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<html>
<head>
<title>
SPARSE_GRID_HERMITE - Sparse Grids Based on Gauss-Hermite Rules
</title>
</head>
<body bgcolor="#EEEEEE" link="#CC0000" alink="#FF3300" vlink="#000055">
<h1 align = "center">
SPARSE_GRID_HERMITE <br> Sparse Grids Based on Gauss-Hermite Rules
</h1>
<hr>
<p>
<b>SPARSE_GRID_HERMITE</b>
is a C++ library which
constructs sparse grids based on 1D Gauss-Hermite rules.
</p>
<p>
Sparse grids are more naturally constructed from a nested family
of quadrature rules. Gauss-Hermite rules are not nested, but
have higher accuracy. Thus, there can be a tradeoff. If we compare
two sparse grids of the same "level", one using Gauss-Hermite rules
and the other a nested rule, then the Gauss-Hermite
sparse grid will have higher accuracy...but also a significantly
greater number of points. When measuring efficiency, we really need
to balance the cost in quadrature points against the accuracy, and
so it is not immediately obvious which choice is best!
</p>
<p>
To slightly complicate matters, Gauss-Hermite rules are very weakly
nested, in that the rules of odd order all include the abscissa
value X=0.0. A sparse grid constructed from Gauss-Hermite rules
will thus have to keep track of this minor point as well.
</p>
<p>
Here is a table showing the number of points in a sparse grid based on
Gauss-Hermite rules, indexed by the spatial dimension, and by the
"level", which is simply an index for the family of sparse grids.
<table border = "1">
<tr>
<th>DIM:</th><th>1</th><th>2</th><th>3</th><th>4</th><th>5</th><th>6</th>
</tr>
<tr>
<th>LEVEL_MAX</th><th> </th><th> </th><th> </th><th> </th><th> </th><th> </th>
</tr>
<tr>
<th>0</th><td>1</td><td>1</td><td>1</td><td>1</td><td>1</td><td>1</td>
</tr>
<tr>
<th>1</th><td>3</td><td>5</td><td>7</td><td>9</td><td>11</td><td>13</td>
</tr>
<tr>
<th>2</th><td>7</td><td>22</td><td>37</td><td>57</td><td>81</td><td>109</td>
</tr>
<tr>
<th>3</th><td>15</td><td>75</td><td>161</td><td>289</td><td>471</td><td>713</td>
</tr>
<tr>
<th>4</th><td>31</td><td>224</td><td>608</td><td>1268</td><td>2341</td><td>3953</td>
</tr>
<tr>
<th>5</th><td>63</td><td>613</td><td>2070</td><td>4994</td><td>10367</td><td>19397</td>
</tr>
<tr>
<th>6</th><td>127</td><td>1570</td><td>6507</td><td>18076</td><td>41957</td><td>86522</td>
</tr>
</table>
</p>
<h3 align = "center">
Web Link:
</h3>
<p>
A version of the sparse grid library is available in
<a href = "http://tasmanian.ornl.gov/">
http://tasmanian.ornl.gov</a>,
the TASMANIAN library, available from Oak Ridge National Laboratory.
</p>
<h3 align = "center">
Licensing:
</h3>
<p>
The code described and made available on this web page is distributed
under the
<a href = "gnu_lgpl.txt">GNU LGPL</a> license.
</p>
<h3 align = "center">
Languages:
</h3>
<p>
<b>SPARSE_GRID_HERMITE</b> is available in
<a href = "../../cpp_src/sparse_grid_hermite/sparse_grid_hermite.html">a C++ version</a> and
<a href = "../../f_src/sparse_grid_hermite/sparse_grid_hermite.html">a FORTRAN90 version</a> and
<a href = "../../m_src/sparse_grid_hermite/sparse_grid_hermite.html">a MATLAB version.</a>
</p>
<h3 align = "center">
Related Data and Programs:
</h3>
<p>
<a href = "../../m_src/cc_display/cc_display.html">
CC_DISPLAY</a>,
a MATLAB library which
can compute and display Clenshaw Curtis grids in two dimensions,
as well as sparse grids formed from sums of Clenshaw Curtis grids.
</p>
<p>
<a href = "../../datasets/quadrature_rules/quadrature_rules.html">
QUADRATURE_RULES</a>,
a dataset directory which
define quadrature rules;
a number of examples of sparse grid quadrature rules are included.
</p>
<p>
<a href = "../../cpp_src/quadrule/quadrule.html">
QUADRULE</a>,
a C++ library which
defines quadrature rules for
various intervals and weight functions.
</p>
<p>
<a href = "../../cpp_src/sgmg/sgmg.html">
SGMG</a>,
a C++ library which
creates a sparse grid dataset based on a mixed set of 1D factor rules,
and experiments with the use of a linear growth rate for the quadrature rules.
</p>
<p>
<a href = "../../cpp_src/sgmga/sgmga.html">
SGMGA</a>,
a C++ library which
creates sparse grids based on a mixture of 1D quadrature rules,
allowing anisotropic weights for each dimension.
</p>
<p>
<a href = "../../c_src/smolpack/smolpack.html">
SMOLPACK</a>,
a C library which
implements Novak and Ritter's method for estimating the integral
of a function over a multidimensional hypercube using sparse grids.
</p>
<p>
<a href = "../../datasets/sparse_grid_cc/sparse_grid_cc.html">
SPARSE_GRID_CC</a>,
a dataset directory which
contains the abscissas of sparse
grids based on a Clenshaw Curtis rule.
</p>
<p>
<a href = "../../datasets/sparse_grid_f2/sparse_grid_f2.html">
SPARSE_GRID_F2</a>,
a dataset directory which
contains the abscissas of sparse
grids based on a Fejer Type 2 rule.
</p>
<p>
<a href = "../../cpp_src/sparse_grid_gl/sparse_grid_gl.html">
SPARSE_GRID_GL</a>,
a C++ library which
computes a sparse grid
based on 1D Gauss-Legendre rules..
</p>
<p>
<a href = "../../datasets/sparse_grid_gp/sparse_grid_gp.html">
SPARSE_GRID_GP</a>,
a dataset directory which
contains the abscissas of sparse
grids based on a Gauss Patterson rule.
</p>
<p>
<a href = "../../datasets/sparse_grid_hermite/sparse_grid_hermite.html">
SPARSE_GRID_HERMITE</a>,
a dataset directory which
contains the abscissas of sparse
grids based on a Gauss Hermite rule.
</p>
<p>
<a href = "../../cpp_src/sparse_grid_mixed/sparse_grid_mixed.html">
SPARSE_GRID_MIXED</a>,
a C++ library which
constructs a sparse grid using different rules in each spatial dimension.
</p>
<p>
<a href = "../../datasets/sparse_grid_ncc/sparse_grid_ncc.html">
SPARSE_GRID_NCC</a>,
a dataset directory which
contains the abscissas of sparse
grids based on a Newton Cotes closed rule.
</p>
<p>
<a href = "../../datasets/sparse_grid_nco/sparse_grid_nco.html">
SPARSE_GRID_NCO</a>,
a dataset directory which
contains the abscissas of sparse
grids based on a Newton Cotes open rule.
</p>
<p>
<a href = "../../cpp_src/sparse_grid_open/sparse_grid_open.html">
SPARSE_GRID_OPEN</a>,
a C++ library which
defines define sparse grids based on open nested quadrature rules.
</p>
<p>
<a href = "../../m_src/toms847/toms847.html">
TOMS847</a>,
a MATLAB program which
uses sparse grids to carry out multilinear hierarchical interpolation.
It is commonly known as SPINTERP, and is by Andreas Klimke.
</p>
<h3 align = "center">
Reference:
</h3>
<p>
<ol>
<li>
Volker Barthelmann, Erich Novak, Klaus Ritter,<br>
High Dimensional Polynomial Interpolation on Sparse Grids,<br>
Advances in Computational Mathematics,<br>
Volume 12, Number 4, 2000, pages 273-288.
</li>
<li>
Thomas Gerstner, Michael Griebel,<br>
Numerical Integration Using Sparse Grids,<br>
Numerical Algorithms,<br>
Volume 18, Number 3-4, 1998, pages 209-232.
</li>
<li>
Albert Nijenhuis, Herbert Wilf,<br>
Combinatorial Algorithms for Computers and Calculators,<br>
Second Edition,<br>
Academic Press, 1978,<br>
ISBN: 0-12-519260-6,<br>
LC: QA164.N54.
</li>
<li>
Fabio Nobile, Raul Tempone, Clayton Webster,<br>
A Sparse Grid Stochastic Collocation Method for Partial Differential
Equations with Random Input Data,<br>
SIAM Journal on Numerical Analysis,<br>
Volume 46, Number 5, 2008, pages 2309-2345.
</li>
<li>
Sergey Smolyak,<br>
Quadrature and Interpolation Formulas for Tensor Products of
Certain Classes of Functions,<br>
Doklady Akademii Nauk SSSR,<br>
Volume 4, 1963, pages 240-243.
</li>
<li>
Dennis Stanton, Dennis White,<br>
Constructive Combinatorics,<br>
Springer, 1986,<br>
ISBN: 0387963472,<br>
LC: QA164.S79.
</li>
</ol>
</p>
<h3 align = "center">
Source Code:
</h3>
<p>
<ul>
<li>
<a href = "sparse_grid_hermite.cpp">sparse_grid_hermite.cpp</a>, the source code.
</li>
<li>
<a href = "sparse_grid_hermite.hpp">sparse_grid_hermite.hpp</a>,
the include file.
</li>
<li>
<a href = "sparse_grid_hermite.sh">sparse_grid_hermite.sh</a>,
commands to compile the source code.
</li>
</ul>
</p>
<h3 align = "center">
Examples and Tests:
</h3>
<p>
<ul>
<li>
<a href = "sparse_grid_hermite_prb.cpp">sparse_grid_hermite_prb.cpp</a>,
a sample calling program.
</li>
<li>
<a href = "sparse_grid_hermite_prb.sh">sparse_grid_hermite_prb.sh</a>,
commands to compile and run the sample program.
</li>
<li>
<a href = "sparse_grid_hermite_prb_output.txt">sparse_grid_hermite_prb_output.txt</a>,
the output file.
</li>
</ul>
</p>
<p>
The sample program creates three files, which
are an example of how a sparse grid quadrature rule for spatial dimension 2
and level 3 can be defined using a Hermite quadrature rule:
<ul>
<li>
<a href = "../../datasets/sparse_grid_hermite/gh_d2_level3_r.txt">gh_d2_level3_r.txt</a>,
the "R" or "region" file.
</li>
<li>
<a href = "../../datasets/sparse_grid_hermite/gh_d2_level3_w.txt">gh_d2_level3_w.txt</a>,
the "W" or "weight" file.
</li>
<li>
<a href = "../../datasets/sparse_grid_hermite/gh_d2_level3_x.txt">gh_d2_level3_x.txt</a>,
the "X" or "abscissa" file.
</li>
</ul>
</p>
<h3 align = "center">
List of Routines:
</h3>
<p>
<ul>
<li>
<b>CHOOSE</b> computes the binomial coefficient C(N,K).
</li>
<li>
<b>COMP_NEXT</b> computes the compositions of the integer N into K parts.
</li>
<li>
<b>HERMITE_ABSCISSA</b> sets abscissas for multidimensional Gauss-Hermite quadrature.
</li>
<li>
<b>HERMITE_INTEGRAL_ND</b> evaluates a multidimensional Hermite polynomial integral.
</li>
<li>
<b>HERMITE_WEIGHTS</b> returns weights for certain Gauss-Hermite quadrature rules.
</li>
<li>
<b>I4_LOG_2</b> returns the integer part of the logarithm base 2 of an I4.
</li>
<li>
<b>I4_MAX</b> returns the maximum of two I4's.
</li>
<li>
<b>I4_MIN</b> returns the smaller of two I4's.
</li>
<li>
<b>I4_MODP</b> returns the nonnegative remainder of I4 division.
</li>
<li>
<b>I4_POWER</b> returns the value of I^J.
</li>
<li>
<b>I4_TO_STRING</b> converts an I4 to a C++ string.
</li>
<li>
<b>I4VEC_PRODUCT</b> multiplies the entries of an I4VEC.
</li>
<li>
<b>INDEX_LEVEL_HERMITE:</b> determine first level at which given index is generated.
</li>
<li>
<b>LEVEL_TO_ORDER_OPEN</b> converts a level to an order for open rules.
</li>
<li>
<b>MONOMIAL_QUADRATURE_HERMITE</b> applies Hermite quadrature to a monomial.
</li>
<li>
<b>MONOMIAL_VALUE</b> evaluates a monomial.
</li>
<li>
<b>MULTIGRID_INDEX_Z</b> returns an indexed multidimensional grid.
</li>
<li>
<b>PRODUCT_WEIGHT_HERMITE:</b> weights for a product Gauss-Hermite rule.
</li>
<li>
<b>R8_EPSILON</b> returns the R8 roundoff unit.
</li>
<li>
<b>R8_FACTORIAL2</b> computes the double factorial function N!!
</li>
<li>
<b>R8_HUGE</b> returns a "huge" R8.
</li>
<li>
<b>R8MAT_WRITE</b> writes an R8MAT file.
</li>
<li>
<b>R8VEC_DIRECT_PRODUCT2</b> creates a direct product of R8VEC's.
</li>
<li>
<b>S_LEN_TRIM</b> returns the length of a string to the last nonblank.
</li>
<li>
<b>SPARSE_GRID_HERMITE</b> computes a sparse grid of Gauss-Hermite points.
</li>
<li>
<b>SPARSE_GRID_HERMITE_INDEX</b> indexes points in a Gauss-Hermite sparse grid.
</li>
<li>
<b>SPARSE_GRID_HERMITE_SIZE</b> sizes a sparse grid of Gauss-Hermite points.
</li>
<li>
<b>TIMESTAMP</b> prints the current YMDHMS date as a time stamp.
</li>
<li>
<b>VEC_COLEX_NEXT2</b> generates vectors in colex order.
</li>
</ul>
</p>
<p>
You can go up one level to <a href = "../cpp_src.html">
the C++ source codes</a>.
</p>
<hr>
<i>
Last revised on 08 November 2009.
</i>
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