# SPARSE_GRID_GL Sparse Grids Based on the Gauss-Legendre Rule

SPARSE_GRID_GL is a MATLAB library 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

### Web Link:

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

### Licensing:

The computer code and data files made available on this web page are distributed under the GNU LGPL license.

### Languages:

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

### Related Data and Programs:

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

SMOLPACK, a C library which implements Novak and Ritter's method for estimating the integral of a function over a multidimensional hypercube using sparse grids.

SPARSE_GRID_CC, a MATLAB lilbrary which computes sparse grids based on a Clenshaw-Curtis rule.

SPARSE_GRID_COMPOSITE, a MATLAB library 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 library which creates sparse grids based on Gauss-Hermite rules.

SPARSE_GRID_HW, a MATLAB 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_MIXED, a MATLAB library which constructs a sparse grid using different rules in each spatial dimension.

SPARSE_GRID_OPEN, a MATLAB library which defines sparse grids based on open nested quadrature rules.

SPQUAD, a MATLAB library 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 program 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,
Advances in Computational Mathematics,
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,
Academic Press, 1978,
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,
Doklady Akademii Nauk SSSR,
Volume 4, 1963, pages 240-243.
6. Dennis Stanton, Dennis White,
Constructive Combinatorics,
Springer, 1986,
ISBN: 0387963472,
LC: QA164.S79.

### Examples and Tests:

You can go up one level to the MATLAB source codes.

Last revised on 02 October 2007.