SPARSE_GRID_LAGUERRE
Sparse Grids Based on Gauss-Laguerre Rules

SPARSE_GRID_LAGUERRE is a MATLAB library which constructs sparse grids based on 1D Gauss-Laguerre rules.

Sparse grids are more naturally constructed from a nested family of quadrature rules. Gauss-Laguerre 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-Laguerre rules and the other a nested rule, then the Gauss-Laguerre 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-Laguerre rules are not nested. A sparse grid constructed from Gauss-Laguerre rules will thus generally have more abscissas than a grid built of nested rules..

Here is a table showing the number of points in a sparse grid based on Gauss-Laguerre 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	7	10	13	16	19
2	7	29	58	95	141	196
3	15	95	255	515	906	1456
4	31	273	945	2309	4746	8722
5	63	723	3120	9065	21503	44758
6	127	1813	9484	32259	87358	204203

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

Related Data and Programs:

CC_DISPLAY, a MATLAB library which can compute and display Clenshaw Curtis grids in two dimensions, as well as sparse grids formed from sums of Clenshaw Curtis grids.

GRID_DISPLAY, a MATLAB library which can display a 2D or 3D grid or sparse grid.

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

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

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 dataset directory which contains the abscissas of 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 which contains the abscissas of sparse grids based on a Fejer Type 2 rule.

SPARSE_GRID_GL, a MATLAB library which computes a sparse grid based on 1D Gauss-Legendre rules.

SPARSE_GRID_GP, a dataset directory which contains the abscissas of sparse grids based on a Gauss Patterson 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_LAGUERRE, a dataset directory which contains the abscissas of sparse grids based on a Gauss-Laguerre rule.

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

SPARSE_GRID_NCC, a dataset directory which contains the abscissas of sparse grids based on a Newton Cotes closed rule.

SPARSE_GRID_NCO, a dataset directory which contains the abscissas of sparse grids based on a Newton Cotes open rule.

SPARSE_GRID_OPEN, a MATLAB library which define define 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.
• 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.
• laguerre_abscissa.m returns Gauss-Laguerre abscissas.
• laguerre_integral_nd.m returns the exact value of a multidimensional Laguerre integral.
• laguerre_weights.m returns Gauss-Laguerre weights.
• level_to_order_open.m converts a level to an order for open rules.
• monomial_quadrature.m applies a quadrature rule to a monomial.
• monomial_value.m evaluates a monomial.
• multigrid_index_one.m returns an indexed multidimensional grid.
• product_weight_laguerre.m computes weights for a Gauss-Laguerre product rule.
• r8_factorial.m returns the factorial function N!.
• r8_huge.m returns a "huge" R8.
• r8mat_write.m writes an R8MAT file.
• r8vec_direct_product2.m direct product of R8VEC's.
• sparse_grid_laguerre.m computes a sparse grid of Gauss-Laguerre points.
• sparse_grid_laguerre_index.m indexes the points forming a sparse grid.
• sparse_grid_laguerre_size.m sizes a sparse grid of Gauss-Laguerre 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_laguerre_test.m, runs all the tests.
• sparse_grid_laguerre_test01.m, tests SPARSE_GRID_LAGUERRE_SIZE.
• sparse_grid_laguerre_test02.m, tests SPARSE_GRID_LAGUERRE_INDEX.
• sparse_grid_laguerre_test03.m, tests SPARSE_GRID_LAGUERRE.
• sparse_grid_laguerre_test04.m, tests computes the weights and sums them.
• sparse_grid_laguerre_test05.m, tests a Gauss-Laguerre sparse grid rule for exactness on monomials.
• sparse_grid_laguerre_test06.m, writes a Gauss-Laguerre sparse grid rule to three files.
• sparse_grid_laguerre_test_output.txt, the output from a run of the sample program.

The test program creates some data files:

• lg_d2_level3_r.txt, the "R" file for a sparse grid quadrature rule for spatial dimension 2 and level 3.
• lg_d2_level3_w.txt, the "W" file for a sparse grid quadrature rule for spatial dimension 2 and level 3.
• lg_d2_level3_x.txt, the "X" file for a sparse grid quadrature rule for spatial dimension 2 and level 3.

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