SPARSE_GRID_OPEN_DATASET
Sparse Grid from Open 1D Quadrature Rule

SPARSE_GRID_OPEN_DATASET is a FORTRAN90 program which computes a sparse quadrature rule for an arbitrary spatial dimension, associated with a particular "level" of the Smolyak construction, and based on an open fully nested 1D quadrature rule.

The program offers a choice of open 1D quadrature rules to be used:

• 2: F2, the Fejer type 2 rule;
• 3: GP, the Gauss-Patterson rule;
• 4: NCO, the Newton-Cotes Open rule;
• 5: TS, the Tanh-Sinh rule;

Usage:

sparse_grid_open_dataset dim_num level_max rule

dim_num

the spatial dimension;

level_max

the level of the Smolyak construction;

rule

the index (2/3/4/5) of the 1D quadrature rule to use.

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_OPEN_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.

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

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 creates a sparse grid dataset based on Gauss-Laguerrre rules.

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 sparse grids based on a Newton Cotes closed rule.

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

SPARSE_GRID_OPEN, a FORTRAN90 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. Philip Davis, Philip Rabinowitz,
   Methods of Numerical Integration,
   Second Edition,
   Dover, 2007,
   ISBN: 0486453391,
   LC: QA299.3.D28.
3. Walter Gautschi,
   Numerical Quadrature in the Presence of a Singularity,
   SIAM Journal on Numerical Analysis,
   Volume 4, Number 3, 1967, pages 357-362.
4. Thomas Gerstner, Michael Griebel,
   Numerical Integration Using Sparse Grids,
   Numerical Algorithms,
   Volume 18, Number 3-4, 1998, pages 209-232.
5. Prem Kythe, Michael Schaeferkotter,
   Handbook of Computational Methods for Integration,
   Chapman and Hall, 2004,
   ISBN: 1-58488-428-2,
   LC: QA299.3.K98.
6. Albert Nijenhuis, Herbert Wilf,
   Combinatorial Algorithms for Computers and Calculators,
   Second Edition,
   Academic Press, 1978,
   ISBN: 0-12-519260-6,
   LC: QA164.N54.
7. 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.
8. Thomas Patterson,
   The Optimal Addition of Points to Quadrature Formulae,
   Mathematics of Computation,
   Volume 22, Number 104, October 1968, pages 847-856.
9. Sergey Smolyak,
   Quadrature and Interpolation Formulas for Tensor Products of Certain Classes of Functions,
   Doklady Akademii Nauk SSSR,
   Volume 4, 1963, pages 240-243.
10. Dennis Stanton, Dennis White,
    Constructive Combinatorics,
    Springer, 1986,
    ISBN: 0387963472,
    LC: QA164.S79.

Source Code:

• sparse_grid_open_dataset.f90, the source code.

Examples and Tests:

F2_D2_LEVEL2 is an example computation based on a Fejer type 2 rule in two dimensions and level 2.

• f2_d2_level2.txt, the printed output from a run of the program for DIM_NUM=2 and LEVEL_MAX=2 and RULE=2.
• f2_d2_level2_x.txt, the abscissas of the computed quadrature rule.
• f2_d2_level2_w.txt, the weights of the computed quadrature rule.
• f2_d2_level2_r.txt, the integration region of the computed quadrature rule.

GP_D2_LEVEL2 is an example computation based on a Gauss-Patterson rule in two dimensions and level 2.

• gp_d2_level2.txt, the printed output from a run of the program for DIM_NUM=2 and LEVEL_MAX=2 and RULE=3.
• gp_d2_level2_x.txt, the abscissas of the computed quadrature rule.
• gp_d2_level2_w.txt, the weights of the computed quadrature rule.
• gp_d2_level2_r.txt, the integration region of the computed quadrature rule.

NCO_D2_LEVEL2 is an example computation based on a Newton-Cotes Open rule in two dimensions and level 2.

• nco_d2_level2.txt, the printed output from a run of the program for DIM_NUM=2 and LEVEL_MAX=2 and RULE=4.
• nco_d2_level2_x.txt, the abscissas of the computed quadrature rule.
• nco_d2_level2_w.txt, the weights of the computed quadrature rule.
• nco_d2_level2_r.txt, the integration region of the computed quadrature rule.

TS_D2_LEVEL4 is an example computation based on a tanh-sinh rule in two dimensions and level 4.

• ts_d2_level4.txt, the output file.
• ts_d2_level4_x.txt, the abscissas.
• ts_d2_level4_w.txt, the weights.
• ts_d2_level4_r.txt, the region.

List of Routines:

• MAIN is the main program for SPARSE_GRID_OPEN_DATASET.
• ABSCISSA_LEVEL_OPEN_ND: first level at which given abscissa is generated.
• CHOOSE computes the binomial coefficient C(N,K).
• COMP_NEXT computes the compositions of the integer N into K parts.
• F2_ABSCISSA returns the I-th abscissa for the Fejer type 2 rule.
• F2_WEIGHTS computes weights for a Fejer type 2 rule.
• GET_UNIT returns a free FORTRAN unit number.
• GP_ABSCISSA returns the I-th abscissa for a Gauss-Patterson rule.
• GP_WEIGHTS sets weights for a Gauss-Patterson rule.
• I4_MODP returns the nonnegative remainder of I4 division.
• I4MAT_TRANSPOSE_PRINT_SOME prints some of the transpose of an I4MAT.
• INDEX_TO_LEVEL_OPEN determines the level of a point given its index.
• LEVEL_TO_ORDER_OPEN converts a level to an order for open rules.
• MULTIGRID_INDEX1 returns an indexed multidimensional grid.
• MULTIGRID_SCALE_OPEN renumbers a grid as a subgrid on a higher level.
• NCO_ABSCISSA returns the I-th abscissa for the Newton Cotes Open rule.
• NCO_WEIGHTS computes weights for a Newton Cotes Open rule.
• PRODUCT_WEIGHTS_OPEN: weights for an open product rule.
• R8_HUGE returns a very large 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_BLANK_DELETE removes blanks from a string, left justifying the remainder.
• S_TO_I4 reads an I4 from a string.
• SPARSE_GRID_OFN_SIZE sizes a sparse grid using Open Fully Nested rules.
• SPGRID_OPEN_INDEX computes open grids with 0 <= LEVEL <= LEVEL_MAX.
• SPGRID_OPEN_WEIGHTS gathers the weights.
• TIMESTAMP prints the current YMDHMS date as a time stamp.
• TS_ABSCISSA returns the I-th abscissa for the tanh-sinh rule.
• TS_WEIGHTS computes weights for a tanh-sinh rule.
• VEC_COLEX_NEXT2 generates vectors in colex order.

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