SPARSE_GRID_HERMITE
Sparse Grids Based on Gauss-Hermite Rules

SPARSE_GRID_HERMITE is a MATLAB 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

Licensing:

The computer code and data files described and made available on this web page are 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:

SGMGA, a MATLAB 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 MATLAB library which computes sparse grids based on a Clenshaw-Curtis rule.

SPARSE_GRID_COMPOSITE, a MATLAB library which creates sparse grids based on 1D composite rules (currently only of order 1).

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

SPARSE_GRID_HERMITE, a dataset directory which contains sparse grids based on a Gauss-Hermite rule.

SPARSE_GRID_HW, a MATLAB library which creates sparse grids based on Gauss-Legendre, Gauss-Hermite, Gauss-Patterson, or a nested variation of Gauss-Hermite rules, by Florian Heiss and Viktor Winschel.

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

SPARSE_GRID_OPEN, a MATLAB library which defines 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:

• comp_next.m computes the compositions of the integer N into K parts.
• hermite_abscissa.m returns Gauss-Hermite abscissas.
• hermite_integral_nd.m returns the exact value of a multidimensional Hermite integral.
• hermite_weights.m returns Gauss-Hermite weights.
• i4_choose.m computes the binomial coefficient C(N,K).
• i4_log_2.m returns the logarithm base 2 of an I4.
• i4_modp.m returns the nonnegative remainder of integer division.
• index_level_herm.m determines the first level at which a given index occurs.
• level_to_order_open.m converts a level to an order for open rules.
• monomial_quadrature.m applies a quadrature rule to a monomial.
• monomial_value.m evaluates a monomial.
• multigrid_index_z.m returns an indexed multidimensional grid.
• product_weight_herm.m computes weights for a Gauss-Hermite product rule.
• r8_factorial2.m returns the double factorial function N!!.
• r8_huge.m returns a "huge" R8.
• r8mat_write.m writes an R8MAT file.
• r8vec_direct_product2.m direct product of R8VEC's.
• sparse_grid_herm.m computes a sparse grid of Gauss-Hermite points.
• sparse_grid_herm_index.m indexes the points forming a sparse grid.
• sparse_grid_herm_size.m sizes a sparse grid of Gauss-Hermite points.
• timestamp.m prints the current YMDHMS date as a time stamp.
• vec_colex_next2.m generates vectors in colex order.

Examples and Tests:

• sparse_grid_hermite_test.m, runs all the tests.
• sparse_grid_hermite_test01.m, tests SPARSE_GRID_HERM_SIZE.
• sparse_grid_hermite_test02.m, tests SPARSE_GRID_HERM_INDEX.
• sparse_grid_hermite_test03.m, tests SPARSE_GRID_HERM.
• sparse_grid_hermite_test04.m, tests computes the weights and sums them.
• sparse_grid_hermite_test05.m, tests a Gauss-Hermite sparse grid rule for exactness on monomials.
• sparse_grid_hermite_test06.m, writes a Gauss-Hermite sparse grid rule to three files.
• sparse_grid_hermite_test_output.txt, the output from a run of the sample program.

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.

You can go up one level to the MATLAB source codes.