# sparse_grid_gl

sparse_grid_gl, a MATLAB 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:123456
LEVEL_MAX
0111111
135791113
2921375781109
32373159289471713
453225597126523413953
5115637203149691036319397
624116936405179454191386517

### Languages:

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

### Related Data and Programs:

sparse_grid_cc, a MATLAB lilbrary 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_f2, a dataset directory of sparse grids based on a fejer type 2 rule.

sparse_grid_gl, a dataset directory which contains sparse grids based on a gauss-legendre rule.

sparse_grid_hermite, a MATLAB code which creates sparse grids based on gauss-hermite rules.

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.

### 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,