jburkardt

SPARSE_GRID_OPEN_DATASET\ Sparse Grid from Open 1D Quadrature Rule {#sparse_grid_open_dataset-sparse-grid-from-open-1d-quadrature-rule align=”center”} ========================================

SPARSE_GRID_OPEN_DATASET is a C++ 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 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: {#usage align=”center”}

sparse_grid_open_dataset dim_num level_max rule

where

dim_num is the spatial dimension;
level_max is the level of the Smolyak construction;
rule is the index (2/3/4/5) of the 1D quadrature rule to use.

Licensing: {#licensing align=”center”}

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

Languages: {#languages align=”center”}

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

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

SPARSE_GRID_CLOSED_DATASET, a C++ program which creates a sparse grid dataset based on closed rules (Clenshaw-Curtis, Newton-Cotes-Closed).

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

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

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

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, a C++ library which constructs a sparse grid using different rules in each spatial dimension.

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

SPARSE_GRID_MIXED_GROWTH, a C++ library which creates a sparse grid dataset based on a mixed set of 1D factor rules, and experiments with the use of a linear growth rate for the quadrature 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 C++ library which defines 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: {#reference align=”center”}

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

Source Code: {#source-code align=”center”}

sparse_grid_open_dataset.cpp, the source code.

Examples and Tests: {#examples-and-tests align=”center”}

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

f2_d2_level2_output.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_output.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_output.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_output.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: {#list-of-routines align=”center”}

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.
GP_ABSCISSA returns the I-th abscissa for a Gauss-Patterson rule.
GP_WEIGHTS sets weights for a Gauss-Patterson rule.
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.
I4MAT_TRANSPOSE_PRINT_SOME prints some of an I4MAT, transposed.
I4_TO_STRING converts an I4 to a C++ string.
I4VEC_PRODUCT multiplies the entries of an I4VEC.
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_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_COPY copies an R8VEC.
R8VEC_DIRECT_PRODUCT2 creates a direct product of R8VEC’s.
R8VEC_PRINT_SOME prints “some” of an R8VEC.
R8VEC_SUM returns the sum of an R8VEC.
S_LEN_TRIM returns the length of a string to the last nonblank.
SPARSE_GRID_OFN_SIZE sizes a sparse grid using Open Fully Nested rules.
LEVELS_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 C++ source codes.

Last revised on 23 December 2009.