SPARSE_GRID_MIXED Sparse Grids using Mixed Factors

SPARSE_GRID_MIXED is a C++ library which can be used to construct a sparse grid whose factors are possibly distinct 1D quadrature rules.

SPARSE_GRID_MIXED calls many routines from the SANDIA_RULES library. Source code or compiled copies of both libraries must be available when a program wishes to use the SPARSE_GRID_MIXED library.

The 1D quadrature rules are designed to approximate an integral of the form:

Integral ( A < X < B ) F(X) W(X) dX
where W(X) is a weight function, by the quadrature sum:
Sum ( 1 <= I <= ORDER) F(X(I)) * W(I)
where the set of X values are known as abscissas and the set of W values are known as weights.

Note that the letter W, unfortunately, is used to denote both the weight function in the original integral, and the vector of weight values in the quadrature sum.

Index Name Abbreviation Default Growth Rule Interval Weight function
1 Clenshaw-Curtis Exponential Growth CC Exponential [-1,+1] 1
2 Fejer Type 2, Exponential Growth F2 Exponential [-1,+1] 1
3 Gauss Patterson, Exponential Growth GP Exponential [-1,+1] 1
4 Gauss-Legendre GL Linear [-1,+1] 1
5 Gauss-Hermite GH Linear (-oo,+oo) e-x*x
6 Generalized Gauss-Hermite GGH Linear (-oo,+oo) |x|alpha e-x*x
7 Gauss-Laguerre LG Linear [0,+oo) e-x
8 Generalized Gauss-Laguerre GLG Linear [0,+oo) xalpha e-x
9 Gauss-Jacobi GJ Linear [-1,+1] (1-x)alpha (1+x)beta
10 Golub-Welsch GW ? ? ?
11 Clenshaw-Curtis, Slow Exponential Growth CC_SE Slow exponential [-1,+1] 1
12 Fejer Type 2, Slow Exponential Growth F2_SE Slow exponential [-1,+1] 1
13 Gauss Patterson, Slow Exponential Growth GP_SE Slow exponential [-1,+1] 1
14 Clenshaw-Curtis, Moderate Exponential Growth CC_ME Moderate exponential [-1,+1] 1
15 Fejer Type 2, Moderate Exponential Growth F2_ME Moderate exponential [-1,+1] 1
16 Gauss Patterson, Moderate Exponential Growth GP_ME Moderate exponential [-1,+1] 1
17 Clenshaw-Curtis Nested, linear growth CCN Linear (2*L+1) [-1,+1] 1

A sparse grid is a quadrature rule for a multidimensional integral. It is formed by taking a certain linear combination of lower-order product rules. The product rules, in turn, are formed as direct products of 1D quadrature rules. It is common to form a sparse grid in which the 1D component quadrature rules are the same. This package, however, is intended to produce sparse grids based on sums of product rules for which the rule chosen for each spatial dimension may be freely chosen from the set listed above.

These sparse grids are still indexed by a number known as level, and assembled as a sum of low order product rules. As the value of level increases, the point growth becomes more complicated. This is because the 1D rules have somewhat varying point growth patterns to begin with, and the varying levels of nestedness have a dramatic effect on the sparsity of the total grid.

Since a sparse grid is made up of a combination of product grids, it is frequently the case that many of the product grids include the same point. For efficiency, it is usually desirable to merge or consolidate such duplicate points when expressing the resulting sparse grid rule. It is possible to "logically" determine when a duplicate point will be generated; however, this logic changes depending on the specific 1-dimensional rules being used, and the tests can become quite elaborate. Moreover, for rules which include real parameters, the determination of duplication can require a numerical tolerance.

In order to simplify the matter of the detection of duplicate points, the codes presented begin by generating the entire "naive" set of points. Then a user-specified tolerance TOL is used to determine when two points are equal. If the maximum difference between any two components is less than or equal to TOL, the points are declared to be equal.

A reasonable value for TOL might be the square root of the machine precision. Setting TOL to zero means that only points which are identical to the last significant digit are taken to be duplicates. Setting TOL to a negative value means that no duplicate points will be eliminated - in other words, this choice produces the full or "naive" grid.

A version of the sparse grid library is available in http://tasmanian.ornl.gov, the TASMANIAN library, available from Oak Ridge National Laboratory.

Languages:

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

Related Data and Programs:

NINT_EXACTNESS_MIXED, a C++ program which measures the polynomial exactness of a multidimensional quadrature rule based on a mixture of 1D quadrature rule factors.

QUADRULE, a C++ library which defines quadrature rules for various intervals and weight functions.

SANDIA_RULES, a C++ library which produces 1D quadrature rules of Chebyshev, Clenshaw Curtis, Fejer 2, Gegenbauer, generalized Hermite, generalized Laguerre, Hermite, Jacobi, Laguerre, Legendre and Patterson types.

SANDIA_SPARSE, a C++ library which computes the points and weights of a Smolyak sparse grid, based on a variety of 1-dimensional quadrature rules.

SGMG, a C++ library which is the successor to SPARSE_GRID_MIXED. It includes the ability to specify a specific growth rule in each dimension.

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_GL, a C++ library which creates sparse grids based on Gauss-Legendre rules.

SPARSE_GRID_HERMITE, a C++ library which creates sparse grids based on Gauss-Hermite rules.

SPARSE_GRID_MIXED, a dataset directory which contains multidimensional Smolyak sparse grids based on a mixed set of 1D factor rules.

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

1. Milton Abramowitz, Irene Stegun,
Handbook of Mathematical Functions,
National Bureau of Standards, 1964,
ISBN: 0-486-61272-4,
LC: QA47.A34.
2. Charles Clenshaw, Alan Curtis,
A Method for Numerical Integration on an Automatic Computer,
Numerische Mathematik,
Volume 2, Number 1, December 1960, pages 197-205.
3. Philip Davis, Philip Rabinowitz,
Methods of Numerical Integration,
Second Edition,
Dover, 2007,
ISBN: 0486453391,
LC: QA299.3.D28.
4. Walter Gautschi,
Numerical Quadrature in the Presence of a Singularity,
SIAM Journal on Numerical Analysis,
Volume 4, Number 3, September 1967, pages 357-362.
5. Thomas Gerstner, Michael Griebel,
Numerical Integration Using Sparse Grids,
Numerical Algorithms,
Volume 18, Number 3-4, 1998, pages 209-232.
6. Gene Golub, John Welsch,
Mathematics of Computation,
Volume 23, Number 106, April 1969, pages 221-230.
7. Prem Kythe, Michael Schaeferkotter,
Handbook of Computational Methods for Integration,
Chapman and Hall, 2004,
ISBN: 1-58488-428-2,
LC: QA299.3.K98.
8. Albert Nijenhuis, Herbert Wilf,
Combinatorial Algorithms for Computers and Calculators,
Second Edition,
ISBN: 0-12-519260-6,
LC: QA164.N54.
9. 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.
10. Thomas Patterson,
Mathematics of Computation,
Volume 22, Number 104, October 1968, pages 847-856.
11. Sergey Smolyak,
Quadrature and Interpolation Formulas for Tensor Products of Certain Classes of Functions,
Volume 4, 1963, pages 240-243.
12. Arthur Stroud, Don Secrest,
Prentice Hall, 1966,
LC: QA299.4G3S7.
13. Joerg Waldvogel,
Fast Construction of the Fejer and Clenshaw-Curtis Quadrature Rules,
BIT Numerical Mathematics,
Volume 43, Number 1, 2003, pages 1-18.

Examples and Tests:

COMP_NEXT_PRB tests COMP_NEXT.

PRODUCT_MIXED_WEIGHT_PRB tests PRODUCT_MIXED_WEIGHT.

SPARSE_GRID_MIXED_INDEX_PRB tests SPARSE_GRID_MIXED_INDEX.

SPARSE_GRID_MIXED_POINT_PRB tests SPARSE_GRID_MIXED_POINT.

SPARSE_GRID_MIXED_SIZE_PRB tests SPARSE_GRID_MIXED_SIZE and SPARSE_GRID_MIXED_SIZE_TOTAL.

SPARSE_GRID_MIXED_SIZE_TABLE makes a point count table.

SPARSE_GRID_MIXED_UNIQUE_INDEX_PRB tests SPARSE_GRID_MIXED_UNIQUE_INDEX.

SPARSE_GRID_MIXED_WEIGHT_PRB tests SPARSE_GRID_MIXED_WEIGHT.

SPARSE_GRID_MIXED_WRITE_PRB tests SPARSE_GRID_MIXED_WRITE.

List of Routines:

• SPARSE_GRID_MIXED_INDEX indexes a sparse grid made from mixed 1D rules.
• SPARSE_GRID_MIXED_POINT computes the points of a sparse grid rule.
• SPARSE_GRID_MIXED_SIZE sizes a sparse grid, discounting duplicate points.
• SPARSE_GRID_MIXED_SIZE_TOTAL sizes a sparse grid, counting duplicates.
• SPARSE_GRID_MIXED_UNIQUE_INDEX maps nonunique points to unique points.
• SPARSE_GRID_MIXED_WEIGHT: sparse grid weights based on a mix of 1D rules.
• SPARSE_GRID_MIXED_WRITE writes a sparse grid rule to five files.

You can go up one level to the C++ source codes.

Last revised on 23 February 2010.