SPARSE_GRID_COMPOSITE
Sparse Grids Based on 1D Composite Rules

SPARSE_GRID_COMPOSITE is a MATLAB library which can be used to compute the points and weights of a Smolyak sparse grid, based on a 1-dimensional composite quadrature rule.

Currently, the library only allows composite rules of the 1 point midpoint formula. A 1D composite rule of the midpoint rule is suitable for integration of functions with low or variable smoothness. A sparse grid formed from such rules can better handle multidimensional integrands with limited smoothness.

Of course, for both the 1D composite rule and the sparse grid rule, the repeated use of the midpoint rule limits the order of accuracy of the quadrature; therefore, if the integrand is known to be of higher smoothness, other approaches will produce an accurate answer more quickly.

Licensing:

The computer code and data files described and made available on this web page are distributed under the GNU LGPL license.

Related Data and Programs:

GRID_DISPLAY, a MATLAB library which can display a 2D or 3D grid or sparse grid.

QUADRATURE_RULES, a dataset directory of files which define quadrature rules; a number of examples of sparse grid quadrature rules are included.

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

SGMGA, a MATLAB 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, a MATLAB library which creates sparse grids based on Clenshaw-Curtis rules.

SPARSE_GRID_COMPOSITE, a dataset directory which contains examples of sparse grids based on 1D composite rules (currently only of order 1).

SPARSE_GRID_F2, a dataset directory which contains the abscissas of sparse grids based on a Fejer Type 2 rule.

SPARSE_GRID_GL, a MATLAB library which computes sparse grids based on a Gauss-Legendre rule.

SPARSE_GRID_GP, a dataset directory which contains the abscissas of sparse grids based on a Gauss Patterson rule.

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

SPARSE_GRID_HW, a MATLAB library which creates sparse grids based on Gauss-Legendre, Gauss-Hermite, Gauss-Patterson, or a nested variation of Gauss-Hermite rules, by Florian Heiss and Viktor Winschel.

SPARSE_GRID_MIXED, a MATLAB library which constructs a sparse grid using different rules in each spatial dimension.

SPARSE_GRID_NCC, a dataset directory which contains the abscissas of sparse grids based on a Newton Cotes closed rule.

SPARSE_GRID_NCO, a dataset directory which contains the abscissas of sparse grids based on a Newton Cotes open rule.

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. 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.
2. Thomas Gerstner, Michael Griebel,
   Numerical Integration Using Sparse Grids,
   Numerical Algorithms,
   Volume 18, Number 3-4, 1998, pages 209-232.
3. Albert Nijenhuis, Herbert Wilf,
   Combinatorial Algorithms for Computers and Calculators,
   Second Edition,
   Academic Press, 1978,
   ISBN: 0-12-519260-6,
   LC: QA164.N54.
4. 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.
5. Sergey Smolyak,
   Quadrature and Interpolation Formulas for Tensor Products of Certain Classes of Functions,
   Doklady Akademii Nauk SSSR,
   Volume 4, 1963, pages 240-243.
6. Dennis Stanton, Dennis White,
   Constructive Combinatorics,
   Springer, 1986,
   ISBN: 0387963472,
   LC: QA164.S79.

Source Code:

• abscissa_level_closed_nd.m first level at which an abscissa is generated.
• comp_next.m computes the compositions of the integer N into K parts.
• composite_abscissa.m returns the I-th abscissa of the composite rule.
• composite_weights.m computes composite rule weights.
• i4_choose.m computes the binomial coefficient C(N,K).
• i4_modp.m returns the nonnegative remainder of integer division.
• index_to_level_closed.m determines the level of a point given its index.
• level_to_order_closed.m converts a level to an order for closed rules.
• monomial_int01.m returns the integral of a monomial over the [0,1] hypercube.
• monomial_quadrature.m applies a quadrature rule to a monomial.
• monomial_value.m evaluates a monomial.
• multigrid_index0.m returns an indexed multidimensional grid.
• multigrid_scale_closed.m renumbers a grid as a subgrid on a higher level.
• product_weights_composite.m computes weights for a composite product rule.
• r8mat_write.m writes a table file.
• r8vec_direct_product2.m direct product of double precision real vectors.
• sparse_grid_composite.m computes a sparse grid of composite rule points.
• sparse_grid_composite_index.m indexes the points forming a sparse grid.
• sparse_grid_composite_size.m sizes a sparse grid of composite rule points.
• sparse_grid_composite_weights.m gathers the weights.
• timestamp.m prints the current YMDHMS date as a time stamp.
• vec_colex_next2.m generates vectors in colex order.

Examples and Tests:

• sparse_grid_composite_test.m, runs all the tests.
• sparse_grid_composite_test01.m, tests SPARSE_GRID_COMPOSITE_SIZE.
• sparse_grid_composite_test02.m, tests SPARSE_GRID_COMPOSITE_INDEX.
• sparse_grid_composite_test03.m, tests SPARSE_GRID_COMPOSITE.
• sparse_grid_composite_test04.m, computes the weights and sums them.
• sparse_grid_composite_test05.m, tests a COMPOSITE sparse grid rule for exactness on monomials.
• sparse_grid_composite_test06.m, writes a COMPOSITE sparse grid rule to three files.
• composite_d2_level3_r.txt, the "R" data for a sparse COMPOSITE quadrature rule.
• composite_d2_level3_w.txt, the "W" data for a sparse COMPOSITE quadrature rule.
• composite_d2_level3_x.txt, the "X" data for a sparse COMPOSITE quadrature rule.
• sparse_grid_composite_test_output.txt, the output from a run of the sample program.

You can go up one level to the MATLAB source codes.