SANDIA_SPARSE
Sparse Grids for Sandia
SANDIA_SPARSE
is a C++ library which
can be used to compute the points and weights of a Smolyak sparse
grid, based on a variety of 1dimensional quadrature rules.
The sparse grids that can be created may be based on any one of
a variety of 1D quadrature rules. However, only isotropic
grids are generated, that is, the same 1D quadrature rule is used
in each dimension, and the same maximum order is used in each dimension.
This is a limitation of this library, and not an inherent limitation
of the sparse grid method.
The 1D quadrature rules that can be used to construct sparse grids include:

CFN, Closed Fully Nested rules:

CC, ClenshawCurtis:
defined on [1,+1], with w(x)=1.

OFN, Open Fully Nested rules:

F1, Fejer Type 1:
defined on (1,+1), with w(x)=1.

F2, Fejer Type 2:
defined on (1,+1), with w(x)=1.

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.

OWN, Open Weakly Nested rules:

GL, Gauss Legendre:
defined on (1,+1), with w(x)=1.

GH, Gauss Hermite:
defined on (oo,+oo), with w(x)=exp(x*x).

ONN, Open NonNested rules:

LG, Gauss Laguerre:
defined on (0,+oo) with w(x)=exp(x).
Point Growth of 1D Rules
A major advantage of sparse grids is that they can achieve accuracy that
is comparable to a corresponding product rule, while using far fewer points,
that is, evaluations of the function that is to be integrated. We will leave
aside the issue of comparing accuracy for now, and simply focus on the pattern
of point growth.
A sparse grid is essentially a linear combination of lower order product
grids. One way point growth is controlled is to only use product grids
based on a set of factors that are nested. In other words, the underlying
1D rules are selected so that, when we increase the order of such a rule,
all the points of the current rule are included in the new one.
The exact details of how this works depend on the particular 1D rule being
used and the nesting behavior it satisfies. We classify the cases as follows:

CFN, "Closed, Fully Nested", based on Clenshaw Curtis.

OFN, "Open, Fully Nested", based on Fejer Type 1, Fejer Type 2, or
Gauss Patterson.

OWN, "Open, Weakly Nested", based on Gauss Legendre or
Gauss Hermite rules.

ONN, "Open, NonNested", based on Gauss Laguerre rules;
For CFN rules we have the following relationship between the level (index of
the grid) and the 1D order (the number of points in the 1D rule.)
order = 2^{level} + 1
except that for the special case of level=0 we assign
order=1.
For OFN, OWN and ONN rules, the relationship between level
and 1D order is:
order = 2^{level+1}  1
Thus, as we allow level to grow, the order of the 1D closed
and open rules behaves as follows:
Level  CFN  OFN/OWN/ONN 
0  1  1 
1  3  3 
2  5  7 
3  9  15 
4  17  31 
5  33  63 
6  65  127 
7  129  255 
8  257  511 
9  513  1,023 
10  1,025  2,057 
When we move to multiple dimensions, the counting becomes more complicated. This is because
a multidimensional sparse grid is made up of a logical sum of product grids. A multidimensional
sparse grid has a multidimensional level, which is a single number. Each product grid
that forms part of this sparse grid has a multidimensional level which is the sum of the
1D levels of its factors. A sparse grid whose multidimensional level is represented by
LEVEL includes all product grids whose level ranges LEVEL+1DIM
and LEVEL.
Thus, as one example, if DIM is 2, the sparse grid of level 3, formed from
a CFN rule, will be formed from the following product rules.
level  level 1  level 2  order 1  order 2  order 
1  0  1  1  3  3 
1  1  0  3  1  3 
2  0  2  1  5  5 
2  1  1  3  3  9 
2  2  0  5  1  5 
Because of the nesting pattern for CFN rules, instead of 25 points (the sum of the orders),
we will actually have just 13 unique points.
For a CFN sparse grid, here is the pattern of growth in the number of points,
as a function of spatial dimension and grid level:
DIM  1  2  3  4  5 
Level      
0  1  1  1  1  1 
1  3  5  7  9  11 
2  5  13  25  41  61 
3  9  29  69  137  241 
4  17  65  177  401  801 
5  33  145  441  1,105  2,433 
6  65  321  1,073  2,929  6,993 
7  129  705  2,561  7,537  19,313 
8  257  1,537  6,017  18,945  51,713 
For an OFN sparse grid, here is the pattern of growth in the number of points,
as a function of spatial dimension and grid level:
DIM  1  2  3  4  5 
Level      
0  1  1  1  1  1 
1  3  5  7  9  11 
2  7  17  31  49  71 
3  15  49  111  209  351 
4  31  129  351  769  1,471 
5  63  321  1,023  2,561  5,503 
6  127  769  2,815  7,937  18,943 
7  255  1,793  7,423  23,297  61,183 
8  511  4,097  18,943  65,537  187,903 
For an OWN sparse grid, here is the pattern of growth in the number of points,
as a function of spatial dimension and grid level:
DIM  1  2  3  4  5 
Level      
0  1  1  1  1  1 
1  3  5  7  9  11 
2  7  21  37  57  81 
3  15  73  159  289  471 
4  31  225  597  1,265  2,341 
5  63  637  2,031  4,969  10,363 
6  127  1,693  6,405  17,945  41,913 
7  255  4,289  19,023  60,577  157,583 
8  511  10,473  53,829  193,457  557,693 
For an ONN sparse grid, here is the pattern of growth in the number of points,
as a function of spatial dimension and grid level:
DIM  1  2  3  4  5 
Level      
0  1  1  1  1  1 
1  3  7  10  13  16 
2  7  29  58  95  141 
3  15  95  255  515  906 
4  31  273  945  2,309  4,746 
5  63  723  3,120  9,065  21,503 
6  127  1,813  9,484  32,259  87,358 
7  255  4,375  27,109  106,455  325,943 
8  511  10,265  73,915  330,985  1,135,893 
Usage:
To integrate a function f(x) over a multidimensional cube [1,+1]^DIM using
a sparse grid based on a Clenshaw Curtis rule, we might use a program something
like the following:
dim = 2;
level = 3;
rule = 1;
point_num = levels_index_size ( dim, level, rule );
w = new double[point_num];
x = new double[dim*point_num];
sparse_grid ( dim, level, rule, point_num, w, x );
quad = 0.0;
for ( j = 0; j < point_num; j++ )
{
quad = quad + w[j] * f ( x+j*dim );
}
Licensing:
The code described and made available on this web page is distributed
under the
GNU LGPL license.
Languages:
SANDIA_SPARSE is available in
a C++ version and
a FORTRAN90 version and
a MATLAB version.
Related Data and Programs:
SANDIA_RULES,
a C++ library which
generates Gauss quadrature rules of various orders and types.
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,
by Knut Petras.
SPARSE_GRID_CC,
a C++ library which
can define a multidimensional sparse grid based on a 1D Clenshaw Curtis rule.
SPARSE_GRID_CC_DATASET,
a C++ program which
reads user input, creates a multidimensional sparse grid based on a
1D Clenshaw Curtis rule and writes it to three files that define a
quadrature rule.
SPINTERP,
a MATLAB library which
uses a sparse grid to perform multilinear hierarchical interpolation,
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 273288.

Charles Clenshaw, Alan Curtis,
A Method for Numerical Integration on an Automatic Computer,
Numerische Mathematik,
Volume 2, Number 1, December 1960, pages 197205.

Philip Davis, Philip Rabinowitz,
Methods of Numerical Integration,
Second Edition,
Dover, 2007,
ISBN: 0486453391,
LC: QA299.3.D28.

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

Albert Nijenhuis, Herbert Wilf,
Combinatorial Algorithms for Computers and Calculators,
Second Edition,
Academic Press, 1978,
ISBN: 0125192606,
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 23092345.

Fabio Nobile, Raul Tempone, Clayton Webster,
An Anisotropic Sparse Grid Stochastic Collocation Method for Partial Differential
Equations with Random Input Data,
SIAM Journal on Numerical Analysis,
Volume 46, Number 5, 2008, pages 24112442.

Sergey Smolyak,
Quadrature and Interpolation Formulas for Tensor Products of
Certain Classes of Functions,
Doklady Akademii Nauk SSSR,
Volume 4, 1963, pages 240243.

Dennis Stanton, Dennis White,
Constructive Combinatorics,
Springer, 1986,
ISBN: 0387963472,
LC: QA164.S79.
Source Code:
Examples and Tests:
List of Routines:

ABSCISSA_LEVEL_CLOSED_ND: first level at which an abscissa is generated.

ABSCISSA_LEVEL_OPEN_ND: first level at which given abscissa is generated.

CC_ABSCISSA returns the Ith abscissa of the Clenshaw Curtis rule.

CC_WEIGHTS computes Clenshaw Curtis weights.

COMP_NEXT computes the compositions of the integer N into K parts.

F1_ABSCISSA returns the Ith abscissa for the Fejer type 1 rule.

F1_WEIGHTS computes weights for a Fejer type 1 rule.

F2_ABSCISSA returns the Ith abscissa for the Fejer type 2 rule.

F2_WEIGHTS computes weights for a Fejer type 2 rule.

GH_ABSCISSA sets abscissas for multidimensional GaussHermite quadrature.

GH_WEIGHTS returns weights for certain GaussHermite quadrature rules.

GL_ABSCISSA sets abscissas for multidimensional GaussLegendre quadrature.

GL_WEIGHTS returns weights for certain GaussLegendre quadrature rules.

GP_ABSCISSA returns the Ith abscissa for a GaussPatterson rule.

GP_WEIGHTS sets weights for a GaussPatterson rule.

I4_LOG_2 returns the integer part of the logarithm base 2 of an I4.

I4_MAX returns the maximum of two I4's.

I4_MIN returns the minimum of two I4's.

I4_MODP returns the nonnegative remainder of I4 division.

I4_POWER returns the value of I^J.

I4VEC_PRODUCT multiplies the entries of an integer vector.

INDEX_LEVEL_OWN: determine first level at which given index is generated.

INDEX_TO_LEVEL_CLOSED determines the level of a point given its index.

INDEX_TO_LEVEL_OPEN determines the level of a point given its index.

LEVEL_TO_ORDER_CLOSED converts a level to an order for closed rules.

LEVEL_TO_ORDER_OPEN converts a level to an order for open rules.

LEVELS_INDEX indexes a sparse grid.

LEVELS_INDEX_CFN indexes a sparse grid made from CFN 1D rules.

LEVELS_INDEX_OFN indexes a sparse grid made from OFN 1D rules.

LEVELS_INDEX_ONN indexes a sparse grid made from ONN 1D rules.

LEVELS_INDEX_OWN indexes a sparse grid made from OWN 1D rules.

LEVELS_INDEX_SIZE sizes a sparse grid.

LEVELS_INDEX_SIZE_CFN sizes a sparse grid made from CFN 1D rules.

LEVELS_INDEX_SIZE_ONN sizes a sparse grid made from ONN 1D rules.

LEVELS_INDEX_SIZE_OWN sizes a sparse grid made from OWN 1D rules.

LG_ABSCISSA sets abscissas for multidimensional GaussLaguerre quadrature.

LG_WEIGHTS returns weights for certain GaussLaguerre quadrature rules.

MONOMIAL_INTEGRAL_HERMITE integrates a Hermite mononomial.

MONOMIAL_INTEGRAL_LAGUERRE integrates a Laguerre monomial.

MONOMIAL_INTEGRAL_LEGENDRE integrates a Legendre monomial.

MONOMIAL_QUADRATURE applies a quadrature rule to a monomial.

MONOMIAL_VALUE evaluates a monomial.

MULTIGRID_INDEX_CFN indexes a sparse grid based on CFN 1D rules.

MULTIGRID_INDEX_OFN indexes a sparse grid based on OFN 1D rules.

MULTIGRID_INDEX_ONN indexes a sparse grid based on ONN 1D rules.

MULTIGRID_INDEX_OWN returns an indexed multidimensional grid.

MULTIGRID_SCALE_CLOSED renumbers a grid as a subgrid on a higher level.

MULTIGRID_SCALE_OPEN renumbers a grid as a subgrid on a higher level.

PRODUCT_WEIGHTS computes the weights of a product rule.

R8_ABS returns the absolute value of an R8.

R8_CHOOSE computes the binomial coefficient C(N,K) as an R8.

R8_FACTORIAL computes the factorial of N.

R8_FACTORIAL2 computes the double factorial function.

R8_HUGE returns a "huge" R8.

R8_MOP returns the Ith power of 1 as an R8 value.

R8VEC_DIRECT_PRODUCT2 creates a direct product of R8VEC's.

SPARSE_GRID computes a sparse grid.

SPARSE_GRID_CC_SIZE sizes a sparse grid using Clenshaw Curtis rules.

SPARSE_GRID_CFN computes a sparse grid based on a CFN 1D rule.

SPARSE_GRID_OFN computes a sparse grid based on an OFN 1D rule.

SPARSE_GRID_OFN_SIZE sizes a sparse grid using Open Fully Nested rules.

SPARSE_GRID_ONN computes a sparse grid based on a ONN 1D rule.

SPARSE_GRID_OWN computes a sparse grid based on an OWN 1D rule.

SPARSE_GRID_WEIGHTS_CFN computes sparse grid weights based on a CFN 1D rule.

SPARSE_GRID_WEIGHTS_OFN computes sparse grid weights based on a OFN 1D rule.

TIMESTAMP prints the current YMDHMS date as a time stamp.

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.