SPARSE_GRID_OPEN_DATASET
Sparse Grid from Open 1D Quadrature Rule
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;
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4: NCO, the Newton-Cotes Open rule;
-
5: TS, the Tanh-Sinh rule;
Usage:
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:
The code described and made available on this web page is 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.
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:
-
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:
Examples and Tests:
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:
-
MAIN is the main program for SPARSE_GRID_OPEN_DATASET.
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ABSCISSA_LEVEL_OPEN_ND: first level at which given abscissa is generated.
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CHOOSE computes the binomial coefficient C(N,K).
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COMP_NEXT computes the compositions of the integer N into K parts.
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F2_ABSCISSA returns the I-th abscissa for the Fejer type 2 rule.
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F2_WEIGHTS computes weights for a Fejer type 2 rule.
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GP_ABSCISSA returns the I-th abscissa for a Gauss-Patterson rule.
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GP_WEIGHTS sets weights for a Gauss-Patterson rule.
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I4_MAX returns the maximum of two I4's.
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I4_MIN returns the smaller of two I4's.
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I4_MODP returns the nonnegative remainder of I4 division.
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I4_POWER returns the value of I^J.
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I4MAT_TRANSPOSE_PRINT_SOME prints some of an I4MAT, transposed.
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I4_TO_STRING converts an I4 to a C++ string.
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I4VEC_PRODUCT multiplies the entries of an I4VEC.
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INDEX_TO_LEVEL_OPEN determines the level of a point given its index.
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LEVEL_TO_ORDER_OPEN converts a level to an order for open rules.
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MULTIGRID_INDEX1 returns an indexed multidimensional grid.
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MULTIGRID_SCALE_OPEN renumbers a grid as a subgrid on a higher level.
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NCO_ABSCISSA returns the I-th abscissa for the Newton Cotes open rule.
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NCO_WEIGHTS computes weights for a Newton-Cotes Open rule.
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PRODUCT_WEIGHTS_OPEN: weights for an open product rule.
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R8_EPSILON returns the R8 roundoff unit.
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R8_HUGE returns a "huge" R8.
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R8MAT_TRANSPOSE_PRINT_SOME prints some of an R8MAT, transposed.
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R8MAT_WRITE writes an R8MAT file.
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R8VEC_COPY copies an R8VEC.
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R8VEC_DIRECT_PRODUCT2 creates a direct product of R8VEC's.
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R8VEC_PRINT_SOME prints "some" of an R8VEC.
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R8VEC_SUM returns the sum of an R8VEC.
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S_LEN_TRIM returns the length of a string to the last nonblank.
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SPARSE_GRID_OFN_SIZE sizes a sparse grid using Open Fully Nested rules.
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LEVELS_OPEN_INDEX computes open grids with 0 <= LEVEL <= LEVEL_MAX.
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SPGRID_OPEN_WEIGHTS gathers the weights.
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TIMESTAMP prints the current YMDHMS date as a time stamp.
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TS_ABSCISSA returns the I-th abscissa for the tanh-sinh rule.
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TS_WEIGHTS computes weights for a tanh-sinh rule.
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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.