sparse_grid_gl, a C++ code which constructs sparse grids based on 1D Gauss-Legendre rules.
Sparse grids are more naturally constructed from a nested family of quadrature rules. Gauss-Legendre rules are not nested, but have higher accuracy. Thus, there is a tradeoff. If we compare two sparse grids of the same "level", one using Gauss-Legendre rules and the other, say, Clenshaw-Curtis rules, then the Gauss-Legendre 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-Legendre 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-Legendre 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-Legendre 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 |
|---|---|---|---|---|---|
| LEVEL_MAX | |||||
| 0 | 1 | 1 | 1 | 1 | 1 |
| 1 | 3 | 5 | 7 | 9 | 11 |
| 2 | 9 | 21 | 37 | 57 | 81 |
| 3 | 23 | 73 | 159 | 289 | 471 |
| 4 | 53 | 225 | 597 | 1265 | 2341 |
| 5 | 115 | 637 | 2031 | 4969 | 10363 |
| 6 | 241 | 1693 | 6405 | 17945 | 41913 |
| DIM: | 6 | 7 | 8 | 9 | 10 |
| LEVEL_MAX | |||||
| 0 | 1 | 1 | 1 | 1 | 1 |
| 1 | 13 | 15 | 17 | 19 | 21 |
| 2 | 109 | 141 | 177 | 217 | 261 |
| 3 | 713 | 1023 | 1409 | 1879 | 2441 |
| 4 | 3953 | 6245 | 9377 | 13525 | 18881 |
| 5 | 19397 | 33559 | 54673 | 84931 | 126925 |
| DIM: | 100 | ||||
| LEVEL_MAX | |||||
| 0 | 1 | ||||
| 1 | 201 | ||||
| 2 | 20601 |
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_gl is available in a C++ version and a Fortran90 version and a MATLAB version.
quadrature_rules, a dataset directory which contains quadrature rules; a number of examples of sparse grid quadrature rules are included.
quad_rule, a C++ code which defines quadrature rules for various intervals and weight functions.
sparse_grid_cc, a C++ code which define sparse grids based on 1D Clenshaw Curtis quadrature rules.
sparse_grid_hermite, a C++ code which creates sparse grids based on Gauss-Hermite rules.
sparse_grid_laguerre, a C++ code which creates sparse grids based on Gauss-Laguerre rules.
sparse_grid_mixed, a C++ code which constructs a sparse grid using different rules in each spatial dimension.