# sparse_grid_hermite

sparse_grid_hermite, a FORTRAN90 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 is constructed by summing a set of product rules. The weak nesting of the Gauss-Hermite rules means that a few of the points will be repeated across several sparse grids, and so an efficient code will try to group those points together and sum their weights.

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:12345
LEVEL_MAX
011111
1357911
2721375781
31573159289471
43122159712652341
5636092031496910363
6127157363971794541913

A version of the sparse grid library is available in http://tasmanian.ornl.gov, the TASMANIAN library, available from Oak Ridge National Laboratory.

### Languages

sparse_grid_hermite is available in a C++ version and a FORTRAN90 version and a MATLAB version.

### Related Data and Programs:

QUADRATURE_RULES, a dataset directory which defines quadrature rules; a number of examples of sparse grid quadrature rules are included.

quad_rule, a FORTRAN90 library which defines quadrature rules for various intervals and weight functions.

SGMGA, a FORTRAN90 library which creates sparse grids based on a mixture of 1D quadrature rules, allowing anisotropic weights for each dimension.

SPARSE_GRID_CC, a FORTRAN90 library which creates sparse grids based on Clenshaw-Curtis rules.

SPARSE_GRID_F2, a dataset directory which contains the abscissas of sparse grids based on a Fejer Type 2 rule.

SPARSE_GRID_GL, a FORTRAN90 library which computes a sparse grid based on 1D Gauss-Legendre rules.

SPARSE_GRID_GP, a dataset directory which contains the abscissas of sparse grids based on a Gauss Patterson rule.

SPARSE_GRID_HERMITE, a dataset directory which contains the abscissas of sparse grids based on a Gauss Hermite rule.

SPARSE_GRID_HW, a FORTRAN90 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_LAGUERRE, a FORTRAN90 library which creates sparse grids based on Gauss-Laguerre rules.

SPARSE_GRID_MIXED, a FORTRAN90 library which constructs a sparse grid using different rules in each spatial dimension.

### Reference:

1. Volker Barthelmann, Erich Novak, Klaus Ritter,
High Dimensional Polynomial Interpolation on Sparse Grids,
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,
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,