SPARSE_GRID_HERMITE_DATASET
Create Gauss-Hermite Sparse Grid Dataset

SPARSE_GRID_HERMITE_DATASET is a FORTRAN90 program which creates a Gauss-Hermite sparse grid dataset.

Usage:

sparse_grid_hermite_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_HERMITE_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, a dataset directory which contains the abscissas of sparse grids based on a Clenshaw Curtis rule.

SPARSE_GRID_CC_DATASET, a FORTRAN90 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_DATASET, a FORTRAN90 program which creates a sparse grid dataset based on 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_DATASET, a FORTRAN90 program which creates a sparse grid dataset based on Gauss-Hermite rules.

SPARSE_GRID_HERMITE, a dataset directory which contains Gauss-Hermite sparse grids.

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_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_hermite_dataset.f90, the source code.

Examples and Tests:

Here are the three quadrature files created by the command

sparse_grid_hermite_dataset 2 3

• sparse_grid_hermite_dataset.txt, printed output from the command.
• gh_d2_level3_r.txt, the "R" or "region" file.
• gh_d2_level3_w.txt, the "W" or "weight" file.
• gh_d2_level3_x.txt, the "X" or "abscissa" file.

List of Routines:

• MAIN is the main program for SPARSE_GRID_HERMITE_DATASET.
• COMP_NEXT computes the compositions of the integer N into K parts.
• GET_UNIT returns a free FORTRAN unit number.
• HERMITE_COMPUTE computes a Gauss-Hermite quadrature rule.
• HERMITE_COMPUTE_POINTS computes points of a Hermite quadrature rule.
• HERMITE_COMPUTE_WEIGHTS computes weights of a Hermite quadrature rule.
• IMTQLX diagonalizes a symmetric tridiagonal matrix.
• LEVEL_TO_ORDER_HERMITE: default growth for Hermite sparse grids.
• MONOMIAL_INTEGRAL_HERMITE integrates a Hermite monomial.
• MONOMIAL_QUADRATURE_HERMITE applies a quadrature rule to a monomial.
• MONOMIAL_VALUE evaluates a monomial.
• PRODUCT_HERMITE_WEIGHT computes the weights of a Hermite product rule.
• R8_CHOOSE computes the binomial coefficient C(N,K) as an R8.
• R8_FACTORIAL2 computes the double factorial function.
• R8_GAMMA evaluates Gamma(X) for a real argument.
• R8_HUGE returns a very large R8.
• R8_MOP returns the I-th power of -1 as an R8.
• R8COL_COMPARE compares columns in an R8COL.
• R8COL_SORT_HEAP_A ascending heapsorts an R8COL.
• R8COL_SORT_HEAP_INDEX_A does an indexed heap ascending sort of an R8COL.
• R8COL_SORTED_UNIQUE_COUNT counts unique elements in a sorted R8COL.
• R8COL_SWAP swaps columns I and J of an R8COL.
• R8COL_TOL_UNIQUE_COUNT counts tolerably unique entries in an R8COL.
• R8COL_UNDEX returns unique sorted indexes for an R8COL.
• R8MAT_PRINT prints an R8MAT.
• R8MAT_PRINT_SOME prints some of an R8MAT.
• R8MAT_TRANSPOSE_PRINT prints an R8MAT, transposed.
• R8MAT_TRANSPOSE_PRINT_SOME prints some of an R8MAT, transposed.
• R8MAT_WRITE writes an R8MAT file.
• R8VEC_COMPARE compares two R8VEC's.
• 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.
• SORT_HEAP_EXTERNAL externally sorts a list of items into ascending order.
• SPARSE_GRID_HERMITE computes a Hermite sparse grid.
• SPARSE_GRID_HERMITE_INDEX indexes a Hermite sparse grid.
• SPARSE_GRID_HERMITE_POINT computes the points of a Hermite sparse grid.
• SPARSE_GRID_HERMITE_SIZE sizes a Hermite sparse grid, discounting duplicates.
• SPARSE_GRID_HERMITE_SIZE_TOTAL Hermite sparse grid size counting duplicates.
• SPARSE_GRID_HERMITE_UNIQUE_INDEX maps nonunique points to unique points.
• SPARSE_GRID_HERMITE_WEIGHT computes Hermite sparse grid weights.
• SPARSE_GRID_HERMITE_WRITE writes a Hermite sparse grid rule to files.
• TIMESTAMP prints the current YMDHMS date as a time stamp.
• VEC_COLEX_NEXT3 generates vectors in colex order.

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