sparse_grid_hermite, a C++ code 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 https://tasmanian.ornl.gov, the TASMANIAN library, available from Oak Ridge National Laboratory.
The code described and made available on this web page is distributed under the GNU LGPL license.
sparse_grid_hermite is available in a C++ version and a Fortran90 version and a MATLAB version.
sparse_grid_cc, a dataset directory which contains the abscissas of sparse grids based on a Clenshaw Curtis rule.
sparse_grid_gl, a C++ code which computes a sparse grid based on 1D Gauss-Legendre rules.
sparse_grid_mixed, a C++ code which constructs a sparse grid using different rules in each spatial dimension.