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:	1	2	3	4	5	6
LEVEL_MAX
0	1	1	1	1	1	1
1	3	5	7	9	11	13
2	9	21	37	57	81	109
3	23	73	159	289	471	713
4	53	225	597	1265	2341	3953
5	115	637	2031	4969	10363	19397
6	241	1693	6405	17945	41913	86517

Licensing:

The computer code and data files described and 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.

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.

Source Code:

• comp_next.m computes the compositions of the integer N into K parts.
• gl_abscissa.m returns Gauss-Legendre abscissas.
• gl_weights.m returns Gauss-Legendre weights.
• i4_choose.m computes the binomial coefficient C(N,K).
• i4_log_2.m returns the logarithm base 2 of an I4.
• i4_modp.m returns the nonnegative remainder of integer division.
• index_level_gl.m determines the first level at which a given index occurs.
• level_to_order_open.m converts a level to an order for open rules.
• monomial_int01.m returns the integral of a monomial over the [0,1] hypercube.
• monomial_quadrature.m applies a quadrature rule to a monomial.
• monomial_value.m evaluates a monomial.
• multigrid_index_z.m returns an indexed multidimensional grid.
• multigrid_scale_closed.m renumbers a grid as a subgrid on a higher level.
• product_weight_gl.m computes weights for a Gauss-Legendre product rule.
• r8mat_write.m writes an R8MAT file.
• r8vec_direct_product2.m direct product of double precision real vectors.
• sparse_grid_gl.m computes a sparse grid of Gauss-Legendre points.
• sparse_grid_gl_index.m indexes the points forming a sparse grid.
• sparse_grid_gl_size.m sizes a sparse grid of Gauss-Legendre points.
• timestamp.m prints the current YMDHMS date as a time stamp.
• vec_colex_next2.m generates vectors in colex order.

Examples and Tests:

• sparse_grid_gl_test.m, runs all the tests.
• sparse_grid_gl_test01.m, tests SPARSE_GRID_GL_SIZE.
• sparse_grid_gl_test02.m, tests SPARSE_GRID_GL_INDEX.
• sparse_grid_gl_test03.m, tests SPARSE_GRID_GL.
• sparse_grid_gl_test04.m, tests computes the weights and sums them.
• sparse_grid_gl_test05.m, tests a GL sparse grid rule for exactness on monomials.
• sparse_grid_gl_test06.m, writes a GL sparse grid rule to three files.
• gl_d2_level3_r.txt, the "R" data for a sparse GL quadrature rule.
• gl_d2_level3_w.txt, the "W" data for a sparse GL quadrature rule.
• gl_d2_level3_x.txt, the "X" data for a sparse GL quadrature rule.
• sparse_grid_gl_test_output.txt, the output from a run of the sample program.

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