adojnt variables.

    +
    def u_boundary(x, on_boundary):
    +    return on_boundary and ( x[1] < dl.DOLFIN_EPS or x[1] > 1.0 - dl.DOLFIN_EPS)
    +
    +u_bdr = dl.Expression("x[1]", degree=1)
    +u_bdr0 = dl.Constant(0.0)
    +bc = dl.DirichletBC(Vh[STATE], u_bdr, u_boundary)
    +bc0 = dl.DirichletBC(Vh[STATE], u_bdr0, u_boundary)
    +
    +f = dl.Constant(0.0)
    +
    +def pde_varf(u,m,p):
    +    return dl.exp(m)*dl.inner(dl.nabla_grad(u), dl.nabla_grad(p))*dl.dx - f*p*dl.dx
    +
    +pde = PDEVariationalProblem(Vh, pde_varf, bc, bc0, is_fwd_linear=True)
    +
    +

    4. Set up the prior

    +

    To obtain the synthetic true paramter we generate a realization from the prior distribution. Here we assume a Gaussian prior with zero average and covariance matrix . The action of on a field is given by

    +

    + +where is chosen to minimize boundary artifacts. +Here is an s.p.d. anisotropic tensor of the form

    +

    + +

    +
    gamma = .1
    +delta = .5
    +
    +anis_diff = dl.Expression(code_AnisTensor2D, degree=1)
    +anis_diff.theta0 = 2.
    +anis_diff.theta1 = .5
    +anis_diff.alpha = math.pi/4
    +
    +prior = BiLaplacianPrior(Vh[PARAMETER], gamma, delta, anis_diff, robin_bc=True)
    +mtrue = true_model(prior)
    +
    +print("Prior regularization: (delta_x - gamma*Laplacian)^order: delta={0}, gamma={1}, order={2}".format(delta, gamma,2))    
    +
    +objs = [dl.Function(Vh[PARAMETER],mtrue), dl.Function(Vh[PARAMETER],prior.mean)]
    +mytitles = ["True Parameter", "Prior mean"]
    +nb.multi1_plot(objs, mytitles)
    +plt.show()
    +
    +model = Model(pde,prior, misfit)
    +
    +
    Prior regularization: (delta_x - gamma*Laplacian)^order: delta=0.5, gamma=0.1, order=2
    +
    +

    png

    +

    5. Set up the misfit functional and generate synthetic observations

    +

    To setup the observation operator , we generate (ntargets in the code below) random locations where to evaluate the value of the state.

    +

    Under the assumption of Gaussian additive noise, the likelihood function has the form

    +

    + +

    +

    where denotes the solution of the forward model at a given parameter .

    +

    The class PointwiseStateObservation implements the evaluation of the log-likelihood function and of its partial derivatives w.r.t. the state and parameter .

    +

    To generate the synthetic observation, we first solve the forward problem using the true parameter . Synthetic observations are obtained by perturbing the state variable at the observation points with a random Gaussian noise. +rel_noise is the signal to noise ratio.

    +
    ntargets = 50
    +rel_noise = 0.01
    +
    +#Targets only on the bottom
    +targets_x = np.random.uniform(0.1,0.9, [ntargets] )
    +targets_y = np.random.uniform(0.1,0.5, [ntargets] )
    +targets = np.zeros([ntargets, ndim])
    +targets[:,0] = targets_x
    +targets[:,1] = targets_y
    +#targets everywhere
    +#targets = np.random.uniform(0.1,0.9, [ntargets, ndim] )
    +print( "Number of observation points: {0}".format(ntargets) )
    +misfit = PointwiseStateObservation(Vh[STATE], targets)
    +
    +utrue = pde.generate_state()
    +x = [utrue, mtrue, None]
    +pde.solveFwd(x[STATE], x, 1e-9)
    +misfit.B.mult(x[STATE], misfit.d)
    +MAX = misfit.d.norm("linf")
    +noise_std_dev = rel_noise * MAX
    +parRandom.normal_perturb(noise_std_dev, misfit.d)
    +misfit.noise_variance = noise_std_dev*noise_std_dev
    +
    +vmax = max( utrue.max(), misfit.d.max() )
    +vmin = min( utrue.min(), misfit.d.min() )
    +
    +plt.figure(figsize=(15,5))
    +nb.plot(dl.Function(Vh[STATE], utrue), mytitle="True State", subplot_loc=121, vmin=vmin, vmax=vmax)
    +nb.plot_pts(targets, misfit.d, mytitle="Observations", subplot_loc=122, vmin=vmin, vmax=vmax)
    +plt.show()
    +
    +
    Number of observation points: 50
    +
    +

    png

    +

    6. Set up the model and test gradient and Hessian

    +

    The model is defined by three component: +- the PDEVariationalProblem pde which provides methods for the solution of the forward problem, adjoint problem, and incremental forward and adjoint problems. +- the Prior prior which provides methods to apply the regularization (precision) operator to a vector or to apply the prior covariance operator (i.e. to solve linear system with the regularization operator) +- the Misfit misfit which provides methods to compute the cost functional and its partial derivatives with respect to the state and parameter variables.

    +

    To test gradient and the Hessian of the model we use forward finite differences.

    +
    model = Model(pde, prior, misfit)
    +
    +m0 = dl.interpolate(dl.Expression("sin(x[0])", degree=5), Vh[PARAMETER])
    +_ = modelVerify(model, m0.vector(), 1e-12)
    +
    +
    (yy, H xx) - (xx, H yy) =  -2.8560765074414784e-14
    +
    +

    png

    +

    7. Compute the MAP point

    +

    We used the globalized Newtown-CG method to compute the MAP point.

    +
    m = prior.mean.copy()
    +solver = ReducedSpaceNewtonCG(model)
    +solver.parameters["rel_tolerance"] = 1e-6
    +solver.parameters["abs_tolerance"] = 1e-12
    +solver.parameters["max_iter"]      = 25
    +solver.parameters["inner_rel_tolerance"] = 1e-15
    +solver.parameters["GN_iter"] = 5
    +solver.parameters["globalization"] = "LS"
    +solver.parameters["LS"]["c_armijo"] = 1e-4
    +
    +
    +x = solver.solve([None, m, None])
    +
    +if solver.converged:
    +    print( "\nConverged in ", solver.it, " iterations.")
    +else:
    +    print( "\nNot Converged")
    +
    +print( "Termination reason: ", solver.termination_reasons[solver.reason] )
    +print( "Final gradient norm: ", solver.final_grad_norm )
    +print( "Final cost: ", solver.final_cost )
    +
    +plt.figure(figsize=(15,5))
    +nb.plot(dl.Function(Vh[STATE], x[STATE]), subplot_loc=121,mytitle="State")
    +nb.plot(dl.Function(Vh[PARAMETER], x[PARAMETER]), subplot_loc=122,mytitle="Parameter")
    +plt.show()
    +
    +
    It  cg_it cost            misfit          reg             (g,dm)          ||g||L2        alpha          tolcg         
    +  1   2    5.676752e+02    5.665943e+02    1.080947e+00   -1.487693e+04   4.008143e+04   1.000000e+00   5.000000e-01
    +  2   2    1.315109e+02    1.292859e+02    2.225043e+00   -8.730801e+02   7.040800e+03   1.000000e+00   4.191210e-01
    +  3   4    5.827697e+01    5.504162e+01    3.235343e+00   -1.565905e+02   2.115036e+03   1.000000e+00   2.297139e-01
    +  4   1    5.257433e+01    4.934371e+01    3.230622e+00   -1.142190e+01   1.827983e+03   1.000000e+00   2.135573e-01
    +  5   6    3.958923e+01    3.507197e+01    4.517257e+00   -3.227624e+01   9.179935e+02   1.000000e+00   1.513381e-01
    +  6   2    3.681315e+01    3.218885e+01    4.624304e+00   -5.501178e+00   7.541736e+02   1.000000e+00   1.371716e-01
    +  7  11    3.276506e+01    2.530117e+01    7.463891e+00   -9.109597e+00   4.788040e+02   1.000000e+00   1.092968e-01
    +  8   1    3.225207e+01    2.478982e+01    7.462243e+00   -1.027328e+00   4.890796e+02   1.000000e+00   1.104634e-01
    +  9  10    3.216228e+01    2.445226e+01    7.710018e+00   -1.799343e-01   7.389810e+01   1.000000e+00   4.293832e-02
    + 10  12    3.216003e+01    2.438903e+01    7.770998e+00   -4.481802e-03   1.291573e+01   1.000000e+00   1.795097e-02
    + 11  18    3.215993e+01    2.438404e+01    7.775888e+00   -2.010595e-04   2.284805e+00   1.000000e+00   7.550105e-03
    + 12  19    3.215993e+01    2.438428e+01    7.775646e+00   -8.282645e-08   6.202767e-02   1.000000e+00   1.244002e-03
    +
    +Converged in  12  iterations.
    +Termination reason:  Norm of the gradient less than tolerance
    +Final gradient norm:  0.0006914733735650482
    +Final cost:  32.15992905025892
    +
    +

    png

    +

    8. Compute the low rank Gaussian approximation of the posterior

    +

    We used the double pass algorithm to compute a low-rank decomposition of the Hessian Misfit. +In particular, we solve

    +

    + +

    +

    The Figure shows the largest k generalized eigenvectors of the Hessian misfit. +The effective rank of the Hessian misfit is the number of eigenvalues above the red line (). +The effective rank is independent of the mesh size.

    +
    model.setPointForHessianEvaluations(x, gauss_newton_approx=False)
    +Hmisfit = ReducedHessian(model, solver.parameters["inner_rel_tolerance"], misfit_only=True)
    +k = 50
    +p = 20
    +print( "Single/Double Pass Algorithm. Requested eigenvectors: {0}; Oversampling {1}.".format(k,p) )
    +
    +Omega = MultiVector(x[PARAMETER], k+p)
    +parRandom.normal(1., Omega)
    +lmbda, V = doublePassG(Hmisfit, prior.R, prior.Rsolver, Omega, k)
    +
    +posterior = GaussianLRPosterior(prior, lmbda, V)
    +posterior.mean = x[PARAMETER]
    +
    +plt.plot(range(0,k), lmbda, 'b*', range(0,k+1), np.ones(k+1), '-r')
    +plt.yscale('log')
    +plt.xlabel('number')
    +plt.ylabel('eigenvalue')
    +
    +nb.plot_eigenvectors(Vh[PARAMETER], V, mytitle="Eigenvector", which=[0,1,2,5,10,15])
    +
    +
    Single/Double Pass Algorithm. Requested eigenvectors: 50; Oversampling 20.
    +
    +

    png

    +

    png

    +

    9. Prior and posterior pointwise variance fields

    +
    compute_trace = True
    +if compute_trace:
    +    post_tr, prior_tr, corr_tr = posterior.trace(method="Randomized", r=200)
    +    print( "Posterior trace {0:5e}; Prior trace {1:5e}; Correction trace {2:5e}".format(post_tr, prior_tr, corr_tr) )
    +post_pw_variance, pr_pw_variance, corr_pw_variance = posterior.pointwise_variance(method="Randomized", r=200)
    +
    +objs = [dl.Function(Vh[PARAMETER], pr_pw_variance),
    +        dl.Function(Vh[PARAMETER], post_pw_variance)]
    +mytitles = ["Prior variance", "Posterior variance"]
    +nb.multi1_plot(objs, mytitles, logscale=False)
    +plt.show()
    +
    +
    Posterior trace 1.014727e+00; Prior trace 1.797376e+00; Correction trace 7.826486e-01
    +
    +

    png

    +

    10. Generate samples from Prior and Posterior

    +
    nsamples = 5
    +noise = dl.Vector()
    +posterior.init_vector(noise,"noise")
    +s_prior = dl.Function(Vh[PARAMETER], name="sample_prior")
    +s_post = dl.Function(Vh[PARAMETER], name="sample_post")
    +
    +pr_max   =  2.5*math.sqrt( pr_pw_variance.max() ) + prior.mean.max()
    +pr_min   = -2.5*math.sqrt( pr_pw_variance.max() ) + prior.mean.min()
    +ps_max   =  2.5*math.sqrt( post_pw_variance.max() ) + posterior.mean.max()
    +ps_min   = -2.5*math.sqrt( post_pw_variance.max() ) + posterior.mean.min()
    +
    +for i in range(nsamples):
    +    parRandom.normal(1., noise)
    +    posterior.sample(noise, s_prior.vector(), s_post.vector())
    +    plt.figure(figsize=(15,5))
    +    nb.plot(s_prior, subplot_loc=121,mytitle="Prior sample", vmin=pr_min, vmax=pr_max)
    +    nb.plot(s_post, subplot_loc=122,mytitle="Posterior sample", vmin=ps_min, vmax=ps_max)
    +    plt.show()
    +
    +

    png

    +

    png

    +

    png

    +

    png

    +

    png

    +

    Copyright (c) 2016-2018, The University of Texas at Austin & University of California, Merced.
    +All Rights reserved.
    +See file COPYRIGHT for details.

    +

    This file is part of the hIPPYlib library. For more information and source code +availability see https://hippylib.github.io.

    +

    hIPPYlib is free software; you can redistribute it and/or modify it under the terms of the GNU General Public License (as published by the Free Software Foundation) version 2.0 dated June 1991.

    +
    +
    + +
    +
    +

    Copyright © 2016-2023 The University of Texas at Austin, University of California Merced.
    Copyright © 2018-2020 The University of Texas at Austin, University of California Merced, Washington University in St. Louis.
    This material is based on work partially supported by the National Science Foundation under Grants No ACI-1550593, ACI-1550547.

    +

    Documentation built with MkDocs.

    +
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    + +
    +
    + +
    + +

    + +

    +

    Bayesian initial condition inversion in an advection-diffusion problem

    +

    In this example we tackle the problem of quantifying the uncertainty in the solution of an inverse problem governed by a parabolic PDE via the Bayesian inference framework. The underlying PDE is a time-dependent advection-diffusion equation in which we seek to infer an unknown initial condition from spatio-temporal point measurements.

    +

    The Bayesian inverse problem:

    +

    Following the Bayesian framework, we utilize +a Gaussian prior measure , +with where is an elliptic differential operator as +described in the PoissonBayesian example, and use an additive +Gaussian noise model. Therefore, the solution of the Bayesian inverse +problem is the posterior measure, with + and .

    +
      +
    • The posterior mean is characterized as the minimizer of
    • +
    +

    + +

    +

    which can also be interpreted as the regularized functional to be +minimized in deterministic inversion. The observation operator extracts the values of the forward solution on a set of +locations at +times .

    +
      +
    • The posterior covariance is the inverse of the Hessian of , i.e.,
    • +
    +

    + +

    +

    The forward problem:

    +

    The parameter-to-observable map maps an initial condition to pointwise spatiotemporal observations of the concentration field through solution of the advection-diffusion equation given by

    +

    + +

    +

    Here, () is a bounded domain, is the diffusion coefficient and is the final +time. The velocity field + is computed by solving the following steady-state +Navier-Stokes equation with the side walls driving the flow:

    +

    + +

    +

    Here, is pressure, is the Reynolds number. The Dirichlet boundary data + is given by + on the left wall of the domain, + on the right wall, and everywhere else.

    +

    The adjoint problem:

    +

    The adjoint problem is a final value problem, since is specified at rather than at . Thus, it is solved backwards in time, which amounts to the solution of the advection-diffusion equation

    +

    + +

    +

    Then, the adjoint of the parameter to observable map is defined by setting +

    +

    1. Load modules

    +
    from __future__ import absolute_import, division, print_function
    +
    +import dolfin as dl
    +import math
    +import numpy as np
    +import matplotlib.pyplot as plt
    +%matplotlib inline
    +import sys
    +import os
    +sys.path.append( os.environ.get('HIPPYLIB_BASE_DIR', "../") )
    +from hippylib import *
    +sys.path.append( os.environ.get('HIPPYLIB_BASE_DIR', "..") + "/applications/ad_diff/" )
    +from model_ad_diff import TimeDependentAD, SpaceTimePointwiseStateObservation
    +
    +import logging
    +logging.getLogger('FFC').setLevel(logging.WARNING)
    +logging.getLogger('UFL').setLevel(logging.WARNING)
    +dl.set_log_active(False)
    +
    +

    2. Construct the velocity field

    +
    def v_boundary(x,on_boundary):
    +    return on_boundary
    +
    +def q_boundary(x,on_boundary):
    +    return x[0] < dl.DOLFIN_EPS and x[1] < dl.DOLFIN_EPS
    +
    +def computeVelocityField(mesh):
    +    Xh = dl.VectorFunctionSpace(mesh,'Lagrange', 2)
    +    Wh = dl.FunctionSpace(mesh, 'Lagrange', 1)
    +    if dlversion() <= (1,6,0):
    +        XW = dl.MixedFunctionSpace([Xh, Wh])
    +    else:
    +        mixed_element = dl.MixedElement([Xh.ufl_element(), Wh.ufl_element()])
    +        XW = dl.FunctionSpace(mesh, mixed_element)
    +
    +    Re = 1e2
    +
    +    g = dl.Expression(('0.0','(x[0] < 1e-14) - (x[0] > 1 - 1e-14)'), degree=1)
    +    bc1 = dl.DirichletBC(XW.sub(0), g, v_boundary)
    +    bc2 = dl.DirichletBC(XW.sub(1), dl.Constant(0), q_boundary, 'pointwise')
    +    bcs = [bc1, bc2]
    +
    +    vq = dl.Function(XW)
    +    (v,q) = dl.split(vq)
    +    (v_test, q_test) = dl.TestFunctions (XW)
    +
    +    def strain(v):
    +        return dl.sym(dl.nabla_grad(v))
    +
    +    F = ( (2./Re)*dl.inner(strain(v),strain(v_test))+ dl.inner (dl.nabla_grad(v)*v, v_test)
    +           - (q * dl.div(v_test)) + ( dl.div(v) * q_test) ) * dl.dx
    +
    +    dl.solve(F == 0, vq, bcs, solver_parameters={"newton_solver":
    +                                         {"relative_tolerance":1e-4, "maximum_iterations":100}})
    +
    +    plt.figure(figsize=(15,5))
    +    vh = dl.project(v,Xh)
    +    qh = dl.project(q,Wh)
    +    nb.plot(nb.coarsen_v(vh), subplot_loc=121,mytitle="Velocity")
    +    nb.plot(qh, subplot_loc=122,mytitle="Pressure")
    +    plt.show()
    +
    +    return v
    +
    +

    3. Set up the mesh and finite element spaces

    +
    mesh = dl.refine( dl.Mesh("ad_20.xml") )
    +wind_velocity = computeVelocityField(mesh)
    +Vh = dl.FunctionSpace(mesh, "Lagrange", 1)
    +print( "Number of dofs: {0}".format( Vh.dim() ) )
    +
    +

    png

    +
    Number of dofs: 2023
    +
    +

    4. Set up model (prior, true/proposed initial condition)

    +
    ic_expr = dl.Expression('min(0.5,exp(-100*(pow(x[0]-0.35,2) +  pow(x[1]-0.7,2))))', element=Vh.ufl_element())
    +true_initial_condition = dl.interpolate(ic_expr, Vh).vector()
    +
    +
    +gamma = 1.
    +delta = 8.
    +prior = BiLaplacianPrior(Vh, gamma, delta, robin_bc=True)
    +print( "Prior regularization: (delta - gamma*Laplacian)^order: delta={0}, gamma={1}, order={2}".format(delta, gamma,2) )
    +prior.mean = dl.interpolate(dl.Constant(0.25), Vh).vector()
    +
    +t_init         = 0.
    +t_final        = 4.
    +t_1            = 1.
    +dt             = .1
    +observation_dt = .2
    +
    +simulation_times = np.arange(t_init, t_final+.5*dt, dt)
    +observation_times = np.arange(t_1, t_final+.5*dt, observation_dt)
    +
    +targets = np.loadtxt('targets.txt')
    +print ("Number of observation points: {0}".format(targets.shape[0]) )
    +misfit = SpaceTimePointwiseStateObservation(Vh, observation_times, targets)
    +
    +problem = TimeDependentAD(mesh, [Vh,Vh,Vh], prior, misfit, simulation_times, wind_velocity, True)
    +
    +objs = [dl.Function(Vh,true_initial_condition),
    +        dl.Function(Vh,prior.mean)]
    +mytitles = ["True Initial Condition", "Prior mean"]
    +nb.multi1_plot(objs, mytitles)
    +plt.show()
    +
    +
    Prior regularization: (delta - gamma*Laplacian)^order: delta=8.0, gamma=1.0, order=2
    +Number of observation points: 80
    +
    +

    png

    +

    5. Generate the synthetic observations

    +
    rel_noise = 0.01
    +utrue = problem.generate_vector(STATE)
    +x = [utrue, true_initial_condition, None]
    +problem.solveFwd(x[STATE], x, 1e-9)
    +misfit.observe(x, misfit.d)
    +MAX = misfit.d.norm("linf", "linf")
    +noise_std_dev = rel_noise * MAX
    +parRandom.normal_perturb(noise_std_dev,misfit.d)
    +misfit.noise_variance = noise_std_dev*noise_std_dev
    +
    +nb.show_solution(Vh, true_initial_condition, utrue, "Solution")
    +
    +

    png

    +

    6. Test the gradient and the Hessian of the cost (negative log posterior)

    +
    m0 = true_initial_condition.copy()
    +_ = modelVerify(problem, m0, 1e-12, is_quadratic=True)
    +
    +
    (yy, H xx) - (xx, H yy) =  -1.174447494183385e-13
    +
    +

    png

    +

    7. Evaluate the gradient

    +
    [u,m,p] = problem.generate_vector()
    +problem.solveFwd(u, [u,m,p], 1e-12)
    +problem.solveAdj(p, [u,m,p], 1e-12)
    +mg = problem.generate_vector(PARAMETER)
    +grad_norm = problem.evalGradientParameter([u,m,p], mg)
    +
    +print( "(g,g) = ", grad_norm)
    +
    +
    (g,g) =  2395071437584061.5
    +
    +

    8. The Gaussian posterior

    +
    H = ReducedHessian(problem, 1e-12, misfit_only=True) 
    +
    +k = 80
    +p = 20
    +print( "Single Pass Algorithm. Requested eigenvectors: {0}; Oversampling {1}.".format(k,p) )
    +Omega = MultiVector(x[PARAMETER], k+p)
    +parRandom.normal(1., Omega)
    +lmbda, V = singlePassG(H, prior.R, prior.Rsolver, Omega, k)
    +
    +
    +posterior = GaussianLRPosterior( prior, lmbda, V )
    +
    +plt.plot(range(0,k), lmbda, 'b*', range(0,k+1), np.ones(k+1), '-r')
    +plt.yscale('log')
    +plt.xlabel('number')
    +plt.ylabel('eigenvalue')
    +
    +nb.plot_eigenvectors(Vh, V, mytitle="Eigenvector", which=[0,1,2,5,10,20,30,45,60])
    +
    +
    Single Pass Algorithm. Requested eigenvectors: 80; Oversampling 20.
    +
    +

    png

    +

    png

    +

    9. Compute the MAP point

    +
    H.misfit_only = False
    +
    +solver = CGSolverSteihaug()
    +solver.set_operator(H)
    +solver.set_preconditioner( posterior.Hlr )
    +solver.parameters["print_level"] = 1
    +solver.parameters["rel_tolerance"] = 1e-6
    +solver.solve(m, -mg)
    +problem.solveFwd(u, [u,m,p], 1e-12)
    +
    +total_cost, reg_cost, misfit_cost = problem.cost([u,m,p])
    +print( "Total cost {0:5g}; Reg Cost {1:5g}; Misfit {2:5g}".format(total_cost, reg_cost, misfit_cost) )
    +
    +posterior.mean = m
    +
    +plt.figure(figsize=(7.5,5))
    +nb.plot(dl.Function(Vh, m), mytitle="Initial Condition")
    +plt.show()
    +
    +nb.show_solution(Vh, m, u, "Solution")
    +
    +
     Iterartion :  0  (B r, r) =  1225123.9338492027
    + Iteration :  1  (B r, r) =  74.94198007028052
    + Iteration :  2  (B r, r) =  0.4132940000407347
    + Iteration :  3  (B r, r) =  0.005765443940190191
    + Iteration :  4  (B r, r) =  2.243708962553132e-05
    + Iteration :  5  (B r, r) =  1.4026897765563288e-07
    +Relative/Absolute residual less than tol
    +Converged in  5  iterations with final norm  0.0003745250027109444
    +Total cost 782.818; Reg Cost 153.671; Misfit 629.147
    +
    +

    png

    +

    png

    +

    10. Prior and posterior pointwise variance fields

    +
    compute_trace = True
    +if compute_trace:
    +    post_tr, prior_tr, corr_tr = posterior.trace(method="Randomized", r=300)
    +    print( "Posterior trace {0:5g}; Prior trace {1:5g}; Correction trace {2:5g}".format(post_tr, prior_tr, corr_tr) )
    +post_pw_variance, pr_pw_variance, corr_pw_variance = posterior.pointwise_variance(method="Randomized", r=300)
    +
    +objs = [dl.Function(Vh, pr_pw_variance),
    +        dl.Function(Vh, post_pw_variance)]
    +mytitles = ["Prior Variance", "Posterior Variance"]
    +nb.multi1_plot(objs, mytitles, logscale=False)
    +plt.show()
    +
    +
    Posterior trace 0.000269147; Prior trace 0.00809706; Correction trace 0.00782792
    +
    +

    png

    +

    11. Draw samples from the prior and posterior distributions

    +
    nsamples = 5
    +noise = dl.Vector()
    +posterior.init_vector(noise,"noise")
    +s_prior = dl.Function(Vh, name="sample_prior")
    +s_post = dl.Function(Vh, name="sample_post")
    +
    +pr_max =  2.5*math.sqrt( pr_pw_variance.max() ) + prior.mean.max()
    +pr_min = -2.5*math.sqrt( pr_pw_variance.min() ) + prior.mean.min()
    +ps_max =  2.5*math.sqrt( post_pw_variance.max() ) + posterior.mean.max()
    +ps_min = -2.5*math.sqrt( post_pw_variance.max() ) + posterior.mean.min()
    +
    +for i in range(nsamples):
    +    parRandom.normal(1., noise)
    +    posterior.sample(noise, s_prior.vector(), s_post.vector())
    +    plt.figure(figsize=(15,5))
    +    nb.plot(s_prior, subplot_loc=121,mytitle="Prior sample", vmin=pr_min, vmax=pr_max)
    +    nb.plot(s_post, subplot_loc=122,mytitle="Posterior sample", vmin=ps_min, vmax=ps_max)
    +    plt.show()
    +
    +

    png

    +

    png

    +

    png

    +

    png

    +

    png

    +

    Copyright (c) 2016-2018, The University of Texas at Austin & University of California, Merced.
    +All Rights reserved.
    +See file COPYRIGHT for details.

    +

    This file is part of the hIPPYlib library. For more information and source code +availability see https://hippylib.github.io.

    +

    hIPPYlib is free software; you can redistribute it and/or modify it under the terms of the GNU General Public License (as published by the Free Software Foundation) version 2.0 dated June 1991.

    +
    +
    + +
    +
    +

    Copyright © 2016-2023 The University of Texas at Austin, University of California Merced.
    Copyright © 2018-2020 The University of Texas at Austin, University of California Merced, Washington University in St. Louis.
    This material is based on work partially supported by the National Science Foundation under Grants No ACI-1550593, ACI-1550547.

    +

    Documentation built with MkDocs.

    +
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+++ b/tutorials_v2.3.0/5_HessianSpectrum/5_HessianSpectrum/index.html @@ -0,0 +1,600 @@ + + + + + + + + + + + Hessian Spectrum - hIPPYlib + + + + + + + + + +

    + +
    +
    + +
    + +

    Spectrum of the preconditioned Hessian misfit operator

    +

    The linear source inversion problem

    +

    We consider the following linear source inversion problem. +Find the state and the source (parameter) that solves

    +

    + +

    +

    Here:

    +
      +
    • +

      + is a finite dimensional vector that denotes noisy observations of the state in locations . Specifically, , where are i.i.d. .

      +
    • +
    • +

      + is the linear operator that evaluates the state at the observation locations .

      +
    • +
    • +

      + and are the parameters of the regularization penalizing the and norm of , respectively.

      +
    • +
    • +

      +, , are given coefficients representing the diffusivity coefficient, the advective velocity and the reaction term, respectively.

      +
    • +
    • +

      +, represents the subdomain of where we impose Dirichlet or Neumann boundary conditions, respectively.

      +
    • +
    +

    1. Load modules

    +
    from __future__ import absolute_import, division, print_function
    +
    +import dolfin as dl
    +import numpy as np
    +import matplotlib.pyplot as plt
    +%matplotlib inline
    +
    +import sys
    +import os
    +sys.path.append( os.environ.get('HIPPYLIB_BASE_DIR', "../") )
    +from hippylib import *
    +
    +import logging
    +logging.getLogger('FFC').setLevel(logging.WARNING)
    +logging.getLogger('UFL').setLevel(logging.WARNING)
    +dl.set_log_active(False)
    +
    +

    2. The linear source inversion problem

    +
    def pde_varf(u,m,p):
    +    return k*dl.inner(dl.nabla_grad(u), dl.nabla_grad(p))*dl.dx \
    +           + dl.inner(dl.nabla_grad(u), v*p)*dl.dx \
    +           + c*u*p*dl.dx \
    +           - m*p*dl.dx
    +
    +def u_boundary(x, on_boundary):
    +    return on_boundary and x[1] < dl.DOLFIN_EPS
    +
    +def solve(nx,ny, targets, rel_noise, gamma, delta, verbose=True):
    +    myRandom = Random()
    +    mesh = dl.UnitSquareMesh(nx, ny)
    +    Vh1 = dl.FunctionSpace(mesh, 'Lagrange', 1)
    +
    +    Vh = [Vh1, Vh1, Vh1]
    +    if verbose:
    +        print( "Number of dofs: STATE={0}, PARAMETER={1}, ADJOINT={2}".format(Vh[STATE].dim(), Vh[PARAMETER].dim(), Vh[ADJOINT].dim()) )
    +
    +
    +    u_bdr = dl.Constant(0.0)
    +    u_bdr0 = dl.Constant(0.0)
    +    bc = dl.DirichletBC(Vh[STATE], u_bdr, u_boundary)
    +    bc0 = dl.DirichletBC(Vh[STATE], u_bdr0, u_boundary)
    +
    +    mtrue = dl.interpolate( dl.Expression('min(0.5,exp(-100*(pow(x[0]-0.35,2) +  pow(x[1]-0.7,2))))',degree=5), Vh[PARAMETER]).vector()
    +    m0 = dl.interpolate(dl.Constant(0.0), Vh[PARAMETER]).vector()
    +
    +    pde = PDEVariationalProblem(Vh, pde_varf, bc, bc0, is_fwd_linear=True)
    +
    +    if verbose:
    +        print( "Number of observation points: {0}".format(targets.shape[0]) )
    +
    +    misfit = PointwiseStateObservation(Vh[STATE], targets)
    +
    +    reg = LaplacianPrior(Vh[PARAMETER], gamma, delta)
    +
    +    #Generate synthetic observations
    +    utrue = pde.generate_state()
    +    x = [utrue, mtrue, None]
    +    pde.solveFwd(x[STATE], x, 1e-9)
    +    misfit.B.mult(x[STATE], misfit.d)
    +    MAX = misfit.d.norm("linf")
    +    noise_std_dev = rel_noise * MAX
    +    myRandom.normal_perturb(noise_std_dev, misfit.d)
    +    misfit.noise_variance = noise_std_dev*noise_std_dev
    +
    +    if verbose:
    +        plt.figure(figsize=(18,4))
    +        nb.plot(dl.Function(Vh[PARAMETER], mtrue), mytitle = "True source", subplot_loc=131)
    +        nb.plot(dl.Function(Vh[STATE], utrue), mytitle="True state", subplot_loc=132)
    +        nb.plot_pts(targets, misfit.d,mytitle="Observations", subplot_loc=133)
    +        plt.show()
    +
    +    model = Model(pde, reg, misfit)
    +    u = model.generate_vector(STATE)
    +    m = m0.copy()
    +    p = model.generate_vector(ADJOINT)
    +    x = [u,m,p]
    +    mg = model.generate_vector(PARAMETER)
    +    model.solveFwd(u, x)
    +    model.solveAdj(p, x)
    +    model.evalGradientParameter(x, mg)
    +    model.setPointForHessianEvaluations(x, gauss_newton_approx=False)
    +
    +    H = ReducedHessian(model, 1e-12)
    +
    +    solver = CGSolverSteihaug()
    +    solver.set_operator(H)
    +    solver.set_preconditioner( reg.Rsolver )
    +    solver.parameters["print_level"] = -1
    +    solver.parameters["rel_tolerance"] = 1e-9
    +    solver.solve(m, -mg)
    +
    +    if solver.converged:
    +        if verbose:
    +            print( "CG converged in ", solver.iter, " iterations." )
    +    else:
    +        print( "CG did not converged." )
    +        raise
    +
    +    model.solveFwd(u, x, 1e-12)
    +
    +    total_cost, reg_cost, misfit_cost = model.cost(x)
    +
    +    if verbose:
    +        plt.figure(figsize=(18,4))
    +        nb.plot(dl.Function(Vh[PARAMETER], m), mytitle = "Reconstructed source", subplot_loc=131)
    +        nb.plot(dl.Function(Vh[STATE], u), mytitle="Reconstructed state", subplot_loc=132)
    +        nb.plot_pts(targets, misfit.B*u - misfit.d, mytitle="Misfit", subplot_loc=133)
    +        plt.show()
    +
    +    H.misfit_only = True
    +    k_evec = 80
    +    p_evec = 5
    +    if verbose:
    +        print( "Double Pass Algorithm. Requested eigenvectors: {0}; Oversampling {1}.".format(k_evec,p_evec) )
    +    Omega = MultiVector(x[PARAMETER], k_evec+p_evec)
    +    myRandom.normal(1., Omega)
    +    lmbda, V = doublePassG(H, reg.R, reg.Rsolver, Omega, k_evec)
    +
    +    if verbose:
    +        plt.figure()
    +        nb.plot_eigenvalues(lmbda, mytitle="Generalized Eigenvalues")
    +        nb.plot_eigenvectors(Vh[PARAMETER], V, mytitle="Eigenvectors", which=[0,1,2,5,10,15])
    +        plt.show()
    +
    +    return lmbda, V, Vh[PARAMETER], solver.iter
    +
    +
    +

    3. Solution of the source inversion problem

    +
    ndim = 2
    +nx = 32
    +ny = 32
    +
    +ntargets = 300
    +np.random.seed(seed=1)
    +targets = np.random.uniform(0.1,0.9, [ntargets, ndim] )
    +rel_noise = 0.01
    +
    +gamma = 70.
    +delta = 1e-1
    +
    +k = dl.Constant(1.0)
    +v = dl.Constant((0.0, 0.0))
    +c = dl.Constant(0.)
    +
    +lmbda, V, Vm, nit = solve(nx,ny, targets, rel_noise, gamma, delta)
    +
    +
    Number of dofs: STATE=1089, PARAMETER=1089, ADJOINT=1089
    +Number of observation points: 300
    +
    +

    png

    +
    CG converged in  70  iterations.
    +
    +

    png

    +
    Double Pass Algorithm. Requested eigenvectors: 80; Oversampling 5.
    +
    +

    png

    +

    png

    +

    4. Mesh independence of the spectrum of the preconditioned Hessian misfit

    +
    gamma = 70.
    +delta = 1e-1
    +
    +k = dl.Constant(1.0)
    +v = dl.Constant((0.0, 0.0))
    +c = dl.Constant(0.)
    +
    +n = [16,32,64]
    +lmbda1, V1, Vm1, niter1 = solve(n[0],n[0], targets, rel_noise, gamma, delta,verbose=False)
    +lmbda2, V2, Vm2, niter2 = solve(n[1],n[1], targets, rel_noise, gamma, delta,verbose=False)
    +lmbda3, V3, Vm3, niter3 = solve(n[2],n[2], targets, rel_noise, gamma, delta,verbose=False)
    +
    +print( "Number of Iterations: ", niter1, niter2, niter3 )
    +plt.figure(figsize=(18,4))
    +nb.plot_eigenvalues(lmbda1, mytitle="Eigenvalues Mesh {0} by {1}".format(n[0],n[0]), subplot_loc=131)
    +nb.plot_eigenvalues(lmbda2, mytitle="Eigenvalues Mesh {0} by {1}".format(n[1],n[1]), subplot_loc=132)
    +nb.plot_eigenvalues(lmbda3, mytitle="Eigenvalues Mesh {0} by {1}".format(n[2],n[2]), subplot_loc=133)
    +
    +nb.plot_eigenvectors(Vm1, V1, mytitle="Mesh {0} by {1} Eigen".format(n[0],n[0]), which=[0,1,5])
    +nb.plot_eigenvectors(Vm2, V2, mytitle="Mesh {0} by {1} Eigen".format(n[1],n[1]), which=[0,1,5])
    +nb.plot_eigenvectors(Vm3, V3, mytitle="Mesh {0} by {1} Eigen".format(n[2],n[2]), which=[0,1,5])
    +
    +plt.show()
    +
    +
    Number of Iterations:  69 68 67
    +
    +

    png

    +

    png

    +

    png

    +

    png

    +

    5. Dependence on the noise level

    +

    We solve the problem for different noise levels. The higher the noise level the more important becomes the effect of the regularization.

    +
    gamma = 70.
    +delta = 1e-1
    +
    +k = dl.Constant(1.0)
    +v = dl.Constant((0.0, 0.0))
    +c = dl.Constant(0.)
    +
    +rel_noise = [1e-3,1e-2,1e-1]
    +lmbda1, V1, Vm1, niter1 = solve(nx,ny, targets, rel_noise[0], gamma, delta,verbose=False)
    +lmbda2, V2, Vm2, niter2 = solve(nx,ny, targets, rel_noise[1], gamma, delta,verbose=False)
    +lmbda3, V3, Vm3, niter3 = solve(nx,ny, targets, rel_noise[2], gamma, delta,verbose=False)
    +
    +print( "Number of Iterations: ", niter1, niter2, niter3 )
    +plt.figure(figsize=(18,4))
    +nb.plot_eigenvalues(lmbda1, mytitle="Eigenvalues rel_noise {0:g}".format(rel_noise[0]), subplot_loc=131)
    +nb.plot_eigenvalues(lmbda2, mytitle="Eigenvalues rel_noise {0:g}".format(rel_noise[1]), subplot_loc=132)
    +nb.plot_eigenvalues(lmbda3, mytitle="Eigenvalues rel_noise {0:g}".format(rel_noise[2]), subplot_loc=133)
    +
    +nb.plot_eigenvectors(Vm1, V1, mytitle="rel_noise {0:g} Eigen".format(rel_noise[0]), which=[0,1,5])
    +nb.plot_eigenvectors(Vm2, V2, mytitle="rel_noise {0:g} Eigen".format(rel_noise[1]), which=[0,1,5])
    +nb.plot_eigenvectors(Vm3, V3, mytitle="rel_noise {0:g} Eigen".format(rel_noise[2]), which=[0,1,5])
    +
    +plt.show()
    +
    +
    Number of Iterations:  166 69 23
    +
    +

    png

    +

    png

    +

    png

    +

    png

    +

    6. Dependence on the PDE coefficients

    +

    Assume a constant reaction term , and we consider different values for the diffusivity coefficient .

    +

    The smaller the value of the slower the decay in the spectrum.

    +
    rel_noise = 0.01
    +
    +k = dl.Constant(1.0)
    +v = dl.Constant((0.0, 0.0))
    +c = dl.Constant(1.0)
    +
    +lmbda1, V1, Vm1, niter1 = solve(nx,ny, targets, rel_noise, gamma, delta,verbose=False)
    +k = dl.Constant(0.1)
    +lmbda2, V2, Vm2, niter2 = solve(nx,ny, targets, rel_noise, gamma, delta,verbose=False)
    +k = dl.Constant(0.01)
    +lmbda3, V3, Vm3, niter3 = solve(nx,ny, targets, rel_noise, gamma, delta,verbose=False)
    +
    +print( "Number of Iterations: ", niter1, niter2, niter3 )
    +plt.figure(figsize=(18,4))
    +nb.plot_eigenvalues(lmbda1, mytitle="Eigenvalues k=1.0", subplot_loc=131)
    +nb.plot_eigenvalues(lmbda2, mytitle="Eigenvalues k=0.1", subplot_loc=132)
    +nb.plot_eigenvalues(lmbda3, mytitle="Eigenvalues k=0.01", subplot_loc=133)
    +
    +nb.plot_eigenvectors(Vm1, V1, mytitle="k=1. Eigen", which=[0,1,5])
    +nb.plot_eigenvectors(Vm2, V2, mytitle="k=0.1 Eigen", which=[0,1,5])
    +nb.plot_eigenvectors(Vm3, V3, mytitle="k=0.01 Eigen", which=[0,1,5])
    +
    +plt.show()
    +
    +
    Number of Iterations:  80 148 256
    +
    +

    png

    +

    png

    +

    png

    +

    png

    +

    Copyright (c) 2016-2018, The University of Texas at Austin & University of California, Merced.
    +All Rights reserved.
    +See file COPYRIGHT for details.

    +

    This file is part of the hIPPYlib library. For more information and source code +availability see https://hippylib.github.io.

    +

    hIPPYlib is free software; you can redistribute it and/or modify it under the terms of the GNU General Public License (as published by the Free Software Foundation) version 2.0 dated June 1991.

    +
    +
    + +
    +
    +

    Copyright © 2016-2023 The University of Texas at Austin, University of California Merced.
    Copyright © 2018-2020 The University of Texas at Austin, University of California Merced, Washington University in St. Louis.
    This material is based on work partially supported by the National Science Foundation under Grants No ACI-1550593, ACI-1550547.

    +

    Documentation built with MkDocs.

    +
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b/tutorials_v3.0.0/1_FEniCS101/index.html new file mode 100644 index 0000000..800c1c3 --- /dev/null +++ b/tutorials_v3.0.0/1_FEniCS101/index.html @@ -0,0 +1,607 @@ + + + + + + + + + + + FEniCS101 - hIPPYlib + + + + + + + + + +

      + +
      +
      + +
      + +

      FEniCS101 Tutorial

      +

      In this tutorial we consider the boundary value problem (BVP)

      +

      + +

      +

      where , and are the union of +the left and right, and top and bottom boundaries of , +respectively.

      +

      Here + +

      +

      The exact solution is + +

      +

      Weak formulation

      +

      Let us define the Hilbert spaces as + + +

      +

      To obtain the weak formulation, we multiply the PDE by an arbitrary function and integrate over the domain leading to

      +

      + +

      +

      Then, integration by parts the non-conforming term gives

      +

      + +

      +

      Finally by recalling that on and that on , we find the weak formulation:

      +

      Find * such that* + +

      +

      1. Load modules

      +

      To start we load the following modules:

      +
        +
      • +

        dolfin and ufl: the python/C++ interface to FEniCS

        +
      • +
      • +

        math: the python module for mathematical functions

        +
      • +
      • +

        numpy: a python package for linear algebra

        +
      • +
      • +

        matplotlib: a python package used for plotting the results

        +
      • +
      +
      import dolfin as dl
      +import ufl
      +
      +import math
      +import numpy as np
      +import logging
      +
      +import matplotlib.pyplot as plt
      +%matplotlib inline
      +
      +import sys
      +import os
      +sys.path.append( os.environ.get('HIPPYLIB_BASE_DIR', "../") )
      +from hippylib import nb
      +
      +logging.getLogger('FFC').setLevel(logging.WARNING)
      +logging.getLogger('UFL').setLevel(logging.WARNING)
      +dl.set_log_active(False)
      +
      +

      2. Define the mesh and the finite element space

      +

      We construct a triangulation (mesh) of the computational domain with n elements in each direction.

      +

      On the mesh , we then define the finite element space consisting of globally continuous piecewise polynomials. The degree variable defines the polynomial degree.

      +
      n = 16
      +degree = 1
      +mesh = dl.UnitSquareMesh(n, n)
      +nb.plot(mesh)
      +
      +Vh  = dl.FunctionSpace(mesh, 'Lagrange', degree)
      +print( "dim(Vh) = ", Vh.dim() )
      +
      +
      dim(Vh) =  289
      +
      +

      png

      +

      3. Define boundary labels

      +

      To partition the boundary of in the subdomains , , , we assign a unique label boundary_parts to each of part of .

      +
      class TopBoundary(dl.SubDomain):
      +    def inside(self, x, on_boundary):
      +        return on_boundary and abs(x[1] - 1) < dl.DOLFIN_EPS
      +
      +class BottomBoundary(dl.SubDomain):
      +    def inside(self, x, on_boundary):
      +        return on_boundary and abs(x[1]) < dl.DOLFIN_EPS
      +
      +class LeftBoundary(dl.SubDomain):
      +    def inside(self, x, on_boundary):
      +        return on_boundary and abs(x[0]) < dl.DOLFIN_EPS
      +
      +class RightBoundary(dl.SubDomain):
      +    def inside(self, x, on_boundary):
      +        return on_boundary and abs(x[0] - 1) < dl.DOLFIN_EPS
      +
      +boundary_parts = dl.MeshFunction("size_t", mesh, 1)
      +boundary_parts.set_all(0)
      +
      +Gamma_top = TopBoundary()
      +Gamma_top.mark(boundary_parts, 1)
      +Gamma_bottom = BottomBoundary()
      +Gamma_bottom.mark(boundary_parts, 2)
      +Gamma_left = LeftBoundary()
      +Gamma_left.mark(boundary_parts, 3)
      +Gamma_right = RightBoundary()
      +Gamma_right.mark(boundary_parts, 4)
      +
      +

      4. Define the coefficients of the PDE and the boundary conditions

      +

      We first define the coefficients of the PDE using the dl.Constant and dl.Expression classes. dl.Constant is used to define coefficients that do not depend on the space coordinates, dl.Expression is used to define coefficients that are a known function of the space coordinates x[0] (x-axis direction) and x[1] (y-axis direction).

      +

      In the finite element method community, Dirichlet boundary conditions are also known as essential boundary conditions since they are imposed directly in the definition of the finite element space. In FEniCS, we use the class dl.DirichletBC to indicate this type of condition.

      +

      On the other hand, Newman boundary conditions are also known as natural boundary conditions since they are weakly imposed as boundary integrals in the variational formulation (weak form). In FEniCS, we create a new boundary measure ds[i] to integrate over the portion of the boundary marked with label i.

      +
      u_L = dl.Constant(0.)
      +u_R = dl.Constant(0.)
      +
      +sigma_bottom = dl.Expression('-(pi/2.0)*sin(2*pi*x[0])', degree=5)
      +sigma_top    = dl.Constant(0.)
      +
      +f = dl.Expression('(4.0*pi*pi+pi*pi/4.0)*(sin(2*pi*x[0])*sin((pi/2.0)*x[1]))', degree=5)
      +
      +bcs = [dl.DirichletBC(Vh, u_L, boundary_parts, 3),
      +       dl.DirichletBC(Vh, u_R, boundary_parts, 4)]
      +
      +ds = dl.Measure("ds", subdomain_data=boundary_parts)
      +
      +

      5. Define and solve the variational problem

      +

      We also define two special types of functions: the dl.TrialFunction u and the dl.TestFunction v. These special types of function are used by FEniCS to generate the finite element vectors and matrices which stem from the weak formulation of the PDE.

      +

      More specifically, by denoting by the finite element basis for the space , a function can be written as + +where represents the coefficients in the finite element expansion of .

      +

      We then define

      +
        +
      • +

        the bilinear form ;

        +
      • +
      • +

        the linear form .

        +
      • +
      +

      We can then solve the variational problem

      +

      Find + such that + +

      +

      using directly the built-in dl.solve method in FEniCS.

      +

      NOTE: As an alternative one can also assemble the finite element matrix and the right hand side that stems from the discretization of and , and then solve the linear system + +where

      +
        +
      • +

        + is the vector collecting the coefficients of the finite element expasion of ,

        +
      • +
      • +

        the entries of the matrix A are such that ,

        +
      • +
      • +

        the entries of the right hand side b are such that .

        +
      • +
      +
      u = dl.TrialFunction(Vh)
      +v = dl.TestFunction(Vh)
      +a = ufl.inner(ufl.grad(u), ufl.grad(v))*ufl.dx
      +L = f*v*ufl.dx + sigma_top*v*ds(1) + sigma_bottom*v*ds(2)
      +
      +uh = dl.Function(Vh)
      +
      +#dl.solve(a == L, uh, bcs=bcs)
      +A, b = dl.assemble_system(a,L, bcs=bcs)
      +dl.solve(A, uh.vector(), b, "cg")
      +
      +nb.plot(uh)
      +
      +
      <matplotlib.collections.TriMesh at 0x11b389ac8>
      +
      +

      png

      +

      6. Compute the discretization error

      +

      For this problem, the exact solution is known. +We can therefore compute the following norms of the discretization error (i.e. the difference between the finite element solution and the exact solution ) + +and + +

      +
      u_e = dl.Expression('sin(2*pi*x[0])*sin((pi/2.0)*x[1])', degree=5)
      +grad_u_e = dl.Expression( ('2*pi*cos(2*pi*x[0])*sin((pi/2.0)*x[1])',
      +                           'pi/2.0*sin(2*pi*x[0])*cos((pi/2.0)*x[1])'), degree=5)
      +
      +err_L2 = math.sqrt( dl.assemble( (uh-u_e)**2*ufl.dx ) )
      +err_grad = math.sqrt( dl.assemble( ufl.inner(ufl.grad(uh) - grad_u_e, ufl.grad(uh) - grad_u_e)*ufl.dx ) )
      +err_H1 = math.sqrt( err_L2**2 + err_grad**2)
      +
      +print ("|| u_h - u_e ||_L2 = ", err_L2)
      +print ("|| u_h - u_e ||_H1 = ", err_H1)
      +
      +
      || u_h - u_e ||_L2 =  0.008805176703139152
      +|| u_h - u_e ||_H1 =  0.3967189507839944
      +
      +

      7. Convergence of the finite element method

      +

      We now verify numerically a well-known convergence result for the finite element method.

      +

      Let denote the polynomial degree of the finite element space, and assume that the solution is at least in . Then we have + +

      +

      In the code below, the function compute(n, degree) solves the PDE using a mesh with n elements in each direction and finite element spaces of polynomial order degree.

      +

      The figure below shows the discretization errors in the and as a function of the mesh size () for piecewise linear (P1, ) and piecewise quadratic (P2, ) finite elements. We observe that numerical results are consistent with the finite element convergence theory. In particular:

      +
        +
      • +

        for piecewise linear finite element P1 we observe first order convergence in the -norm and second order convergence in the -norm;

        +
      • +
      • +

        for piecewise quadratic finite element P2 we observe second order convergence in the -norm and third order convergence in the -norm.

        +
      • +
      +
      def compute(n, degree):
      +    mesh = dl.UnitSquareMesh(n, n)
      +    Vh  = dl.FunctionSpace(mesh, 'Lagrange', degree)
      +    boundary_parts = dl.MeshFunction("size_t", mesh,1)
      +    boundary_parts.set_all(0)
      +
      +    Gamma_top = TopBoundary()
      +    Gamma_top.mark(boundary_parts, 1)
      +    Gamma_bottom = BottomBoundary()
      +    Gamma_bottom.mark(boundary_parts, 2)
      +    Gamma_left = LeftBoundary()
      +    Gamma_left.mark(boundary_parts, 3)
      +    Gamma_right = RightBoundary()
      +    Gamma_right.mark(boundary_parts, 4)
      +
      +    bcs = [dl.DirichletBC(Vh, u_L, boundary_parts, 3), 
      +           dl.DirichletBC(Vh, u_R, boundary_parts, 4)]
      +    ds = dl.Measure("ds", subdomain_data=boundary_parts)
      +
      +    u = dl.TrialFunction(Vh)
      +    v = dl.TestFunction(Vh)
      +    a = ufl.inner(ufl.grad(u), ufl.grad(v))*ufl.dx
      +    L = f*v*ufl.dx + sigma_top*v*ds(1) + sigma_bottom*v*ds(2)
      +    uh = dl.Function(Vh)
      +    dl.solve(a == L, uh, bcs=bcs)
      +    err_L2 = math.sqrt( dl.assemble( (uh-u_e)**2*ufl.dx ) )
      +    err_grad = math.sqrt( dl.assemble( ufl.inner(ufl.grad(uh) - grad_u_e, ufl.grad(uh) - grad_u_e)*ufl.dx ) )
      +    err_H1 = math.sqrt( err_L2**2 + err_grad**2)
      +
      +    return err_L2, err_H1
      +
      +nref = 5
      +n = 8*np.power(2,np.arange(0,nref))
      +h = 1./n
      +
      +err_L2_P1 = np.zeros(nref)
      +err_H1_P1 = np.zeros(nref)
      +err_L2_P2 = np.zeros(nref)
      +err_H1_P2 = np.zeros(nref)
      +
      +for i in range(nref):
      +    err_L2_P1[i], err_H1_P1[i] = compute(n[i], 1)
      +    err_L2_P2[i], err_H1_P2[i] = compute(n[i], 2)
      +
      +plt.figure(figsize=(15,5))
      +
      +plt.subplot(121)
      +plt.loglog(h, err_H1_P1, '-or', label = "H1 error")
      +plt.loglog(h, err_L2_P1, '-*b', label = "L2 error")
      +plt.loglog(h, h*.5*err_H1_P1[0]/h[0], '--g', label = "First Order")
      +plt.loglog(h, np.power(h,2)*.5*np.power( err_L2_P1[0]/h[0], 2), '-.k', label = "Second Order")
      +plt.xlabel("Mesh size h")
      +plt.ylabel("Error")
      +plt.title("P1 Finite Element")
      +plt.legend(loc='lower right')
      +
      +
      +plt.subplot(122)
      +plt.loglog(h, err_H1_P2, '-or', label = "H1 error")
      +plt.loglog(h, err_L2_P2, '-*b', label = "L2 error")
      +plt.loglog(h, np.power(h/h[0],2)*.5*err_H1_P2[0], '--g', label = "Second Order")
      +plt.loglog(h, np.power(h/h[0],3)*.5*err_L2_P2[0], '-.k', label = "Third Order")
      +plt.xlabel("Mesh size h")
      +plt.ylabel("Error")
      +plt.title("P2 Finite Element")
      +plt.legend(loc='lower right')
      +
      +plt.show()
      +
      +

      png

      +

      Copyright (c) 2016-2018, The University of Texas at Austin & University of California, Merced.
      +Copyright (c) 2019-2020, The University of Texas at Austin, University of California--Merced, Washington University in St. Louis.
      +All Rights reserved.
      +See file COPYRIGHT for details.

      +

      This file is part of the hIPPYlib library. For more information and source code +availability see https://hippylib.github.io.

      +

      hIPPYlib is free software; you can redistribute it and/or modify it under the terms of the GNU General Public License (as published by the Free Software Foundation) version 2.0 dated June 1991.

      +
      +
      + +
      +
      +

      Copyright © 2016-2023 The University of Texas at Austin, University of California Merced.
      Copyright © 2018-2020 The University of Texas at Austin, University of California Merced, Washington University in St. Louis.
      This material is based on work partially supported by the National Science Foundation under Grants No ACI-1550593, ACI-1550547.

      +

      Documentation built with MkDocs.

      +
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zs;Sb+Mx~I=#?J*#26vDosE-xCBI5rp8ZG^WYK^p9`;Kl8lBksd#&-8@-$VdN+wkI3Z5pb%EM|LOd9ts3PbJ{JuY&|}czkXxzUAd==-Hd9hXzQ= zn~aXrbFmEpr|m7VA4$0ZCyCKGlHA;XuVYwb4Xl-2!@8RCm91EUT)pZ-EAZk05NEc8 z@~$s}?{d##XtMPSV;);(a9$C8s3AFpvum3dQ_;E8Olub00TDgPcigi9!MH<|L0ie z+rP(PO(z*MOiohA@q+ejCb62$ar3y|1l&YkJ1Id1M}9g!1zyTLcv$$73MIz~y9k1; z;W!zP`&9BpWRv{0^}k?$oW_d{*45EXtge5^W#(tMWOejBWStZuupI?oEVMBv1nc)2>SKYM^$J4OE;-vH5;r&R1FzW1tHgc;v#j=i`S%R}k3WDoWaT%LI7 + + + + + + + + + + Poisson Deterministic - hIPPYlib + + + + + + + + + + + +
      +
      + +
      + +

      Coefficient field inversion in an elliptic partial differential equation

      +

      We consider the estimation of a coefficient in an elliptic partial +differential equation as a model problem. Depending on the +interpretation of the unknowns and the type of measurements, this +model problem arises, for instance, in inversion for groundwater flow +or heat conductivity. It can also be interpreted as finding a +membrane with a certain spatially varying stiffness. Let +, be an open, bounded +domain and consider the following problem:

      +

      + +

      +

      where is the solution of

      +

      + +

      +

      Here the unknown coefficient field, denotes (possibly noisy) data, a given force, and the regularization parameter.

      +

      The variational (or weak) form of the state equation:

      +

      Find such that

      +

      + +

      +

      where is the space of functions vanishing on with square integrable derivatives. Here, denotes the -inner product, i.e, for scalar functions we denote

      +

      + +

      +

      Gradient evaluation:

      +

      The Lagrangian functional is given by

      +

      + +

      +

      Then the gradient of the cost functional with respect to the parameter is

      +

      + +

      +

      where is the solution of the forward problem,

      +

      + +

      +

      and is the solution of the adjoint problem,

      +

      + +

      +

      Hessian action:

      +

      To evaluate the action of the Hessian in a given direction , we consider variations of the meta-Lagrangian functional

      +

      + +

      +

      Then the action of the Hessian in a given direction is

      +

      + +

      +

      where

      +
        +
      • +

        + and are the solution of the forward and adjoint problem, respectively;

        +
      • +
      • +

        + is the solution of the incremental forward problem,

        +
      • +
      +

      + +

      +
        +
      • and is the solution of the incremental adjoint problem, + +
      • +
      +

      Inexact Newton-CG:

      +

      Written in abstract form, the Newton Method computes an update direction by solving the linear system

      +

      + +

      +

      where the evaluation of the gradient involve the solution and of the forward and adjoint problem (respectively) for . +Similarly, the Hessian action requires to additional solve the incremental forward and adjoint problems.

      +

      Discrete Newton system:

      +

      + + + +

      +

      Let us denote the vectors corresponding to the discretization of the functions by and of the functions by .

      +

      Then, the discretization of the above system is given by the following symmetric linear system:

      +

      + +

      +

      The gradient is computed using the following three steps

      +
        +
      • Given we solve the forward problem
      • +
      +

      + +

      +

      where stems from the discretization , and stands for the discretization of the right hand side .

      +
        +
      • Given and solve the adjoint problem
      • +
      +

      + +

      +

      where stems from the discretization of , is the mass matrix corresponding to the inner product in the state space, and stems from the data.

      +
        +
      • Define the gradient
      • +
      +

      + +

      +

      where is the matrix stemming from discretization of the regularization operator , and stems from discretization of the term .

      +

      Similarly the action of the Hessian in a direction (by using the CG algorithm we only need the action of to solve the Newton step) is given by

      +
        +
      • Solve the incremental forward problem
      • +
      +

      + +

      +

      where stems from discretization of .

      +
        +
      • Solve the incremental adjoint problem
      • +
      +

      + +

      +

      where stems for the discretization of .

      +
        +
      • Define the Hessian action
      • +
      +

      + +

      +

      Goals:

      +

      By the end of this notebook, you should be able to:

      +
        +
      • solve the forward and adjoint Poisson equations
      • +
      • understand the inverse method framework
      • +
      • visualise and understand the results
      • +
      • modify the problem and code
      • +
      +

      Mathematical tools used:

      +
        +
      • Finite element method
      • +
      • Derivation of gradiant and Hessian via the adjoint method
      • +
      • Inexact Newton-CG
      • +
      • Armijo line search
      • +
      +

      List of software used:

      +
        +
      • FEniCS, a parallel finite element element library for the discretization of partial differential equations
      • +
      • PETSc, for scalable and efficient linear algebra operations and solvers
      • +
      • Matplotlib, a python package used for plotting the results
      • +
      +

      Set up

      +

      Import dependencies

      +
      import dolfin as dl
      +import ufl
      +
      +import sys
      +import os
      +sys.path.append( os.environ.get('HIPPYLIB_BASE_DIR', "../") )
      +from hippylib import *
      +
      +import logging
      +import math
      +
      +import matplotlib.pyplot as plt
      +%matplotlib inline
      +
      +logging.getLogger('FFC').setLevel(logging.WARNING)
      +logging.getLogger('UFL').setLevel(logging.WARNING)
      +dl.set_log_active(False)
      +
      +

      Model set up:

      +

      As in the introduction, the first thing we need to do is set up the numerical model. In this cell, we set the mesh, the finite element functions corresponding to state, adjoint and coefficient/gradient variables, and the corresponding test functions and the parameters for the optimization.

      +
      # create mesh and define function spaces
      +nx = 64
      +ny = 64
      +mesh = dl.UnitSquareMesh(nx, ny)
      +Vm = dl.FunctionSpace(mesh, 'Lagrange', 1)
      +Vu = dl.FunctionSpace(mesh, 'Lagrange', 2)
      +
      +# The true and inverted parameter
      +mtrue_expression = dl.Expression(
      +    'std::log(2 + 7*(std::pow(std::pow(x[0] - 0.5,2) + std::pow(x[1] - 0.5,2),0.5) > 0.2))',
      +    degree=5)
      +mtrue = dl.interpolate(mtrue_expression,Vm)
      +m = dl.interpolate(dl.Expression("std::log(2.0)", degree=1),Vm)
      +
      +# define function for state and adjoint
      +u = dl.Function(Vu)
      +p = dl.Function(Vu)
      +
      +# define Trial and Test Functions
      +u_trial, p_trial, m_trial = dl.TrialFunction(Vu), dl.TrialFunction(Vu), dl.TrialFunction(Vm)
      +u_test, p_test, m_test = dl.TestFunction(Vu), dl.TestFunction(Vu), dl.TestFunction(Vm)
      +
      +# initialize input functions
      +f = dl.Constant(1.0)
      +u0 = dl.Constant(0.0)
      +
      +# plot
      +plt.figure(figsize=(15,5))
      +nb.plot(mesh,subplot_loc=121, mytitle="Mesh", show_axis='on')
      +nb.plot(mtrue,subplot_loc=122, mytitle="True parameter field")
      +plt.show()
      +
      +

      png

      +
      # set up dirichlet boundary conditions
      +def boundary(x,on_boundary):
      +    return on_boundary
      +
      +bc_state = dl.DirichletBC(Vu, u0, boundary)
      +bc_adj = dl.DirichletBC(Vu, dl.Constant(0.), boundary)
      +
      +

      Set up synthetic observations:

      +
        +
      • Propose a coefficient field shown above
      • +
      • +

        The weak form of the pde: + Find such that .

        +
      • +
      • +

        Perturb the solution: , where +

        +
      • +
      +
      # noise level
      +noise_level = 0.05
      +
      +# weak form for setting up the synthetic observations
      +a_goal = ufl.inner(ufl.exp(mtrue) * ufl.grad(u_trial), ufl.grad(u_test)) * ufl.dx
      +L_goal = f * u_test * ufl.dx
      +
      +# solve the forward/state problem to generate synthetic observations
      +goal_A, goal_b = dl.assemble_system(a_goal, L_goal, bc_state)
      +
      +utrue = dl.Function(Vu)
      +dl.solve(goal_A, utrue.vector(), goal_b)
      +
      +ud = dl.Function(Vu)
      +ud.assign(utrue)
      +
      +# perturb state solution and create synthetic measurements ud
      +# ud = u + ||u||/SNR * random.normal
      +MAX = ud.vector().norm("linf")
      +noise = dl.Vector()
      +goal_A.init_vector(noise,1)
      +parRandom.normal(noise_level * MAX, noise)
      +bc_adj.apply(noise)
      +
      +ud.vector().axpy(1., noise)
      +
      +# plot
      +nb.multi1_plot([utrue, ud], ["State solution with mtrue", "Synthetic observations"])
      +plt.show()
      +
      +

      png

      +

      The cost function evaluation:

      +

      + +

      +

      In the code below, and are symmetric positive definite matrices that stem from finite element discretization of the misfit and regularization component of the cost functional, respectively.

      +
      # regularization parameter
      +gamma = 1e-8
      +
      +# weak for for setting up the misfit and regularization compoment of the cost
      +W_equ   = ufl.inner(u_trial, u_test) * ufl.dx
      +R_equ   = gamma * ufl.inner(ufl.grad(m_trial), ufl.grad(m_test)) * ufl.dx
      +
      +W = dl.assemble(W_equ)
      +R = dl.assemble(R_equ)
      +
      +# refine cost function
      +def cost(u, ud, m, W, R):
      +    diff = u.vector() - ud.vector()
      +    reg = 0.5 * m.vector().inner(R*m.vector() ) 
      +    misfit = 0.5 * diff.inner(W * diff)
      +    return [reg + misfit, misfit, reg]
      +
      +

      Setting up the state equations, right hand side for the adjoint and the necessary matrices:

      +
      # weak form for setting up the state equation
      +a_state = ufl.inner(ufl.exp(m) * ufl.grad(u_trial), ufl.grad(u_test)) * ufl.dx
      +L_state = f * u_test * ufl.dx
      +
      +# weak form for setting up the adjoint equation
      +a_adj = ufl.inner(ufl.exp(m) * ufl.grad(p_trial), ufl.grad(p_test)) * ufl.dx
      +L_adj = -ufl.inner(u - ud, p_test) * ufl.dx
      +
      +# weak form for setting up matrices
      +Wum_equ = ufl.inner(ufl.exp(m) * m_trial * ufl.grad(p_test), ufl.grad(p)) * ufl.dx
      +C_equ   = ufl.inner(ufl.exp(m) * m_trial * ufl.grad(u), ufl.grad(u_test)) * ufl.dx
      +Wmm_equ = ufl.inner(ufl.exp(m) * m_trial * m_test *  ufl.grad(u),  ufl.grad(p)) * ufl.dx
      +
      +M_equ   = ufl.inner(m_trial, m_test) * ufl.dx
      +
      +# assemble matrix M
      +M = dl.assemble(M_equ)
      +
      +

      Initial guess

      +

      We solve the state equation and compute the cost functional for the initial guess of the parameter m_ini

      +
      # solve state equation
      +state_A, state_b = dl.assemble_system (a_state, L_state, bc_state)
      +dl.solve (state_A, u.vector(), state_b)
      +
      +# evaluate cost
      +[cost_old, misfit_old, reg_old] = cost(u, ud, m, W, R)
      +
      +# plot
      +plt.figure(figsize=(15,5))
      +nb.plot(m,subplot_loc=121, mytitle="m_ini", vmin=mtrue.vector().min(), vmax=mtrue.vector().max())
      +nb.plot(u,subplot_loc=122, mytitle="u(m_ini)")
      +plt.show()
      +
      +

      png

      +

      The reduced Hessian apply to a vector :

      +

      Here we describe how to apply the reduced Hessian operator to a vector . For an opportune choice of the regularization, the reduced Hessian operator evaluated in a neighborhood of the solution is positive define, whereas far from the solution the reduced Hessian may be indefinite. On the constrary, the Gauss-Newton approximation of the Hessian is always positive defined.

      +

      For this reason, it is beneficial to perform a few initial Gauss-Newton steps (5 in this particular example) to accelerate the convergence of the inexact Newton-CG algorithm.

      +

      The Hessian apply reads: + +

      +

      The Gauss-Newton Hessian apply is obtained by dropping the second derivatives operators , , and : + +

      +
      # Class HessianOperator to perform Hessian apply to a vector
      +class HessianOperator():
      +    cgiter = 0
      +    def __init__(self, R, Wmm, C, A, adj_A, W, Wum, gauss_newton_approx=False):
      +        self.R = R
      +        self.Wmm = Wmm
      +        self.C = C
      +        self.A = A
      +        self.adj_A = adj_A
      +        self.W = W
      +        self.Wum = Wum
      +        self.gauss_newton_approx = gauss_newton_approx
      +
      +        # incremental state
      +        self.du = dl.Vector()
      +        self.A.init_vector(self.du,0)
      +
      +        # incremental adjoint
      +        self.dp = dl.Vector()
      +        self.adj_A.init_vector(self.dp,0)
      +
      +        # auxiliary vectors
      +        self.CT_dp = dl.Vector()
      +        self.C.init_vector(self.CT_dp, 1)
      +        self.Wum_du = dl.Vector()
      +        self.Wum.init_vector(self.Wum_du, 1)
      +
      +    def init_vector(self, v, dim):
      +        self.R.init_vector(v,dim)
      +
      +    # Hessian performed on v, output as generic vector y
      +    def mult(self, v, y):
      +        self.cgiter += 1
      +        y.zero()
      +        if self.gauss_newton_approx:
      +            self.mult_GaussNewton(v,y)
      +        else:
      +            self.mult_Newton(v,y)
      +
      +    # define (Gauss-Newton) Hessian apply H * v
      +    def mult_GaussNewton(self, v, y):
      +
      +        # incremental forward
      +        rhs = -(self.C * v)
      +        bc_adj.apply(rhs)
      +        dl.solve (self.A, self.du, rhs)
      +
      +        # incremental adjoint
      +        rhs = - (self.W * self.du)
      +        bc_adj.apply(rhs)
      +        dl.solve (self.adj_A, self.dp, rhs)
      +
      +        # Reg/Prior term
      +        self.R.mult(v,y)
      +
      +        # Misfit term
      +        self.C.transpmult(self.dp, self.CT_dp)
      +        y.axpy(1, self.CT_dp)
      +
      +    # define (Newton) Hessian apply H * v
      +    def mult_Newton(self, v, y):
      +
      +        # incremental forward
      +        rhs = -(self.C * v)
      +        bc_adj.apply(rhs)
      +        dl.solve (self.A, self.du, rhs)
      +
      +        # incremental adjoint
      +        rhs = -(self.W * self.du) -  self.Wum * v
      +        bc_adj.apply(rhs)
      +        dl.solve (self.adj_A, self.dp, rhs)
      +
      +        # Reg/Prior term
      +        self.R.mult(v,y)
      +        y.axpy(1., self.Wmm*v)
      +
      +        # Misfit term
      +        self.C.transpmult(self.dp, self.CT_dp)
      +        y.axpy(1., self.CT_dp)
      +        self.Wum.transpmult(self.du, self.Wum_du)
      +        y.axpy(1., self.Wum_du)
      +
      + +

      We solve the constrained optimization problem using the inexact Newton-CG method with Armijo line search.

      +

      The stopping criterion is based on a relative reduction of the norm of the gradient (i.e. ).

      +

      First, we compute the gradient by solving the state and adjoint equation for the current parameter , and then substituing the current state , parameter and adjoint variables in the weak form expression of the gradient: + +

      +

      Then, we compute the Newton direction by iteratively solving . +The Newton system is solved inexactly by early termination of conjugate gradient iterations via Eisenstat–Walker (to prevent oversolving) and Steihaug (to avoid negative curvature) criteria.

      +

      Finally, the Armijo line search uses backtracking to find such that a sufficient reduction in the cost functional is achieved. +More specifically, we use backtracking to find such that: + +

      +
      # define parameters for the optimization
      +tol = 1e-8
      +c = 1e-4
      +maxiter = 12
      +plot_on = False
      +
      +# initialize iter counters
      +iter = 1
      +total_cg_iter = 0
      +converged = False
      +
      +# initializations
      +g, m_delta = dl.Vector(), dl.Vector()
      +R.init_vector(m_delta,0)
      +R.init_vector(g,0)
      +
      +m_prev = dl.Function(Vm)
      +
      +print ("Nit   CGit   cost          misfit        reg           sqrt(-G*D)    ||grad||       alpha  tolcg")
      +
      +while iter <  maxiter and not converged:
      +
      +    # assemble matrix C
      +    C =  dl.assemble(C_equ)
      +
      +    # solve the adoint problem
      +    adjoint_A, adjoint_RHS = dl.assemble_system(a_adj, L_adj, bc_adj)
      +    dl.solve(adjoint_A, p.vector(), adjoint_RHS)
      +
      +    # assemble W_ua and R
      +    Wum = dl.assemble (Wum_equ)
      +    Wmm = dl.assemble (Wmm_equ)
      +
      +    # evaluate the  gradient
      +    CT_p = dl.Vector()
      +    C.init_vector(CT_p,1)
      +    C.transpmult(p.vector(), CT_p)
      +    MG = CT_p + R * m.vector()
      +    dl.solve(M, g, MG)
      +
      +    # calculate the norm of the gradient
      +    grad2 = g.inner(MG)
      +    gradnorm = math.sqrt(grad2)
      +
      +    # set the CG tolerance (use Eisenstat–Walker termination criterion)
      +    if iter == 1:
      +        gradnorm_ini = gradnorm
      +    tolcg = min(0.5, math.sqrt(gradnorm/gradnorm_ini))
      +
      +    # define the Hessian apply operator (with preconditioner)
      +    Hess_Apply = HessianOperator(R, Wmm, C, state_A, adjoint_A, W, Wum, gauss_newton_approx=(iter<6) )
      +    P = R + gamma * M
      +    Psolver = dl.PETScKrylovSolver("cg", amg_method())
      +    Psolver.set_operator(P)
      +
      +    solver = CGSolverSteihaug()
      +    solver.set_operator(Hess_Apply)
      +    solver.set_preconditioner(Psolver)
      +    solver.parameters["rel_tolerance"] = tolcg
      +    solver.parameters["zero_initial_guess"] = True
      +    solver.parameters["print_level"] = -1
      +
      +    # solve the Newton system H a_delta = - MG
      +    solver.solve(m_delta, -MG)
      +    total_cg_iter += Hess_Apply.cgiter
      +
      +    # linesearch
      +    alpha = 1
      +    descent = 0
      +    no_backtrack = 0
      +    m_prev.assign(m)
      +    while descent == 0 and no_backtrack < 10:
      +        m.vector().axpy(alpha, m_delta )
      +
      +        # solve the state/forward problem
      +        state_A, state_b = dl.assemble_system(a_state, L_state, bc_state)
      +        dl.solve(state_A, u.vector(), state_b)
      +
      +        # evaluate cost
      +        [cost_new, misfit_new, reg_new] = cost(u, ud, m, W, R)
      +
      +        # check if Armijo conditions are satisfied
      +        if cost_new < cost_old + alpha * c * MG.inner(m_delta):
      +            cost_old = cost_new
      +            descent = 1
      +        else:
      +            no_backtrack += 1
      +            alpha *= 0.5
      +            m.assign(m_prev)  # reset a
      +
      +    # calculate sqrt(-G * D)
      +    graddir = math.sqrt(- MG.inner(m_delta) )
      +
      +    sp = ""
      +    print( "%2d %2s %2d %3s %8.5e %1s %8.5e %1s %8.5e %1s %8.5e %1s %8.5e %1s %5.2f %1s %5.3e" % \
      +        (iter, sp, Hess_Apply.cgiter, sp, cost_new, sp, misfit_new, sp, reg_new, sp, \
      +         graddir, sp, gradnorm, sp, alpha, sp, tolcg) )
      +
      +    if plot_on:
      +        nb.multi1_plot([m,u,p], ["m","u","p"], same_colorbar=False)
      +        plt.show()
      +
      +    # check for convergence
      +    if gradnorm < tol and iter > 1:
      +        converged = True
      +        print( "Newton's method converged in ",iter,"  iterations")
      +        print( "Total number of CG iterations: ", total_cg_iter)
      +
      +    iter += 1
      +
      +if not converged:
      +    print( "Newton's method did not converge in ", maxiter, " iterations")
      +
      +
      Nit   CGit   cost          misfit        reg           sqrt(-G*D)    ||grad||       alpha  tolcg
      + 1     1     1.12916e-05   1.12916e-05   1.34150e-11   1.56616e-02   3.79614e-04    1.00   5.000e-01
      + 2     1     7.83206e-07   7.83170e-07   3.68430e-11   4.68686e-03   5.35269e-05    1.00   3.755e-01
      + 3     1     3.12292e-07   3.12243e-07   4.92462e-11   9.73515e-04   7.14570e-06    1.00   1.372e-01
      + 4     6     1.91985e-07   1.61584e-07   3.04009e-08   4.54547e-04   1.00594e-06    1.00   5.148e-02
      + 5     1     1.86421e-07   1.56004e-07   3.04171e-08   1.05501e-04   6.25400e-07    1.00   4.059e-02
      + 6    13     1.80334e-07   1.36992e-07   4.33419e-08   1.12181e-04   2.14958e-07    1.00   2.380e-02
      + 7     5     1.80267e-07   1.38198e-07   4.20693e-08   1.15139e-05   3.92325e-08    1.00   1.017e-02
      + 8    15     1.80266e-07   1.38243e-07   4.20235e-08   1.48892e-06   3.20470e-09    1.00   2.906e-03
      +Newton's method converged in  8   iterations
      +Total number of CG iterations:  43
      +
      +
      nb.multi1_plot([mtrue, m], ["mtrue", "m"])
      +nb.multi1_plot([u,p], ["u","p"], same_colorbar=False)
      +plt.show()
      +
      +

      png

      +

      png

      +

      Copyright (c) 2016-2018, The University of Texas at Austin & University of California, Merced.
      +Copyright (c) 2019-2020, The University of Texas at Austin, University of California--Merced, Washington University in St. Louis.
      +All Rights reserved.
      +See file COPYRIGHT for details.

      +

      This file is part of the hIPPYlib library. For more information and source code +availability see https://hippylib.github.io.

      +

      hIPPYlib is free software; you can redistribute it and/or modify it under the terms of the GNU General Public License (as published by the Free Software Foundation) version 2.0 dated June 1991.

      +
      +
      + +
      +
      +

      Copyright © 2016-2023 The University of Texas at Austin, University of California Merced.
      Copyright © 2018-2020 The University of Texas at Austin, University of California Merced, Washington University in St. Louis.
      This material is based on work partially supported by the National Science Foundation under Grants No ACI-1550593, ACI-1550547.

      +

      Documentation built with MkDocs.

      +
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      + +
      +
      + +
      + +

      + +

      +

      Bayesian quantification of parameter uncertainty:

      +

      Estimating the Gaussian approximation of posterior pdf of the coefficient parameter field in an elliptic PDE

      +

      In this example we tackle the problem of quantifying the +uncertainty in the solution of an inverse problem governed by an +elliptic PDE via the Bayesian inference framework. +Hence, we state the inverse problem as a +problem of statistical inference over the space of uncertain +parameters, which are to be inferred from data and a physical +model. The resulting solution to the statistical inverse problem +is a posterior distribution that assigns to any candidate set of +parameter fields our belief (expressed as a probability) that a +member of this candidate set is the ``true'' parameter field that +gave rise to the observed data.

      +

      For simplicity, in what follows we give finite-dimensional expressions (i.e., after +discretization of the parameter space) for the Bayesian +formulation of the inverse problem.

      +

      Bayes' Theorem:

      +

      The posterior probability distribution combines the prior pdf + over the parameter space, which encodes +any knowledge or assumptions about the parameter space that we may +wish to impose before the data are considered, with a likelihood pdf +, which explicitly +represents the probability that a given set of parameters +might give rise to the observed data , namely:

      +

      + +

      +

      Note that infinite-dimensional analog of Bayes' formula requires the use Radon-Nikodym derivatives instead of probability density functions.

      +

      Gaussian prior and noise:

      +

      The prior:

      +

      We consider a Gaussian prior with mean and covariance . The covariance is given by the discretization of the inverse of differential operator , where , control the correlation length and the variance of the prior operator. This choice of prior ensures that it is a trace-class operator, guaranteeing bounded pointwise variance and a well-posed infinite-dimensional Bayesian inverse problem.

      +

      The likelihood:

      +

      + +

      +

      + +

      +

      Here is the parameter-to-observable map that takes a parameter vector and maps +it to the space observation vector .

      +

      The posterior:

      +

      + +

      +

      The Laplace approximation to posterior: +

      +

      The mean of this posterior distribution, , is the +parameter vector maximizing the posterior, and +is known as the maximum a posteriori (MAP) point. It can be found +by minimizing the negative log of the posterior, which amounts to +solving a deterministic inverse problem with appropriately weighted norms,

      +

      + +

      +

      The posterior covariance matrix is then given by the inverse of +the Hessian matrix of at , namely

      +

      + +

      +

      The generalized eigenvalue problem:

      +

      + +

      +

      where +contains the generalized eigenvalues and the columns of the generalized eigenvectors such that +.

      +

      Randomized eigensolvers to construct the approximate spectral decomposition:

      +

      When the generalized eigenvalues decay rapidly, we can +extract a low-rank approximation of by retaining only the +largest eigenvalues and corresponding eigenvectors,

      +

      + +

      +

      Here, contains only the +generalized eigenvectors of that correspond to the largest eigenvalues, +which are assembled into the diagonal matrix .

      +

      The approximate posterior covariance:

      +

      Using the Sherman–Morrison–Woodbury formula, we write

      +

      + +

      +

      where . The last term in this expression captures the +error due to truncation in terms of the discarded eigenvalues; this +provides a criterion for truncating the spectrum, namely that is +chosen such that is small relative to 1.

      +

      Therefore we can approximate the posterior covariance as

      +

      + +

      +

      Drawing samples from a Gaussian distribution with covariance +

      +

      Let be a sample for the prior distribution, i.e. , then, using the low rank approximation of the posterior covariance, we compute a sample as

      +

      + +

      +

      This tutorial shows:

      +
        +
      • Description of the inverse problem (the forward problem, the prior, and the misfit functional)
      • +
      • Convergence of the inexact Newton-CG algorithm
      • +
      • Low-rank-based approximation of the posterior covariance (built on a low-rank +approximation of the Hessian of the data misfit)
      • +
      • How to construct the low-rank approximation of the Hessian of the data misfit
      • +
      • How to apply the inverse and square-root inverse Hessian to a vector efficiently
      • +
      • Samples from the Gaussian approximation of the posterior
      • +
      +

      Goals:

      +

      By the end of this notebook, you should be able to:

      +
        +
      • Understand the Bayesian inverse framework
      • +
      • Visualise and understand the results
      • +
      • Modify the problem and code
      • +
      +

      Mathematical tools used:

      +
        +
      • Finite element method
      • +
      • Derivation of gradient and Hessian via the adjoint method
      • +
      • inexact Newton-CG
      • +
      • Armijo line search
      • +
      • Bayes' formula
      • +
      • randomized eigensolvers
      • +
      +

      List of software used:

      +
        +
      • FEniCS, a parallel finite element element library for the discretization of partial differential equations
      • +
      • PETSc, for scalable and efficient linear algebra operations and solvers
      • +
      • Matplotlib, A great python package that I used for plotting many of the results
      • +
      • Numpy, A python package for linear algebra. While extensive, this is mostly used to compute means and sums in this notebook.
      • +
      +

      1. Load modules

      +
      import dolfin as dl
      +import ufl
      +import math
      +import numpy as np
      +import matplotlib.pyplot as plt
      +%matplotlib inline
      +import sys
      +import os
      +sys.path.append( os.environ.get('HIPPYLIB_BASE_DIR', "../") )
      +from hippylib import *
      +
      +import logging
      +logging.getLogger('FFC').setLevel(logging.WARNING)
      +logging.getLogger('UFL').setLevel(logging.WARNING)
      +dl.set_log_active(False)
      +
      +np.random.seed(seed=1)
      +
      +

      2. Generate the true parameter

      +

      This function generates a random field with a prescribed anysotropic covariance function.

      +
      def true_model(prior):
      +    noise = dl.Vector()
      +    prior.init_vector(noise,"noise")
      +    parRandom.normal(1., noise)
      +    mtrue = dl.Vector()
      +    prior.init_vector(mtrue, 0)
      +    prior.sample(noise,mtrue)
      +    return mtrue
      +
      +

      3. Set up the mesh and finite element spaces

      +

      We compute a two dimensional mesh of a unit square with nx by ny elements. +We define a P2 finite element space for the state and adjoint variable and P1 for the parameter.

      +
      ndim = 2
      +nx = 64
      +ny = 64
      +mesh = dl.UnitSquareMesh(nx, ny)
      +Vh2 = dl.FunctionSpace(mesh, 'Lagrange', 2)
      +Vh1 = dl.FunctionSpace(mesh, 'Lagrange', 1)
      +Vh = [Vh2, Vh1, Vh2]
      +print( "Number of dofs: STATE={0}, PARAMETER={1}, ADJOINT={2}".format(
      +    Vh[STATE].dim(), Vh[PARAMETER].dim(), Vh[ADJOINT].dim()) )
      +
      +
      Number of dofs: STATE=16641, PARAMETER=4225, ADJOINT=16641
      +
      +

      4. Set up the forward problem

      +

      To set up the forward problem we use the PDEVariationalProblem class, which requires the following inputs +- the finite element spaces for the state, parameter, and adjoint variables Vh +- the pde in weak form pde_varf +- the boundary conditions bc for the forward problem and bc0 for the adjoint and incremental problems.

      +

      The PDEVariationalProblem class offer the following functionality: +- solving the forward/adjoint and incremental problems +- evaluate first and second partial derivative of the forward problem with respect to the state, parameter, and adojnt variables.

      +
      def u_boundary(x, on_boundary):
      +    return on_boundary and ( x[1] < dl.DOLFIN_EPS or x[1] > 1.0 - dl.DOLFIN_EPS)
      +
      +u_bdr = dl.Expression("x[1]", degree=1)
      +u_bdr0 = dl.Constant(0.0)
      +bc = dl.DirichletBC(Vh[STATE], u_bdr, u_boundary)
      +bc0 = dl.DirichletBC(Vh[STATE], u_bdr0, u_boundary)
      +
      +f = dl.Constant(0.0)
      +
      +def pde_varf(u,m,p):
      +    return ufl.exp(m)*ufl.inner(ufl.grad(u), ufl.grad(p))*ufl.dx - f*p*ufl.dx
      +
      +pde = PDEVariationalProblem(Vh, pde_varf, bc, bc0, is_fwd_linear=True)
      +
      +

      4. Set up the prior

      +

      To obtain the synthetic true paramter we generate a realization from the prior distribution. Here we assume a Gaussian prior with zero average and covariance matrix . The action of on a field is given by

      +

      + +where is chosen to minimize boundary artifacts. +Here is an s.p.d. anisotropic tensor of the form

      +

      + +

      +
      gamma = .1
      +delta = .5
      +
      +theta0 = 2.
      +theta1 = .5
      +alpha  = math.pi/4
      +
      +anis_diff = dl.CompiledExpression(ExpressionModule.AnisTensor2D(), degree = 1)
      +anis_diff.set(theta0, theta1, alpha)
      +
      +prior = BiLaplacianPrior(Vh[PARAMETER], gamma, delta, anis_diff, robin_bc=True)
      +mtrue = true_model(prior)
      +
      +print("Prior regularization: (delta_x - gamma*Laplacian)^order: delta={0}, gamma={1}, order={2}".format(delta, gamma,2))    
      +
      +objs = [dl.Function(Vh[PARAMETER],mtrue), dl.Function(Vh[PARAMETER],prior.mean)]
      +mytitles = ["True Parameter", "Prior mean"]
      +nb.multi1_plot(objs, mytitles)
      +plt.show()
      +
      +
      Prior regularization: (delta_x - gamma*Laplacian)^order: delta=0.5, gamma=0.1, order=2
      +
      +

      png

      +

      5. Set up the misfit functional and generate synthetic observations

      +

      To setup the observation operator , we generate (ntargets in the code below) random locations where to evaluate the value of the state.

      +

      Under the assumption of Gaussian additive noise, the likelihood function has the form

      +

      + +

      +

      where denotes the solution of the forward model at a given parameter .

      +

      The class PointwiseStateObservation implements the evaluation of the log-likelihood function and of its partial derivatives w.r.t. the state and parameter .

      +

      To generate the synthetic observation, we first solve the forward problem using the true parameter . Synthetic observations are obtained by perturbing the state variable at the observation points with a random Gaussian noise. +rel_noise is the signal to noise ratio.

      +
      ntargets = 50
      +rel_noise = 0.01
      +
      +#Targets only on the bottom
      +targets_x = np.random.uniform(0.1,0.9, [ntargets] )
      +targets_y = np.random.uniform(0.1,0.5, [ntargets] )
      +targets = np.zeros([ntargets, ndim])
      +targets[:,0] = targets_x
      +targets[:,1] = targets_y
      +#targets everywhere
      +#targets = np.random.uniform(0.1,0.9, [ntargets, ndim] )
      +print( "Number of observation points: {0}".format(ntargets) )
      +misfit = PointwiseStateObservation(Vh[STATE], targets)
      +
      +utrue = pde.generate_state()
      +x = [utrue, mtrue, None]
      +pde.solveFwd(x[STATE], x)
      +misfit.B.mult(x[STATE], misfit.d)
      +MAX = misfit.d.norm("linf")
      +noise_std_dev = rel_noise * MAX
      +parRandom.normal_perturb(noise_std_dev, misfit.d)
      +misfit.noise_variance = noise_std_dev*noise_std_dev
      +
      +vmax = max( utrue.max(), misfit.d.max() )
      +vmin = min( utrue.min(), misfit.d.min() )
      +
      +plt.figure(figsize=(15,5))
      +nb.plot(dl.Function(Vh[STATE], utrue), mytitle="True State", subplot_loc=121, vmin=vmin, vmax=vmax)
      +nb.plot_pts(targets, misfit.d, mytitle="Observations", subplot_loc=122, vmin=vmin, vmax=vmax)
      +plt.show()
      +
      +
      Number of observation points: 50
      +
      +

      png

      +

      6. Set up the model and test gradient and Hessian

      +

      The model is defined by three component: +- the PDEVariationalProblem pde which provides methods for the solution of the forward problem, adjoint problem, and incremental forward and adjoint problems. +- the Prior prior which provides methods to apply the regularization (precision) operator to a vector or to apply the prior covariance operator (i.e. to solve linear system with the regularization operator) +- the Misfit misfit which provides methods to compute the cost functional and its partial derivatives with respect to the state and parameter variables.

      +

      To test gradient and the Hessian of the model we use forward finite differences.

      +
      model = Model(pde, prior, misfit)
      +
      +m0 = dl.interpolate(dl.Expression("sin(x[0])", degree=5), Vh[PARAMETER])
      +_ = modelVerify(model, m0.vector())
      +
      +
      (yy, H xx) - (xx, H yy) =  -2.7824662881775175e-14
      +
      +

      png

      +

      7. Compute the MAP point

      +

      We used the globalized Newtown-CG method to compute the MAP point.

      +
      m = prior.mean.copy()
      +solver = ReducedSpaceNewtonCG(model)
      +solver.parameters["rel_tolerance"] = 1e-6
      +solver.parameters["abs_tolerance"] = 1e-12
      +solver.parameters["max_iter"]      = 25
      +solver.parameters["GN_iter"] = 5
      +solver.parameters["globalization"] = "LS"
      +solver.parameters["LS"]["c_armijo"] = 1e-4
      +
      +
      +x = solver.solve([None, m, None])
      +
      +if solver.converged:
      +    print( "\nConverged in ", solver.it, " iterations.")
      +else:
      +    print( "\nNot Converged")
      +
      +print( "Termination reason: ", solver.termination_reasons[solver.reason] )
      +print( "Final gradient norm: ", solver.final_grad_norm )
      +print( "Final cost: ", solver.final_cost )
      +
      +plt.figure(figsize=(15,5))
      +nb.plot(dl.Function(Vh[STATE], x[STATE]), subplot_loc=121,mytitle="State")
      +nb.plot(dl.Function(Vh[PARAMETER], x[PARAMETER]), subplot_loc=122,mytitle="Parameter")
      +plt.show()
      +
      +
      It  cg_it cost            misfit          reg             (g,dm)          ||g||L2        alpha          tolcg         
      +  1   2    5.676752e+02    5.665943e+02    1.080947e+00   -1.487693e+04   4.008143e+04   1.000000e+00   5.000000e-01
      +  2   2    1.315109e+02    1.292859e+02    2.225043e+00   -8.730801e+02   7.040800e+03   1.000000e+00   4.191210e-01
      +  3   4    5.827697e+01    5.504162e+01    3.235343e+00   -1.565905e+02   2.115036e+03   1.000000e+00   2.297139e-01
      +  4   1    5.257433e+01    4.934371e+01    3.230622e+00   -1.142190e+01   1.827983e+03   1.000000e+00   2.135573e-01
      +  5   6    3.958912e+01    3.507186e+01    4.517259e+00   -3.227629e+01   9.179935e+02   1.000000e+00   1.513381e-01
      +  6   2    3.681315e+01    3.218885e+01    4.624305e+00   -5.500988e+00   7.541262e+02   1.000000e+00   1.371673e-01
      +  7  11    3.276535e+01    2.530155e+01    7.463799e+00   -9.109597e+00   4.788040e+02   1.000000e+00   1.092968e-01
      +  8   1    3.225221e+01    2.479005e+01    7.462158e+00   -1.027637e+00   4.891427e+02   1.000000e+00   1.104705e-01
      +  9  10    3.216230e+01    2.445205e+01    7.710247e+00   -1.804897e-01   7.393553e+01   1.000000e+00   4.294919e-02
      + 10  10    3.216015e+01    2.439780e+01    7.762350e+00   -4.285017e-03   1.308609e+01   1.000000e+00   1.806897e-02
      + 11  18    3.215993e+01    2.438399e+01    7.775934e+00   -4.424839e-04   3.526254e+00   1.000000e+00   9.379620e-03
      + 12  22    3.215993e+01    2.438428e+01    7.775651e+00   -1.835996e-07   7.735457e-02   1.000000e+00   1.389221e-03
      +
      +Converged in  12  iterations.
      +Termination reason:  Norm of the gradient less than tolerance
      +Final gradient norm:  0.00013206555177717097
      +Final cost:  32.15992905024544
      +
      +

      png

      +

      8. Compute the low rank-based Laplace approximation to posterior

      +

      We used the double pass algorithm to compute a low-rank decomposition of the Hessian Misfit. +In particular, we solve

      +

      + +

      +

      The figure shows the largest k generalized eigenvectors of the Hessian misfit. +The effective rank of the Hessian misfit is the number of eigenvalues above the red line (). +The effective rank is independent of the mesh size.

      +
      model.setPointForHessianEvaluations(x, gauss_newton_approx=False)
      +Hmisfit = ReducedHessian(model, misfit_only=True)
      +k = 50
      +p = 20
      +print( "Single/Double Pass Algorithm. Requested eigenvectors: {0}; Oversampling {1}.".format(k,p) )
      +
      +Omega = MultiVector(x[PARAMETER], k+p)
      +parRandom.normal(1., Omega)
      +lmbda, V = doublePassG(Hmisfit, prior.R, prior.Rsolver, Omega, k)
      +
      +posterior = GaussianLRPosterior(prior, lmbda, V)
      +posterior.mean = x[PARAMETER]
      +
      +plt.plot(range(0,k), lmbda, 'b*', range(0,k+1), np.ones(k+1), '-r')
      +plt.yscale('log')
      +plt.xlabel('number')
      +plt.ylabel('eigenvalue')
      +
      +nb.plot_eigenvectors(Vh[PARAMETER], V, mytitle="Eigenvector", which=[0,1,2,5,10,15])
      +
      +
      Single/Double Pass Algorithm. Requested eigenvectors: 50; Oversampling 20.
      +
      +

      png

      +

      png

      +

      9. Prior and Laplace approximation to posterior pointwise variance fields

      +
      compute_trace = True
      +if compute_trace:
      +    post_tr, prior_tr, corr_tr = posterior.trace(method="Randomized", r=200)
      +    print( "Posterior trace {0:5e}; Prior trace {1:5e}; Correction trace {2:5e}".format(post_tr, prior_tr, corr_tr) )
      +post_pw_variance, pr_pw_variance, corr_pw_variance = posterior.pointwise_variance(method="Randomized", r=200)
      +
      +objs = [dl.Function(Vh[PARAMETER], pr_pw_variance),
      +        dl.Function(Vh[PARAMETER], post_pw_variance)]
      +mytitles = ["Prior variance", "Posterior variance"]
      +nb.multi1_plot(objs, mytitles, logscale=False)
      +plt.show()
      +
      +
      Posterior trace 1.014727e+00; Prior trace 1.797376e+00; Correction trace 7.826487e-01
      +
      +

      png

      +

      10. Generate samples from Prior and Laplace approximation to posterior

      +
      nsamples = 5
      +noise = dl.Vector()
      +posterior.init_vector(noise,"noise")
      +s_prior = dl.Function(Vh[PARAMETER], name="sample_prior")
      +s_post = dl.Function(Vh[PARAMETER], name="sample_post")
      +
      +pr_max   =  2.5*math.sqrt( pr_pw_variance.max() ) + prior.mean.max()
      +pr_min   = -2.5*math.sqrt( pr_pw_variance.max() ) + prior.mean.min()
      +ps_max   =  2.5*math.sqrt( post_pw_variance.max() ) + posterior.mean.max()
      +ps_min   = -2.5*math.sqrt( post_pw_variance.max() ) + posterior.mean.min()
      +
      +for i in range(nsamples):
      +    parRandom.normal(1., noise)
      +    posterior.sample(noise, s_prior.vector(), s_post.vector())
      +    plt.figure(figsize=(15,5))
      +    nb.plot(s_prior, subplot_loc=121,mytitle="Prior sample", vmin=pr_min, vmax=pr_max)
      +    nb.plot(s_post, subplot_loc=122,mytitle="Posterior sample", vmin=ps_min, vmax=ps_max)
      +    plt.show()
      +
      +

      png

      +

      png

      +

      png

      +

      png

      +

      png

      +

      Copyright (c) 2016-2018, The University of Texas at Austin & University of California, Merced.
      +Copyright (c) 2019-2020, The University of Texas at Austin, University of California--Merced, Washington University in St. Louis.
      +All Rights reserved.
      +See file COPYRIGHT for details.

      +

      This file is part of the hIPPYlib library. For more information and source code +availability see https://hippylib.github.io.

      +

      hIPPYlib is free software; you can redistribute it and/or modify it under the terms of the GNU General Public License (as published by the Free Software Foundation) version 2.0 dated June 1991.

      +
      +
      + +
      +
      +

      Copyright © 2016-2023 The University of Texas at Austin, University of California Merced.
      Copyright © 2018-2020 The University of Texas at Austin, University of California Merced, Washington University in St. Louis.
      This material is based on work partially supported by the National Science Foundation under Grants No ACI-1550593, ACI-1550547.

      +

      Documentation built with MkDocs.

      +
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b/tutorials_v3.0.0/4_AdvectionDiffusionBayesian/index.html new file mode 100644 index 0000000..ac5670b --- /dev/null +++ b/tutorials_v3.0.0/4_AdvectionDiffusionBayesian/index.html @@ -0,0 +1,657 @@ + + + + + + + + + + + Advection-Diffusion Bayesian - hIPPYlib + + + + + + + + + +

      + +
      +
      + +
      + +

      + +

      +

      Bayesian initial condition inversion in an advection-diffusion problem

      +

      In this example we tackle the problem of quantifying the uncertainty in the solution of an inverse problem governed by a parabolic PDE via the Bayesian inference framework. The underlying PDE is a time-dependent advection-diffusion equation in which we seek to infer an unknown initial condition from spatio-temporal point measurements.

      +

      The Bayesian inverse problem:

      +

      Following the Bayesian framework, we utilize +a Gaussian prior measure , +with where is an elliptic differential operator as +described in the PoissonBayesian example, and use an additive +Gaussian noise model. Therefore, the solution of the Bayesian inverse +problem is the posterior measure, with + and .

      +
        +
      • The posterior mean is characterized as the minimizer of
      • +
      +

      + +

      +

      which can also be interpreted as the regularized functional to be +minimized in deterministic inversion. The observation operator extracts the values of the forward solution on a set of +locations at +times .

      +
        +
      • The posterior covariance is the inverse of the Hessian of , i.e.,
      • +
      +

      + +

      +

      The forward problem:

      +

      The parameter-to-observable map maps an initial condition to pointwise spatiotemporal observations of the concentration field through solution of the advection-diffusion equation given by

      +

      + +

      +

      Here, () is a bounded domain, is the diffusion coefficient and is the final +time. The velocity field + is computed by solving the following steady-state +Navier-Stokes equation with the side walls driving the flow:

      +

      + +

      +

      Here, is pressure, is the Reynolds number. The Dirichlet boundary data + is given by + on the left wall of the domain, + on the right wall, and everywhere else.

      +

      The adjoint problem:

      +

      The adjoint problem is a final value problem, since is specified at rather than at . Thus, it is solved backwards in time, which amounts to the solution of the advection-diffusion equation

      +

      + +

      +

      Then, the adjoint of the parameter to observable map is defined by setting +

      +

      1. Load modules

      +
      import dolfin as dl
      +import ufl
      +import math
      +import numpy as np
      +import matplotlib.pyplot as plt
      +%matplotlib inline
      +import sys
      +import os
      +sys.path.append( os.environ.get('HIPPYLIB_BASE_DIR', "../") )
      +from hippylib import *
      +sys.path.append( os.environ.get('HIPPYLIB_BASE_DIR', "..") + "/applications/ad_diff/" )
      +from model_ad_diff import TimeDependentAD, SpaceTimePointwiseStateObservation
      +
      +import logging
      +logging.getLogger('FFC').setLevel(logging.WARNING)
      +logging.getLogger('UFL').setLevel(logging.WARNING)
      +dl.set_log_active(False)
      +
      +

      2. Construct the velocity field

      +
      def v_boundary(x,on_boundary):
      +    return on_boundary
      +
      +def q_boundary(x,on_boundary):
      +    return x[0] < dl.DOLFIN_EPS and x[1] < dl.DOLFIN_EPS
      +
      +def computeVelocityField(mesh):
      +    Xh = dl.VectorFunctionSpace(mesh,'Lagrange', 2)
      +    Wh = dl.FunctionSpace(mesh, 'Lagrange', 1)
      +    mixed_element = ufl.MixedElement([Xh.ufl_element(), Wh.ufl_element()])
      +    XW = dl.FunctionSpace(mesh, mixed_element)
      +
      +    Re = dl.Constant(1e2)
      +
      +    g = dl.Expression(('0.0','(x[0] < 1e-14) - (x[0] > 1 - 1e-14)'), degree=1)
      +    bc1 = dl.DirichletBC(XW.sub(0), g, v_boundary)
      +    bc2 = dl.DirichletBC(XW.sub(1), dl.Constant(0), q_boundary, 'pointwise')
      +    bcs = [bc1, bc2]
      +
      +    vq = dl.Function(XW)
      +    (v,q) = ufl.split(vq)
      +    (v_test, q_test) = dl.TestFunctions (XW)
      +
      +    def strain(v):
      +        return ufl.sym(ufl.grad(v))
      +
      +    F = ( (2./Re)*ufl.inner(strain(v),strain(v_test))+ ufl.inner (ufl.nabla_grad(v)*v, v_test)
      +           - (q * ufl.div(v_test)) + ( ufl.div(v) * q_test) ) * ufl.dx
      +
      +    dl.solve(F == 0, vq, bcs, solver_parameters={"newton_solver":
      +                                         {"relative_tolerance":1e-4, "maximum_iterations":100}})
      +
      +    plt.figure(figsize=(15,5))
      +    vh = dl.project(v,Xh)
      +    qh = dl.project(q,Wh)
      +    nb.plot(nb.coarsen_v(vh), subplot_loc=121,mytitle="Velocity")
      +    nb.plot(qh, subplot_loc=122,mytitle="Pressure")
      +    plt.show()
      +
      +    return v
      +
      +

      3. Set up the mesh and finite element spaces

      +
      mesh = dl.refine( dl.Mesh("ad_20.xml") )
      +wind_velocity = computeVelocityField(mesh)
      +Vh = dl.FunctionSpace(mesh, "Lagrange", 1)
      +print( "Number of dofs: {0}".format( Vh.dim() ) )
      +
      +

      png

      +
      Number of dofs: 2023
      +
      +

      4. Set up model (prior, true/proposed initial condition)

      +
      ic_expr = dl.Expression(
      +    'std::min(0.5,std::exp(-100*(std::pow(x[0]-0.35,2) +  std::pow(x[1]-0.7,2))))',
      +    element=Vh.ufl_element())
      +true_initial_condition = dl.interpolate(ic_expr, Vh).vector()
      +gamma = 1.
      +delta = 8.
      +prior = BiLaplacianPrior(Vh, gamma, delta, robin_bc=True)
      +
      +prior.mean = dl.interpolate(dl.Constant(0.25), Vh).vector()
      +
      +t_init         = 0.
      +t_final        = 4.
      +t_1            = 1.
      +dt             = .1
      +observation_dt = .2
      +
      +simulation_times = np.arange(t_init, t_final+.5*dt, dt)
      +observation_times = np.arange(t_1, t_final+.5*dt, observation_dt)
      +
      +targets = np.loadtxt('targets.txt')
      +print ("Number of observation points: {0}".format(targets.shape[0]) )
      +misfit = SpaceTimePointwiseStateObservation(Vh, observation_times, targets)
      +
      +problem = TimeDependentAD(mesh, [Vh,Vh,Vh], prior, misfit, simulation_times, wind_velocity, True)
      +
      +objs = [dl.Function(Vh,true_initial_condition),
      +        dl.Function(Vh,prior.mean)]
      +mytitles = ["True Initial Condition", "Prior mean"]
      +nb.multi1_plot(objs, mytitles)
      +plt.show()
      +
      +
      Number of observation points: 80
      +
      +

      png

      +

      5. Generate the synthetic observations

      +
      rel_noise = 0.01
      +utrue = problem.generate_vector(STATE)
      +x = [utrue, true_initial_condition, None]
      +problem.solveFwd(x[STATE], x)
      +misfit.observe(x, misfit.d)
      +MAX = misfit.d.norm("linf", "linf")
      +noise_std_dev = rel_noise * MAX
      +parRandom.normal_perturb(noise_std_dev,misfit.d)
      +misfit.noise_variance = noise_std_dev*noise_std_dev
      +
      +nb.show_solution(Vh, true_initial_condition, utrue, "Solution")
      +
      +

      png

      +

      6. Test the gradient and the Hessian of the cost (negative log posterior)

      +
      m0 = true_initial_condition.copy()
      +_ = modelVerify(problem, m0, is_quadratic=True)
      +
      +
      (yy, H xx) - (xx, H yy) =  -1.1681934136389683e-13
      +
      +

      png

      +

      7. Evaluate the gradient

      +
      [u,m,p] = problem.generate_vector()
      +problem.solveFwd(u, [u,m,p])
      +problem.solveAdj(p, [u,m,p])
      +mg = problem.generate_vector(PARAMETER)
      +grad_norm = problem.evalGradientParameter([u,m,p], mg)
      +
      +print( "(g,g) = ", grad_norm)
      +
      +
      (g,g) =  2354159843455358.0
      +
      +

      8. The Gaussian posterior

      +
      H = ReducedHessian(problem, misfit_only=True) 
      +
      +k = 80
      +p = 20
      +print( "Single Pass Algorithm. Requested eigenvectors: {0}; Oversampling {1}.".format(k,p) )
      +Omega = MultiVector(x[PARAMETER], k+p)
      +parRandom.normal(1., Omega)
      +lmbda, V = singlePassG(H, prior.R, prior.Rsolver, Omega, k)
      +
      +
      +posterior = GaussianLRPosterior( prior, lmbda, V )
      +
      +plt.plot(range(0,k), lmbda, 'b*', range(0,k+1), np.ones(k+1), '-r')
      +plt.yscale('log')
      +plt.xlabel('number')
      +plt.ylabel('eigenvalue')
      +
      +nb.plot_eigenvectors(Vh, V, mytitle="Eigenvector", which=[0,1,2,5,10,20,30,45,60])
      +
      +
      Single Pass Algorithm. Requested eigenvectors: 80; Oversampling 20.
      +
      +

      png

      +

      png

      +

      9. Compute the MAP point

      +
      H.misfit_only = False
      +
      +solver = CGSolverSteihaug()
      +solver.set_operator(H)
      +solver.set_preconditioner( posterior.Hlr )
      +solver.parameters["print_level"] = 1
      +solver.parameters["rel_tolerance"] = 1e-6
      +solver.solve(m, -mg)
      +problem.solveFwd(u, [u,m,p])
      +
      +total_cost, reg_cost, misfit_cost = problem.cost([u,m,p])
      +print( "Total cost {0:5g}; Reg Cost {1:5g}; Misfit {2:5g}".format(total_cost, reg_cost, misfit_cost) )
      +
      +posterior.mean = m
      +
      +plt.figure(figsize=(7.5,5))
      +nb.plot(dl.Function(Vh, m), mytitle="Initial Condition")
      +plt.show()
      +
      +nb.show_solution(Vh, m, u, "Solution")
      +
      +
       Iterartion :  0  (B r, r) =  1231866.6310868163
      + Iteration :  1  (B r, r) =  66.31734328582567
      + Iteration :  2  (B r, r) =  0.3150082598782035
      + Iteration :  3  (B r, r) =  0.003975346025817018
      + Iteration :  4  (B r, r) =  1.7003714104952925e-05
      + Iteration :  5  (B r, r) =  8.577697903362507e-08
      +Relative/Absolute residual less than tol
      +Converged in  5  iterations with final norm  0.00029287707153962235
      +Total cost 782.968; Reg Cost 153.412; Misfit 629.556
      +
      +

      png

      +

      png

      +

      10. Prior and posterior pointwise variance fields

      +
      compute_trace = True
      +if compute_trace:
      +    post_tr, prior_tr, corr_tr = posterior.trace(method="Randomized", r=300)
      +    print( "Posterior trace {0:5g}; Prior trace {1:5g}; Correction trace {2:5g}".format(post_tr, prior_tr, corr_tr) )
      +post_pw_variance, pr_pw_variance, corr_pw_variance = posterior.pointwise_variance(method="Randomized", r=300)
      +
      +objs = [dl.Function(Vh, pr_pw_variance),
      +        dl.Function(Vh, post_pw_variance)]
      +mytitles = ["Prior Variance", "Posterior Variance"]
      +nb.multi1_plot(objs, mytitles, logscale=False)
      +plt.show()
      +
      +
      Posterior trace 0.000268048; Prior trace 0.00809706; Correction trace 0.00782902
      +
      +

      png

      +

      11. Draw samples from the prior and posterior distributions

      +
      nsamples = 5
      +noise = dl.Vector()
      +posterior.init_vector(noise,"noise")
      +s_prior = dl.Function(Vh, name="sample_prior")
      +s_post = dl.Function(Vh, name="sample_post")
      +
      +pr_max =  2.5*math.sqrt( pr_pw_variance.max() ) + prior.mean.max()
      +pr_min = -2.5*math.sqrt( pr_pw_variance.min() ) + prior.mean.min()
      +ps_max =  2.5*math.sqrt( post_pw_variance.max() ) + posterior.mean.max()
      +ps_min = -2.5*math.sqrt( post_pw_variance.max() ) + posterior.mean.min()
      +
      +for i in range(nsamples):
      +    parRandom.normal(1., noise)
      +    posterior.sample(noise, s_prior.vector(), s_post.vector())
      +    plt.figure(figsize=(15,5))
      +    nb.plot(s_prior, subplot_loc=121,mytitle="Prior sample", vmin=pr_min, vmax=pr_max)
      +    nb.plot(s_post, subplot_loc=122,mytitle="Posterior sample", vmin=ps_min, vmax=ps_max)
      +    plt.show()
      +
      +

      png

      +

      png

      +

      png

      +

      png

      +

      png

      +

      Copyright (c) 2016-2018, The University of Texas at Austin & University of California, Merced.
      +Copyright (c) 2019-2020, The University of Texas at Austin, University of California--Merced, Washington University in St. Louis.
      +All Rights reserved.
      +See file COPYRIGHT for details.

      +

      This file is part of the hIPPYlib library. For more information and source code +availability see https://hippylib.github.io.

      +

      hIPPYlib is free software; you can redistribute it and/or modify it under the terms of the GNU General Public License (as published by the Free Software Foundation) version 2.0 dated June 1991.

      +
      +
      + +
      +
      +

      Copyright © 2016-2023 The University of Texas at Austin, University of California Merced.
      Copyright © 2018-2020 The University of Texas at Austin, University of California Merced, Washington University in St. Louis.
      This material is based on work partially supported by the National Science Foundation under Grants No ACI-1550593, ACI-1550547.

      +

      Documentation built with MkDocs.

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      + +
      +
      + +
      + +

      Spectrum of the preconditioned Hessian misfit operator

      +

      The linear source inversion problem

      +

      We consider the following linear source inversion problem. +Find the state and the source (parameter) that solves

      +

      + +

      +

      Here:

      +
        +
      • +

        + is a finite dimensional vector that denotes noisy observations of the state in locations . Specifically, , where are i.i.d. .

        +
      • +
      • +

        + is the linear operator that evaluates the state at the observation locations .

        +
      • +
      • +

        + and are the parameters of the regularization penalizing the and norm of , respectively.

        +
      • +
      • +

        +, , are given coefficients representing the diffusivity coefficient, the advective velocity and the reaction term, respectively.

        +
      • +
      • +

        +, represents the subdomain of where we impose Dirichlet or Neumann boundary conditions, respectively.

        +
      • +
      +

      1. Load modules

      +
      import dolfin as dl
      +import ufl
      +import numpy as np
      +import matplotlib.pyplot as plt
      +%matplotlib inline
      +
      +import sys
      +import os
      +sys.path.append( os.environ.get('HIPPYLIB_BASE_DIR', "../") )
      +from hippylib import *
      +
      +import logging
      +logging.getLogger('FFC').setLevel(logging.WARNING)
      +logging.getLogger('UFL').setLevel(logging.WARNING)
      +dl.set_log_active(False)
      +
      +

      2. The linear source inversion problem

      +
      def pde_varf(u,m,p):
      +    return k*ufl.inner(ufl.grad(u), ufl.grad(p))*ufl.dx \
      +           + ufl.inner(ufl.grad(u), v*p)*ufl.dx \
      +           + c*u*p*ufl.dx \
      +           - m*p*ufl.dx
      +
      +def u_boundary(x, on_boundary):
      +    return on_boundary and x[1] < dl.DOLFIN_EPS
      +
      +def solve(nx,ny, targets, rel_noise, gamma, delta, verbose=True):
      +    myRandom = Random()
      +    mesh = dl.UnitSquareMesh(nx, ny)
      +    Vh1 = dl.FunctionSpace(mesh, 'Lagrange', 1)
      +
      +    Vh = [Vh1, Vh1, Vh1]
      +    if verbose:
      +        print( "Number of dofs: STATE={0}, PARAMETER={1}, ADJOINT={2}".format(Vh[STATE].dim(), Vh[PARAMETER].dim(), Vh[ADJOINT].dim()) )
      +
      +
      +    u_bdr = dl.Constant(0.0)
      +    u_bdr0 = dl.Constant(0.0)
      +    bc = dl.DirichletBC(Vh[STATE], u_bdr, u_boundary)
      +    bc0 = dl.DirichletBC(Vh[STATE], u_bdr0, u_boundary)
      +
      +    mtrue_expr =  dl.Expression('min(0.5,exp(-100*(pow(x[0]-0.35,2) +  pow(x[1]-0.7,2))))',degree=5)
      +    mtrue = dl.interpolate(mtrue_expr , Vh[PARAMETER]).vector()
      +    m0 = dl.interpolate(dl.Constant(0.0), Vh[PARAMETER]).vector()
      +
      +    pde = PDEVariationalProblem(Vh, pde_varf, bc, bc0, is_fwd_linear=True)
      +
      +    if verbose:
      +        print( "Number of observation points: {0}".format(targets.shape[0]) )
      +
      +    misfit = PointwiseStateObservation(Vh[STATE], targets)
      +
      +    reg = LaplacianPrior(Vh[PARAMETER], gamma, delta)
      +
      +    #Generate synthetic observations
      +    utrue = pde.generate_state()
      +    x = [utrue, mtrue, None]
      +    pde.solveFwd(x[STATE], x)
      +    misfit.B.mult(x[STATE], misfit.d)
      +    MAX = misfit.d.norm("linf")
      +    noise_std_dev = rel_noise * MAX
      +    myRandom.normal_perturb(noise_std_dev, misfit.d)
      +    misfit.noise_variance = noise_std_dev*noise_std_dev
      +
      +    if verbose:
      +        plt.figure(figsize=(18,4))
      +        nb.plot(dl.Function(Vh[PARAMETER], mtrue), mytitle = "True source", subplot_loc=131)
      +        nb.plot(dl.Function(Vh[STATE], utrue), mytitle="True state", subplot_loc=132)
      +        nb.plot_pts(targets, misfit.d,mytitle="Observations", subplot_loc=133)
      +        plt.show()
      +
      +    model = Model(pde, reg, misfit)
      +    u = model.generate_vector(STATE)
      +    m = m0.copy()
      +    p = model.generate_vector(ADJOINT)
      +    x = [u,m,p]
      +    mg = model.generate_vector(PARAMETER)
      +    model.solveFwd(u, x)
      +    model.solveAdj(p, x)
      +    model.evalGradientParameter(x, mg)
      +    model.setPointForHessianEvaluations(x, gauss_newton_approx=False)
      +
      +    H = ReducedHessian(model)
      +
      +    solver = CGSolverSteihaug()
      +    solver.set_operator(H)
      +    solver.set_preconditioner( reg.Rsolver )
      +    solver.parameters["print_level"] = -1
      +    solver.parameters["rel_tolerance"] = 1e-9
      +    solver.solve(m, -mg)
      +
      +    if solver.converged:
      +        if verbose:
      +            print( "CG converged in ", solver.iter, " iterations." )
      +    else:
      +        print( "CG did not converged." )
      +        raise
      +
      +    model.solveFwd(u, x)
      +
      +    total_cost, reg_cost, misfit_cost = model.cost(x)
      +
      +    if verbose:
      +        plt.figure(figsize=(18,4))
      +        nb.plot(dl.Function(Vh[PARAMETER], m), mytitle = "Reconstructed source", subplot_loc=131)
      +        nb.plot(dl.Function(Vh[STATE], u), mytitle="Reconstructed state", subplot_loc=132)
      +        nb.plot_pts(targets, misfit.B*u - misfit.d, mytitle="Misfit", subplot_loc=133)
      +        plt.show()
      +
      +    H.misfit_only = True
      +    k_evec = 80
      +    p_evec = 5
      +    if verbose:
      +        print( "Double Pass Algorithm. Requested eigenvectors: {0}; Oversampling {1}.".format(k_evec,p_evec) )
      +    Omega = MultiVector(x[PARAMETER], k_evec+p_evec)
      +    myRandom.normal(1., Omega)
      +    lmbda, V = doublePassG(H, reg.R, reg.Rsolver, Omega, k_evec)
      +
      +    if verbose:
      +        plt.figure()
      +        nb.plot_eigenvalues(lmbda, mytitle="Generalized Eigenvalues")
      +        nb.plot_eigenvectors(Vh[PARAMETER], V, mytitle="Eigenvectors", which=[0,1,2,5,10,15])
      +        plt.show()
      +
      +    return lmbda, V, Vh[PARAMETER], solver.iter
      +
      +
      +

      3. Solution of the source inversion problem

      +
      ndim = 2
      +nx = 32
      +ny = 32
      +
      +ntargets = 300
      +np.random.seed(seed=1)
      +targets = np.random.uniform(0.1,0.9, [ntargets, ndim] )
      +rel_noise = 0.01
      +
      +gamma = 70.
      +delta = 1e-1
      +
      +k = dl.Constant(1.0)
      +v = dl.Constant((0.0, 0.0))
      +c = dl.Constant(0.)
      +
      +lmbda, V, Vm, nit = solve(nx,ny, targets, rel_noise, gamma, delta)
      +
      +
      Number of dofs: STATE=1089, PARAMETER=1089, ADJOINT=1089
      +Number of observation points: 300
      +
      +

      png

      +
      CG converged in  69  iterations.
      +
      +

      png

      +
      Double Pass Algorithm. Requested eigenvectors: 80; Oversampling 5.
      +
      +

      png

      +

      png

      +

      4. Mesh independence of the spectrum of the preconditioned Hessian misfit

      +
      gamma = 70.
      +delta = 1e-1
      +
      +k = dl.Constant(1.0)
      +v = dl.Constant((0.0, 0.0))
      +c = dl.Constant(0.)
      +
      +n = [16,32,64]
      +lmbda1, V1, Vm1, niter1 = solve(n[0],n[0], targets, rel_noise, gamma, delta,verbose=False)
      +lmbda2, V2, Vm2, niter2 = solve(n[1],n[1], targets, rel_noise, gamma, delta,verbose=False)
      +lmbda3, V3, Vm3, niter3 = solve(n[2],n[2], targets, rel_noise, gamma, delta,verbose=False)
      +
      +print( "Number of Iterations: ", niter1, niter2, niter3 )
      +plt.figure(figsize=(18,4))
      +nb.plot_eigenvalues(lmbda1, mytitle="Eigenvalues Mesh {0} by {1}".format(n[0],n[0]), subplot_loc=131)
      +nb.plot_eigenvalues(lmbda2, mytitle="Eigenvalues Mesh {0} by {1}".format(n[1],n[1]), subplot_loc=132)
      +nb.plot_eigenvalues(lmbda3, mytitle="Eigenvalues Mesh {0} by {1}".format(n[2],n[2]), subplot_loc=133)
      +
      +nb.plot_eigenvectors(Vm1, V1, mytitle="Mesh {0} by {1} Eigen".format(n[0],n[0]), which=[0,1,5])
      +nb.plot_eigenvectors(Vm2, V2, mytitle="Mesh {0} by {1} Eigen".format(n[1],n[1]), which=[0,1,5])
      +nb.plot_eigenvectors(Vm3, V3, mytitle="Mesh {0} by {1} Eigen".format(n[2],n[2]), which=[0,1,5])
      +
      +plt.show()
      +
      +
      Number of Iterations:  67 69 68
      +
      +

      png

      +

      png

      +

      png

      +

      png

      +

      5. Dependence on the noise level

      +

      We solve the problem for different noise levels. The higher the noise level the more important becomes the effect of the regularization.

      +
      gamma = 70.
      +delta = 1e-1
      +
      +k = dl.Constant(1.0)
      +v = dl.Constant((0.0, 0.0))
      +c = dl.Constant(0.)
      +
      +rel_noise = [1e-3,1e-2,1e-1]
      +lmbda1, V1, Vm1, niter1 = solve(nx,ny, targets, rel_noise[0], gamma, delta,verbose=False)
      +lmbda2, V2, Vm2, niter2 = solve(nx,ny, targets, rel_noise[1], gamma, delta,verbose=False)
      +lmbda3, V3, Vm3, niter3 = solve(nx,ny, targets, rel_noise[2], gamma, delta,verbose=False)
      +
      +print( "Number of Iterations: ", niter1, niter2, niter3 )
      +plt.figure(figsize=(18,4))
      +nb.plot_eigenvalues(lmbda1, mytitle="Eigenvalues rel_noise {0:g}".format(rel_noise[0]), subplot_loc=131)
      +nb.plot_eigenvalues(lmbda2, mytitle="Eigenvalues rel_noise {0:g}".format(rel_noise[1]), subplot_loc=132)
      +nb.plot_eigenvalues(lmbda3, mytitle="Eigenvalues rel_noise {0:g}".format(rel_noise[2]), subplot_loc=133)
      +
      +nb.plot_eigenvectors(Vm1, V1, mytitle="rel_noise {0:g} Eigen".format(rel_noise[0]), which=[0,1,5])
      +nb.plot_eigenvectors(Vm2, V2, mytitle="rel_noise {0:g} Eigen".format(rel_noise[1]), which=[0,1,5])
      +nb.plot_eigenvectors(Vm3, V3, mytitle="rel_noise {0:g} Eigen".format(rel_noise[2]), which=[0,1,5])
      +
      +plt.show()
      +
      +
      Number of Iterations:  185 67 23
      +
      +

      png

      +

      png

      +

      png

      +

      png

      +

      6. Dependence on the PDE coefficients

      +

      Assume a constant reaction term , and we consider different values for the diffusivity coefficient .

      +

      The smaller the value of the slower the decay in the spectrum.

      +
      rel_noise = 0.01
      +
      +k = dl.Constant(1.0)
      +v = dl.Constant((0.0, 0.0))
      +c = dl.Constant(1.0)
      +
      +lmbda1, V1, Vm1, niter1 = solve(nx,ny, targets, rel_noise, gamma, delta,verbose=False)
      +k = dl.Constant(0.1)
      +lmbda2, V2, Vm2, niter2 = solve(nx,ny, targets, rel_noise, gamma, delta,verbose=False)
      +k = dl.Constant(0.01)
      +lmbda3, V3, Vm3, niter3 = solve(nx,ny, targets, rel_noise, gamma, delta,verbose=False)
      +
      +print( "Number of Iterations: ", niter1, niter2, niter3 )
      +plt.figure(figsize=(18,4))
      +nb.plot_eigenvalues(lmbda1, mytitle="Eigenvalues k=1.0", subplot_loc=131)
      +nb.plot_eigenvalues(lmbda2, mytitle="Eigenvalues k=0.1", subplot_loc=132)
      +nb.plot_eigenvalues(lmbda3, mytitle="Eigenvalues k=0.01", subplot_loc=133)
      +
      +nb.plot_eigenvectors(Vm1, V1, mytitle="k=1. Eigen", which=[0,1,5])
      +nb.plot_eigenvectors(Vm2, V2, mytitle="k=0.1 Eigen", which=[0,1,5])
      +nb.plot_eigenvectors(Vm3, V3, mytitle="k=0.01 Eigen", which=[0,1,5])
      +
      +plt.show()
      +
      +
      Number of Iterations:  79 143 256
      +
      +

      png

      +

      png

      +

      png

      +

      png

      +

      Copyright (c) 2016-2018, The University of Texas at Austin & University of California, Merced.
      +Copyright (c) 2019-2020, The University of Texas at Austin, University of California--Merced, Washington University in St. Louis.
      +All Rights reserved.
      +See file COPYRIGHT for details.

      +

      This file is part of the hIPPYlib library. For more information and source code +availability see https://hippylib.github.io.

      +

      hIPPYlib is free software; you can redistribute it and/or modify it under the terms of the GNU General Public License (as published by the Free Software Foundation) version 2.0 dated June 1991.

      +
      +
      + +
      +
      +

      Copyright © 2016-2023 The University of Texas at Austin, University of California Merced.
      Copyright © 2018-2020 The University of Texas at Austin, University of California Merced, Washington University in St. Louis.
      This material is based on work partially supported by the National Science Foundation under Grants No ACI-1550593, ACI-1550547.

      +

      Documentation built with MkDocs.

      +
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