SPARSE_GRID_MIXED_DATASET
Create Sparse Grid Using Mixed Rules
SPARSE_GRID_MIXED_DATASET
is a MATLAB program which
creates a sparse grid dataset from a mixture of 1D rules, using
instructions from a user.
SPARSE_GRID_MIXED_DATASET relies on the availability of
two other directories of MATLAB functions, namely
SPARSE_GRID_MIXED and SANDIA_RULES. As written,
it will issue the commands
addpath ( '../sparse_grid_mixed' )
addpath ( '../sandia_rules' )
so these directories would need to be downloaded from the web
and installed in directories that are at the same level as the
directory containing SPARSE_GRID_MIXED_DATASET.
The available 1D quadrature rules are identified by a 2 or 3 letter code,
and include:

CC, ClenshawCurtis:
defined on [1,+1], with w(x)=1,
a closed, fully nested rule.

F2, Fejer Type 2:
defined on (1,+1), with w(x)=1,
an open, fully nested rule.

GP, Gauss Patterson:
defined on (1,+1), with w(x)=1,
a family of the midpoint rule, the 3 point
Gauss Legendre rule, and then successive Patterson refinements,
an open, fully nested rule.

GL, Gauss Legendre:
defined on (1,+1), with w(x)=1,
an open, weakly nested rule.

GH, Gauss Hermite:
defined on (oo,+oo), with w(x)=exp(x*x),
an open, weakly nested rule.

GGH, Generalized Gauss Hermite:
defined on (oo,+oo), with w(x)=x^alpha * exp(x*x),
an open, weakly nested rule.

LG, Gauss Laguerre:
defined on (0,+oo) with w(x)=exp(x),
an open, nonnested rule.

GLG, Generalized Gauss Laguerre:
defined on (0,+oo) with w(x)=x^alpha * exp(x),
an open, nonnested rule.

GJ, Gauss Jacobi:
defined on [1,+1] with w(x)=(1x)^alpha (1+x)^beta
an open, nonnested rule.

GW, Golub Welsch:
a rule defined by the user based on the GolubWelsch algorithm,
with points and weights supplied by external routines,
presumably an open, nonnested rule.
(not supported yet)

CCS, ClenshawCurtis SlowGrowth:
defined on [1,+1], with w(x)=1,
a closed, fully nested rule.

F2S, Fejer Type 2, SlowGrowth:
defined on (1,+1), with w(x)=1,
an open, fully nested rule.

GPS, Gauss Patterson, SlowGrowth:
defined on (1,+1), with w(x)=1,
a family of the midpoint rule, the 3 point
Gauss Legendre rule, and then successive Patterson refinements,
an open, fully nested rule.
Usage:
The user first specifies:

M the spatial dimension of the quadrature region,

L the level that defines the Smolyak grid.
and these two values can actuallyb be specified on the command line.
Then the user specifies rules for each of the M dimensions.
A rule, when specified, may be used for one, or for multiple consecutive
dimensions.

RULE identifies the 1D rule, and must be one of
"CC", "F2", "GP", "GL", "GH", "GGH", "LG", "GLG", "GJ", "GW",
"CCS", "F2S", "GPS".

the number of times the rule is to be used for consecutive dimensions;

ALPHA, a rule parameter, if needed. (Rules GGH, GLG and GJ);

BETA, a rule parameter, if needed. (Rule GJ only).
Finally, the user specifies a "file identifier", that is, the main part
of a string to be used when creating the files.
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_MIXED_DATASET is available in
a CPP version and
a FORTRAN90 version and
a MATLAB version.
Related Data and Programs:
SANDIA_RULES,
a MATLAB library which
produces 1D quadrature rules of
Chebyshev, Clenshaw Curtis, Fejer 2, Gegenbauer, generalized Hermite,
generalized Laguerre, Hermite, Jacobi, Laguerre, Legendre and Patterson types.
SPARSE_GRID_CC_DATASET,
a MATLAB program which
creates a sparse grid dataset based on ClenshawCurtis rules.
SPARSE_GRID_GL_DATASET,
a MATLAB program which
creates a sparse grid dataset based on GaussLegendre rules.
SPARSE_GRID_HERMITE_DATASET,
a MATLAB program which
creates a sparse grid dataset based on GaussHermite rules.
SPARSE_GRID_HW,
a MATLAB library which
creates sparse grids based on GaussLegendre, GaussHermite,
GaussPatterson, or a nested variation of GaussHermite rules,
by Florian Heiss and Viktor Winschel.
SPARSE_GRID_LAGUERRE_DATASET,
a MATLAB program which
creates a sparse grid dataset based on GaussLaguerrre rules.
SPARSE_GRID_MIXED,
a MATLAB library which
creates a sparse grid dataset based on a mixed set of 1D factor rules.
SPARSE_GRID_MIXED,
a dataset directory which
contains multidimensional Smolyak sparse grids
based on a mixed set of 1D factor rules.
SPARSE_GRID_OPEN_DATASET,
a MATLAB program which
creates a sparse grid dataset based on
open rules (Fejer 2, GaussPatterson, NewtonCotesOpen).
Reference:

Thomas Gerstner, Michael Griebel,
Numerical Integration Using Sparse Grids,
Numerical Algorithms,
Volume 18, Number 34, 1998, pages 209232.

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 23092345.

Sergey Smolyak,
Quadrature and Interpolation Formulas for Tensor Products of
Certain Classes of Functions,
Doklady Akademii Nauk SSSR,
Volume 4, 1963, pages 240243.
Source Code:
You can go up one level to
the MATLAB source codes.
Last revised on 15 April 2013.