# sparse_grid_hermite

sparse_grid_hermite, a MATLAB 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:123456
LEVEL_MAX
0111111
135791113
2722375781109
31575161289471713
431224608126823413953
563613207049941036719397
612715706507180764195786522

### Languages:

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

### Related Data and Programs:

sparse_grid_cc, a MATLAB code which computes sparse grids based on a clenshaw-curtis rule.

sparse_grid_composite, a MATLAB code which creates sparse grids based on 1d composite rules (currently only of order 1).

sparse_grid_gl, a MATLAB 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, a MATLAB 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, a MATLAB 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, a MATLAB code which uses sparse grids to carry out multilinear hierarchical interpolation. it is commonly known as spinterp, and is by andreas klimke.

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### 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,