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<html>
<head>
<title>
SANDIA_SGMG - Sparse Grids, Mixed Factors, Growth Rules
</title>
</head>
<body bgcolor="#EEEEEE" link="#CC0000" alink="#FF3300" vlink="#000055">
<h1 align = "center">
SANDIA_SGMG <br> Sparse Grids, Mixed Factors, Growth Rules
</h1>
<hr>
<p>
<b>SANDIA_SGMG</b>
is a C++ library which
constructs a sparse grid whose factors are
possibly distinct 1D quadrature rules.
</p>
<p>
<b>SANDIA_SGMG</b> is a variant of the
SPARSE_GRID_MIXED_GROWTH library. It has the same capabilities as
that library, but it uses a different interface to the SANDIA_RULES routines
which compute points and weights of quadrature rules. This interface is
supplied by a small number of functions in the SANDIA_RULES2 library.
</p>
<p>
Instead of passing
the parameters needed by some of those rules as function arguments, these
values are made available indirectly. This library implements that
indirect storage using a function called <b>parameter</b>, which must
therefore be supplied by the user as part of every program that calls
the library. Refer to the source code of the example programs to see
how <b>parameter</b> is defined and used.
</p>
<p>
For each spatial dimension, the user controls the routines that determine
the order of the 1D quadrature rule, and that evaluate
the quadrature points and weights. This information is supplied in arrays
of pointers to functions. Many common functions for determining the
order, points and weights are available in the SANDIA_RULES library.
</p>
<p>
The 1D quadrature rules are designed to approximate an integral
of the form:
<blockquote><b>
Integral ( A < X < B ) F(X) W(X) dX
</b></blockquote>
where <b>W(X)</b> is a weight function, by the quadrature sum:
<blockquote><b>
Sum ( 1 <= I <= ORDER) F(X(I)) * W(I)
</b></blockquote>
where the set of <b>X</b> values are known as <i>abscissas</i> and
the set of <b>W</b> values are known as <i>weights</i>.
</p>
<p>
Note that the letter <b>W</b>, unfortunately, is used to denote both the
weight function in the original integral, and the vector of weight
values in the quadrature sum.
</p>
<p>
<table border=1>
<tr>
<th>Name</th>
<th>Abbreviation</th>
<th>Interval</th>
<th>Weight function</th>
</tr>
<tr>
<td>Clenshaw-Curtis</td>
<td>CC</td>
<td>[-1,+1]</td>
<td>1</td>
</tr>
<tr>
<td>Fejer Type 2</td>
<td>F2</td>
<td>[-1,+1]</td>
<td>1</td>
</tr>
<tr>
<td>Gauss Patterson</td>
<td>GP</td>
<td>[-1,+1]</td>
<td>1</td>
</tr>
<tr>
<td>Gauss-Legendre</td>
<td>GL</td>
<td>[-1,+1]</td>
<td>1</td>
</tr>
<tr>
<td>Gauss-Hermite</td>
<td>GH</td>
<td>(-oo,+oo)</td>
<td>e<sup>-x*x</sup></td>
</tr>
<tr>
<td>Generalized Gauss-Hermite</td>
<td>GGH</td>
<td>(-oo,+oo)</td>
<td>|x|<sup>alpha</sup> e<sup>-x*x</sup></td>
</tr>
<tr>
<td>Gauss-Laguerre</td>
<td>LG</td>
<td>[0,+oo)</td>
<td>e<sup>-x</sup></td>
</tr>
<tr>
<td>Generalized Gauss-Laguerre</td>
<td>GLG</td>
<td>[0,+oo)</td>
<td>x<sup>alpha</sup> e<sup>-x</sup></td>
</tr>
<tr>
<td>Gauss-Jacobi</td>
<td>GJ</td>
<td>[-1,+1]</td>
<td>(1-x)<sup>alpha</sup> (1+x)<sup>beta</sup></td>
</tr>
<tr>
<td>Hermite Genz-Keister</td>
<td>HGK</td>
<td>(-oo,+oo)</td>
<td>e<sup>-x*x</sup></td>
</tr>
</table>
</p>
<p>
A sparse grid is a quadrature rule for a multidimensional integral.
It is formed by taking a certain linear combination of lower-order
product rules. The product rules, in turn, are formed as direct products
of 1D quadrature rules. It is common to form a sparse grid in which the
1D component quadrature rules are the same. This package, however,
is intended to produce sparse grids based on sums of product rules for
which the rule chosen for each spatial dimension may be freely chosen
from the set listed above.
</p>
<p>
These sparse grids are still indexed by a number known as <b>level</b>,
and assembled as a sum of low order product rules. As the value of
<b>level</b> increases, the point growth becomes more complicated.
This is because the 1D rules have somewhat varying point growth patterns
to begin with, and the varying levels of nestedness have a dramatic
effect on the sparsity of the total grid.
</p>
<p>
Since a sparse grid is made up of a combination of product grids,
it is frequently the case that many of the product grids include the
same point. For efficiency, it is usually desirable to merge or
consolidate such duplicate points when expressing the resulting
sparse grid rule. It is possible to "logically" determine when
a duplicate point will be generated; however, this logic changes
depending on the specific 1-dimensional rules being used, and the
tests can become quite elaborate. Moreover, for rules which include
real parameters, the determination of duplication can require
a numerical tolerance.
</p>
<p>
In order to simplify the matter of the detection of duplicate points,
the codes presented begin by generating the entire "naive" set of
points. Then a user-specified tolerance <b>TOL</b> is used to determine when
two points are equal. If the maximum difference between any two components
is less than or equal to <b>TOL</b>, the points are declared to be equal.
</p>
<p>
A reasonable value for <b>TOL</b> might be the square root of the machine
precision. Setting <b>TOL</b> to zero means that only points which are
identical to the last significant digit are taken to be duplicates. Setting
<b>TOL</b> to a negative value means that no duplicate points will be eliminated -
in other words, this choice produces the full or "naive" grid.
</p>
<p>
A scalar quantity GROWTH selects the rule order O for level L
from the three growth options for each 1D rule. In the case of exponentially
growing rules, the slow and moderate growth rules choose O indirectly,
by imposing a requirement on P, the degree of precision of the rule.
<table border=1>
<tr>
<th>Value</th>
<th>Name</th>
<th>Meaning</th>
</tr>
<tr>
<td>0</td>
<td>Slow</td>
<td>O=L+1 for linear rules, P=2*L+1 for exponential</td>
</tr>
<tr>
<td>1</td>
<td>Moderate</td>
<td>O=2*L+1 for linear rules, P=4*L+1 for exponential</td>
</tr>
<tr>
<td>2</td>
<td>Full</td>
<td>O=2*L+1 for linear rules, O = next rule in sequence</td>
</tr>
</table>
</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>SANDIA_SGMG</b> is available in
<a href = "../../cpp_src/sandia_sgmg/sandia_sgmg.html">a C++ version</a>.
</p>
<h3 align = "center">
Related Data and Programs:
</h3>
<p>
<a href = "../../cpp_src/nint_exactness_mixed/nint_exactness_mixed.html">
NINT_EXACTNESS_MIXED</a>,
a C++ program which
measures the polynomial exactness of a multidimensional quadrature rule
based on a mixture of 1D quadrature rule factors.
</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/sandia_rules/sandia_rules.html">
SANDIA_RULES</a>,
a C++ library which
produces 1D quadrature rules of
Chebyshev, Clenshaw Curtis, Fejer 2, Gegenbauer, generalized Hermite,
generalized Laguerre, Hermite, Jacobi, Laguerre, Legendre and Patterson types.
</p>
<p>
<a href = "../../cpp_src/sandia_rules2/sandia_rules2.html">
SANDIA_RULES2</a>,
a C++ library which
contains a very small selection of functions which serve as an interface
between SANDIA_SGMG or SANDIA_SGMGA and SANDIA_RULES.
</p>
<p>
<a href = "../../cpp_src/sandia_sparse/sandia_sparse.html">
SANDIA_SPARSE</a>,
a C++ library which
computes the points and weights of a Smolyak sparse
grid, based on a variety of 1-dimensional quadrature rules.
</p>
<p>
<a href = "../../datasets/sgmg/sgmg.html">
SGMG</a>,
a dataset directory which
contains multidimensional Smolyak sparse grids
based on a mixed set of 1D factor rules and a choice of growth rules.
</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,
by Knut Petras.
</p>
<p>
<a href = "../../m_src/sparse_grid_display/sparse_grid_display.html">
SPARSE_GRID_DISPLAY</a>,
a MATLAB program which
can display a 2D or 3D sparse grid.
</p>
<p>
<a href = "../../cpp_src/sparse_grid_mixed/sparse_grid_mixed.html">
SPARSE_GRID_MIXED</a>,
a C++ library which
creates sparse grids based on a mix of 1D 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>
Milton Abramowitz, Irene Stegun,<br>
Handbook of Mathematical Functions,<br>
National Bureau of Standards, 1964,<br>
ISBN: 0-486-61272-4,<br>
LC: QA47.A34.
</li>
<li>
Charles Clenshaw, Alan Curtis,<br>
A Method for Numerical Integration on an Automatic Computer,<br>
Numerische Mathematik,<br>
Volume 2, Number 1, December 1960, pages 197-205.
</li>
<li>
Philip Davis, Philip Rabinowitz,<br>
Methods of Numerical Integration,<br>
Second Edition,<br>
Dover, 2007,<br>
ISBN: 0486453391,<br>
LC: QA299.3.D28.
</li>
<li>
Michael Eldred, John Burkardt,<br>
Comparison of Non-Intrusive Polynomial Chaos and Stochastic
Collocation Methods for Uncertainty Quantification,<br>
American Institute of Aeronautics and Astronautics,<br>
Paper 2009-0976,<br>
47th AIAA Aerospace Sciences Meeting Including The New Horizons Forum and Aerospace Exposition,<br>
5 - 8 January 2009, Orlando, Florida.
</li>
<li>
Walter Gautschi,<br>
Numerical Quadrature in the Presence of a Singularity,<br>
SIAM Journal on Numerical Analysis,<br>
Volume 4, Number 3, September 1967, pages 357-362.
</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>
Gene Golub, John Welsch,<br>
Calculation of Gaussian Quadrature Rules,<br>
Mathematics of Computation,<br>
Volume 23, Number 106, April 1969, pages 221-230.
</li>
<li>
Prem Kythe, Michael Schaeferkotter,<br>
Handbook of Computational Methods for Integration,<br>
Chapman and Hall, 2004,<br>
ISBN: 1-58488-428-2,<br>
LC: QA299.3.K98.
</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>
Thomas Patterson,<br>
The Optimal Addition of Points to Quadrature Formulae,<br>
Mathematics of Computation,<br>
Volume 22, Number 104, October 1968, pages 847-856.
</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>
Arthur Stroud, Don Secrest,<br>
Gaussian Quadrature Formulas,<br>
Prentice Hall, 1966,<br>
LC: QA299.4G3S7.
</li>
<li>
Joerg Waldvogel,<br>
Fast Construction of the Fejer and Clenshaw-Curtis
Quadrature Rules,<br>
BIT Numerical Mathematics,<br>
Volume 43, Number 1, 2003, pages 1-18.
</li>
</ol>
</p>
<h3 align = "center">
Source Code:
</h3>
<p>
<ul>
<li>
<a href = "sandia_sgmg.cpp">sandia_sgmg.cpp</a>, the source code.
</li>
<li>
<a href = "sandia_sgmg.hpp">sandia_sgmg.hpp</a>, the include file.
</li>
<li>
<a href = "sandia_sgmg.sh">sandia_sgmg.sh</a>,
commands to compile the source code.
</li>
</ul>
</p>
<h3 align = "center">
Examples and Tests:
</h3>
<p>
<ul>
<li>
<a href = "sandia_sgmg_prb.cpp">sandia_sgmg_prb.cpp</a>,
a sample calling program.
</li>
<li>
<a href = "sandia_sgmg_prb.sh">sandia_sgmg_prb.sh</a>,
commands to compile and run the sample program.
</li>
</ul>
</p>
<p>
<b>COMP_NEXT_PRB</b> tests COMP_NEXT.
<ul>
<li>
<a href = "comp_next_prb.cpp">comp_next_prb.cpp</a>,
a sample calling program.
</li>
<li>
<a href = "comp_next_prb.sh">comp_next_prb.sh</a>,
commands to compile and run the sample program.
</li>
<li>
<a href = "comp_next_prb_output.txt">
comp_next_prb_output.txt</a>,
the output file.
</li>
</ul>
</p>
<p>
<b>PRODUCT_MIXED_GROWTH_WEIGHT_PRB</b> tests PRODUCT_MIXED_GROWTH_WEIGHT.
<ul>
<li>
<a href = "product_mixed_growth_weight_prb.cpp">product_mixed_growth_weight_prb.cpp</a>,
a sample calling program.
</li>
<li>
<a href = "product_mixed_growth_weight_prb.sh">product_mixed_growth_weight_prb.sh</a>,
commands to compile and run the sample program.
</li>
<li>
<a href = "product_mixed_growth_weight_prb_output.txt">
product_mixed_growth_weight_prb_output.txt</a>,
the output file.
</li>
</ul>
</p>
<p>
<b>SANDIA_SGMG_INDEX_PRB</b> tests SANDIA_SGMG_INDEX.
<ul>
<li>
<a href = "sandia_sgmg_index_prb.cpp">sandia_sgmg_index_prb.cpp</a>,
tests SANDIA_SGMG_INDEX.
</li>
<li>
<a href = "sandia_sgmg_index_prb.sh">sandia_sgmg_index_prb.sh</a>,
commands to compile and run the sample program.
</li>
<li>
<a href = "sandia_sgmg_index_prb_output.txt">
sandia_sgmg_index_prb_output.txt</a>,
the output file.
</li>
</ul>
</p>
<p>
<b>SANDIA_SGMG_POINT_PRB</b> tests SANDIA_SGMG_POINT.
<ul>
<li>
<a href = "sandia_sgmg_point_prb.cpp">sandia_sgmg_point_prb.cpp</a>,
tests SANDIA_SGMG_POINT.
</li>
<li>
<a href = "sandia_sgmg_point_prb.sh">sandia_sgmg_point_prb.sh</a>,
commands to compile and run the sample program.
</li>
<li>
<a href = "sandia_sgmg_point_prb_output.txt">
sandia_sgmg_point_prb_output.txt</a>,
the output file.
</li>
</ul>
</p>
<p>
<b>SANDIA_SGMG_SIZE_PRB</b> tests SANDIA_SGMG_SIZE and
SANDIA_SGMG_SIZE_TOTAL.
<ul>
<li>
<a href = "sandia_sgmg_size_prb.cpp">sandia_sgmg_size_prb.cpp</a>,
the test program.
</li>
<li>
<a href = "sandia_sgmg_size_prb.sh">sandia_sgmg_size_prb.sh</a>,
commands to compile and run the sample program.
</li>
<li>
<a href = "sandia_sgmg_size_prb_output.txt">
sandia_sgmg_size_prb_output.txt</a>,
the output file.
</li>
</ul>
</p>
<p>
<b>SANDIA_SGMG_SIZE_TABLE</b> makes a point count table.
<ul>
<li>
<a href = "sandia_sgmg_size_table.cpp">sandia_sgmg_size_table.cpp</a>,
the test program.
</li>
<li>
<a href = "sandia_sgmg_size_table.sh">sandia_sgmg_size_table.sh</a>,
commands to compile and run the sample program.
</li>
<li>
<a href = "sandia_sgmg_size_table_output.txt">
sandia_sgmg_size_table_output.txt</a>,
the output file.
</li>
</ul>
</p>
<p>
<b>SANDIA_SGMG_UNIQUE_INDEX_PRB</b> tests SANDIA_SGMG_UNIQUE_INDEX.
<ul>
<li>
<a href = "sandia_sgmg_unique_index_prb.cpp">sandia_sgmg_unique_index_prb.cpp</a>,
the test program.
</li>
<li>
<a href = "sandia_sgmg_unique_index_prb.sh">sandia_sgmg_unique_index_prb.sh</a>,
commands to compile and run the sample program.
</li>
<li>
<a href = "sandia_sgmg_unique_index_prb_output.txt">
sandia_sgmg_unique_index_prb_output.txt</a>,
the output file.
</li>
</ul>
</p>
<p>
<b>SANDIA_SGMG_WEIGHT_PRB</b> tests SANDIA_SGMG_WEIGHT.
<ul>
<li>
<a href = "sandia_sgmg_weight_prb.cpp">sandia_sgmg_weight_prb.cpp</a>,
the test program.
</li>
<li>
<a href = "sandia_sgmg_weight_prb.sh">sandia_sgmg_weight_prb.sh</a>,
commands to compile and run the sample program.
</li>
<li>
<a href = "sandia_sgmg_weight_prb_output.txt">
sandia_sgmg_weight_prb_output.txt</a>,
the output file.
</li>
</ul>
</p>
<h3 align = "center">
List of Routines:
</h3>
<p>
<ul>
<li>
<b>PRODUCT_MIXED_GROWTH_WEIGHT</b> computes the weights of a mixed product rule.
</li>
<li>
<b>SANDIA_SGMG_INDEX</b> indexes a sparse grid of mixed 1D rules.
</li>
<li>
<b>SANDIA_SGMG_POINT</b> computes the points of a sparse grid rule.
</li>
<li>
<b>SANDIA_SGMG_SIZE</b> sizes a sparse grid, discounting duplicates.
</li>
<li>
<b>SANDIA_SGMG_SIZE_TOTAL</b> sizes a sparse grid, counting duplicates.
</li>
<li>
<b>SANDIA_SGMG_UNIQUE_INDEX</b> maps nonunique to unique points.
</li>
<li>
<b>SANDIA_SGMG_WEIGHT:</b> sparse grid weights for a mix of 1D rules.
</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 01 January 2012.
</i>
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