SPARSE_GRID_GL_DATASET
Create Gauss-Legendre Sparse Grid Dataset

SPARSE_GRID_GL_DATASET is a C++ 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.

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:

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.

QUADRATURE_RULES, a dataset directory which defines 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.

SGMGA, a C++ 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_CC_DATASET, a C++ program which creates a sparse grid dataset based on Clenshaw-Curtis rules.

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

SPARSE_GRID_GL, a C++ 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 C++ program which creates a sparse grid dataset based on Gauss-Hermite rules.

SPARSE_GRID_LAGUERRE_DATASET, a C++ program which creates a sparse grid dataset based on Gauss-Laguerrre rules.

SPARSE_GRID_MIXED_DATASET, a C++ 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 C++ 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.cpp, 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_output.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.
• GL_ABSCISSA sets abscissas for multidimensional Gauss-Legendre quadrature.
• GL_WEIGHTS returns weights for certain Gauss-Legendre quadrature rules.
• I4_LOG_2 returns the integer part of the logarithm base 2 of an I4.
• I4_MAX returns the maximum of two I4's.
• I4_MIN returns the smaller of two I4's.
• I4_MODP returns the nonnegative remainder of I4 division.
• I4_POWER returns the value of I^J.
• I4_TO_STRING converts an I4 to a C++ string.
• I4VEC_PRODUCT multiplies the entries of an I4VEC.
• 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.
• R8_EPSILON returns the R8 roundoff unit.
• R8_HUGE returns a "huge" R8.
• 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_LEN_TRIM returns the length of a string to the last nonblank.
• 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 C++ source codes.