sparse_grid_hermite, an Octave 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 |
The computer code and data files made available on this web page are distributed under the MIT license
sparse_grid_hermite is available in a C++ version and a Fortran90 version and a MATLAB version and an Octave version.
sparse_grid_cc, an Octave code which computes sparse grids based on a Clenshaw-Curtis rule.
sparse_grid_gl, an Octave code 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, an Octave code 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.
spquad, an Octave code 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, an Octave code which uses sparse grids to carry out multilinear hierarchical interpolation. It is commonly known as spinterp(), and is by Andreas Klimke.
-