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”}
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.
GP_D2_LEVEL2 is an example computation based on a Gauss-Patterson
rule in two dimensions and level 2.
NCO_D2_LEVEL2 is an example computation based on a Newton-Cotes
Open rule in two dimensions and level 2.
TS_D2_LEVEL4 is an example computation based on a tanh-sinh rule
in two dimensions and level 4.
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.