SPARSE_GRID_GL
Sparse Grids Based on Gauss-Legendre Rules


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:12345
LEVEL_MAX     
011111
1357911
2921375781
32373159289471
45322559712652341
51156372031496910363
6241169364051794541913
      
DIM:678910
LEVEL_MAX     
011111
11315171921
2109141177217261
37131023140918792441
43953624593771352518881
519397335595467384931126925
      
DIM:100    
LEVEL_MAX     
01    
1201    
220601    

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 code described and made available on this web page is 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:

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

QUADRULE, a C++ library which defines quadrature rules for various intervals and weight functions.

SPARSE_GRID_CC, a C++ library which define sparse grids based on 1D Clenshaw Curtis quadrature rules.

sparse_grid_gl_test

SPARSE_GRID_HERMITE, a C++ library which creates sparse grids based on Gauss-Hermite rules.

SPARSE_GRID_LAGUERRE, a C++ library which creates sparse grids based on Gauss-Laguerre rules.

SPARSE_GRID_MIXED, a C++ 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,
    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.

Source Code:


Last revised on 15 April 2020.