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 |
A version of the sparse grid library is available in http://tasmanian.ornl.gov, the TASMANIAN library, available from Oak Ridge National Laboratory.
The computer code and data files made available on this web page are distributed under the GNU LGPL license.
SPARSE_GRID_HERMITE is available in a C++ version and a FORTRAN90 version and a MATLAB version.
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.
SPQUAD, a MATLAB library which computes the points and weights of a sparse grid quadrature rule for a multidimensional integral, based on the Clenshaw-Curtis quadrature rule, by Greg von Winckel.
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.
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:
You can go up one level to the MATLAB source codes.