SPARSE_GRID_HERMITE
Sparse Grids Based on Gauss-Hermite Rules

SPARSE_GRID_HERMITE is a C++ library which constructs sparse grids based on 1D Gauss-Hermite rules.

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!

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.

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.

DIM:	1	2	3	4	5	6
LEVEL_MAX
0	1	1	1	1	1	1
1	3	5	7	9	11	13
2	7	22	37	57	81	109
3	15	75	161	289	471	713
4	31	224	608	1268	2341	3953
5	63	613	2070	4994	10367	19397
6	127	1570	6507	18076	41957	86522

Web Link:

A version of the sparse grid library is available in http://tasmanian.ornl.gov, the TASMANIAN library, available from Oak Ridge National Laboratory.

Licensing:

The code described and made available on this web page is distributed under the GNU LGPL license.

Languages:

SPARSE_GRID_HERMITE is available in a C++ version and a FORTRAN90 version and a MATLAB version.

Related Data and Programs:

CC_DISPLAY, 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.

QUADRATURE_RULES, a dataset directory which define quadrature rules; a number of examples of sparse grid quadrature rules are included.

QUADRULE, a C++ library which defines quadrature rules for various intervals and weight functions.

SGMG, 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.

SGMGA, a C++ library which creates sparse grids based on a mixture of 1D quadrature rules, allowing anisotropic weights for each dimension.

SMOLPACK, a C library which implements Novak and Ritter's method for estimating the integral of a function over a multidimensional hypercube using sparse grids.

SPARSE_GRID_CC, a dataset directory which contains the abscissas of sparse grids based on a Clenshaw Curtis rule.

SPARSE_GRID_F2, a dataset directory which contains the abscissas of sparse grids based on a Fejer Type 2 rule.

SPARSE_GRID_GL, a C++ library which computes a sparse grid based on 1D Gauss-Legendre rules..

SPARSE_GRID_GP, a dataset directory which contains the abscissas of sparse grids based on a Gauss Patterson rule.

SPARSE_GRID_HERMITE, a dataset directory which contains the abscissas of sparse grids based on a Gauss Hermite rule.

SPARSE_GRID_MIXED, a C++ library which constructs a sparse grid using different rules in each spatial dimension.

SPARSE_GRID_NCC, a dataset directory which contains the abscissas of sparse grids based on a Newton Cotes closed rule.

SPARSE_GRID_NCO, a dataset directory which contains the abscissas of sparse grids based on a Newton Cotes open rule.

SPARSE_GRID_OPEN, a C++ library which defines define sparse grids based on open nested quadrature rules.

TOMS847, a MATLAB program which uses sparse grids to carry out multilinear hierarchical interpolation. It is commonly known as SPINTERP, and is by Andreas Klimke.

Reference:

1. Volker Barthelmann, Erich Novak, Klaus Ritter,
   High Dimensional Polynomial Interpolation on Sparse Grids,
   Advances in Computational Mathematics,
   Volume 12, Number 4, 2000, pages 273-288.
2. Thomas Gerstner, Michael Griebel,
   Numerical Integration Using Sparse Grids,
   Numerical Algorithms,
   Volume 18, Number 3-4, 1998, pages 209-232.
3. Albert Nijenhuis, Herbert Wilf,
   Combinatorial Algorithms for Computers and Calculators,
   Second Edition,
   Academic Press, 1978,
   ISBN: 0-12-519260-6,
   LC: QA164.N54.
4. Fabio Nobile, Raul Tempone, Clayton Webster,
   A Sparse Grid Stochastic Collocation Method for Partial Differential Equations with Random Input Data,
   SIAM Journal on Numerical Analysis,
   Volume 46, Number 5, 2008, pages 2309-2345.
5. Sergey Smolyak,
   Quadrature and Interpolation Formulas for Tensor Products of Certain Classes of Functions,
   Doklady Akademii Nauk SSSR,
   Volume 4, 1963, pages 240-243.
6. Dennis Stanton, Dennis White,
   Constructive Combinatorics,
   Springer, 1986,
   ISBN: 0387963472,
   LC: QA164.S79.

Source Code:

• sparse_grid_hermite.cpp, the source code.
• sparse_grid_hermite.hpp, the include file.

Examples and Tests:

• sparse_grid_hermite_test.cpp, a sample calling program.
• sparse_grid_hermite_test.txt, the output file.

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:

• gh_d2_level3_r.txt, the "R" or "region" file.
• gh_d2_level3_w.txt, the "W" or "weight" file.
• gh_d2_level3_x.txt, the "X" or "abscissa" file.

List of Routines:

• CHOOSE computes the binomial coefficient C(N,K).
• COMP_NEXT computes the compositions of the integer N into K parts.
• HERMITE_ABSCISSA sets abscissas for multidimensional Gauss-Hermite quadrature.
• HERMITE_INTEGRAL_ND evaluates a multidimensional Hermite polynomial integral.
• HERMITE_WEIGHTS returns weights for certain Gauss-Hermite quadrature rules.
• I4_LOG_2 returns the integer part of the logarithm base 2 of an I4.
• I4_MAX returns the maximum of two I4's.
• I4_MIN returns the smaller of two I4's.
• I4_MODP returns the nonnegative remainder of I4 division.
• I4_POWER returns the value of I^J.
• I4_TO_STRING converts an I4 to a C++ string.
• I4VEC_PRODUCT multiplies the entries of an I4VEC.
• INDEX_LEVEL_HERMITE: determine first level at which given index is generated.
• LEVEL_TO_ORDER_OPEN converts a level to an order for open rules.
• MONOMIAL_QUADRATURE_HERMITE applies Hermite quadrature to a monomial.
• MONOMIAL_VALUE evaluates a monomial.
• MULTIGRID_INDEX_Z returns an indexed multidimensional grid.
• PRODUCT_WEIGHT_HERMITE: weights for a product Gauss-Hermite rule.
• R8_EPSILON returns the R8 roundoff unit.
• R8_FACTORIAL2 computes the double factorial function N!!
• R8_HUGE returns a "huge" R8.
• R8MAT_WRITE writes an R8MAT file.
• R8VEC_DIRECT_PRODUCT2 creates a direct product of R8VEC's.
• S_LEN_TRIM returns the length of a string to the last nonblank.
• SPARSE_GRID_HERMITE computes a sparse grid of Gauss-Hermite points.
• SPARSE_GRID_HERMITE_INDEX indexes points in a Gauss-Hermite sparse grid.
• SPARSE_GRID_HERMITE_SIZE sizes a sparse grid of Gauss-Hermite points.
• TIMESTAMP prints the current YMDHMS date as a time stamp.
• VEC_COLEX_NEXT2 generates vectors in colex order.

You can go up one level to the C++ source codes.