SPARSE_GRID_GL_DATASET
Create Gauss-Legendre Sparse Grid Dataset

SPARSE_GRID_GL_DATASET is a FORTRAN90 program which creates a Gauss-Legendre sparse grid dataset.

Usage:

sparse_grid_gl_dataset dim_num level_max

• dim_num is the spatial dimension, typically between 2 and 10;
• level_max is the sparse grid level, typically between 0 and 6, which controls the number of points in the grid. The 1D rules used will have order 2^(level_max)+1.

Each sparse grid is stored using the "quadrature rule" format, that is, as three files:

• an "R" or "region" file, which lists two points that bound the region;
• a "W" or "weight" file, which lists the weight for each abscissa;
• an "X" or "abscissa" file, which lists the abscissas of the rule.

The abscissas are ordered to respect the natural nesting of the sparse grids by level. That is, the file of level 3 points begins by listing the points in the level 2 grid.

Licensing:

The code described and made available on this web page is distributed under the GNU LGPL license.

Languages:

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

Related Data and Programs:

MONTE_CARLO_RULE, a FORTRAN90 program which generates a dataset of N random M-dimensional points, regards it as a quadrature rule for the unit hypercube, and writes out three files of information.

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

QUADRULE, a FORTRAN90 library which defines quadrature rules for various intervals and weight functions.

SGMGA, a FORTRAN90 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_DATASET, a FORTRAN90 program which can create a Clenshaw Curtis sparse grid and write the data to three files.

SPARSE_GRID_F2, a dataset directory which contains the abscissas of sparse grids based on a Fejer Type 2 rule.

SPARSE_GRID_GL, a FORTRAN90 library which can create a Gauss-Legendre sparse grid.

SPARSE_GRID_GL, a dataset directory which contains Gauss-Legendre sparse grids.

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

SPARSE_GRID_HERMITE_DATASET, a FORTRAN90 program which can create a Gauss-Hermite sparse grid and write the data to three files.

SPARSE_GRID_HW, a FORTRAN90 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_DATASET, a FORTRAN90 program which can create a Gauss-Laguerre sparse grid and write the data to three files.

SPARSE_GRID_MIXED_DATASET, a FORTRAN90 program which creates a sparse grid dataset based on a mixture of 1D rules.

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_DATASET, a FORTRAN90 program which creates a sparse grid dataset based on open rules (Fejer 2, Gauss-Patterson, Newton-Cotes-Open).

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:

• sparse_grid_gl_dataset.f90, the source code.

Examples and Tests:

Here are the three quadrature files created by the command

sparse_grid_gl_dataset 2 3

• sparse_grid_gl_dataset.txt, the printed output from the program.
• gl_d2_level3_r.txt, the "R" or "region" file.
• gl_d2_level3_w.txt, the "W" or "weight" file.
• gl_d2_level3_x.txt, the "X" or "abscissa" file.

List of Routines:

• MAIN is the main program for SPARSE_GRID_GL_DATASET.
• CHOOSE computes the binomial coefficient C(N,K).
• COMP_NEXT computes the compositions of the integer N into K parts.
• GET_UNIT returns a free FORTRAN unit number.
• GL_ABSCISSA sets abscissas for multidimensional Gauss-Legendre quadrature.
• GL_WEIGHTS returns weights for certain Gauss-Legendre quadrature rules.
• I4_HUGE returns a "huge" I4.
• I4_LOG_2 returns the integer part of the logarithm base 2 of an I4.
• INDEX_LEVEL_GL: determine first level at which given index is generated.
• LEVEL_TO_ORDER_OPEN converts a level to an order for open rules.
• MULTIGRID_INDEX_Z returns an indexed multidimensional grid.
• PRODUCT_WEIGHT_GL: weights for a product Gauss-Legendre rule.
• R8MAT_TRANSPOSE_PRINT_SOME prints some of an R8MAT, transposed.
• R8MAT_WRITE writes an R8MAT file.
• R8VEC_DIRECT_PRODUCT2 creates a direct product of R8VEC's.
• R8VEC_PRINT_SOME prints "some" of an R8VEC.
• S_BLANK_DELETE removes blanks from a string, left justifying the remainder.
• S_TO_I4 reads an integer value from a string.
• SPARSE_GRID_GL computes a sparse grid of Gauss-Legendre points.
• SPARSE_GRID_GL_INDEX indexes the points forming a sparse grid of GL points.
• SPARSE_GRID_GL_SIZE sizes a sparse grid of Gauss-Legendre points.
• TIMESTAMP prints the current YMDHMS date as a time stamp.
• VEC_COLEX_NEXT2 generates vectors in colex order.

You can go up one level to the FORTRAN90 source codes.