SPINTERP is a MATLAB library which can determine points defining a sparse grid in a multidimensional space, and given specific values at those points, can construct an interpolating function that can be evaluated anywhere.
The program can plot the interpolating function, perform optimization (seeking minima or maxima) and can integrate the function.
An early version of this library was documented and released as ACM TOMS Algorithm 847.
The sparse grid is constructed using Smolyak's construction. In fact, a nested series of grids is defined, each a refinement of the previous one, but chosen in such a way that the typical exponential growth in order does not occur. Moreover, because the grids are nested, the procedure produces a hierarchical series of piecewise linear interpolants. The nesting of these interpolants allows the estimation of the interpolation error.
The package includes several choices for the underlying one dimensional rule used to construct the sparse grids. The recommended rule is a Newton Cotes Closed rule, which produces grids of uniformly spaced points in [0,1], including the endpoints, of orders 1, 3, 5, 9, 17, 33, 65, and in general (2^I)+1. Note that the first rule is a special case (it doesn't include the endpoints, and the number of points in the rule is not equal to (2^0)+1!). Also note that the authors denote this rule as the "Clenshaw Curtis" or "CC" rule, although that name is more properly associated with the grid obtained by taking the cosine of the points given by the Newton Cotes Closed rule!
Another 1D rule is denoted by the authors as the "NB" or "no boundary" rule. This is simply a Newton Cotes Open rule which produces grids of uniformly spaced points in [0,1], omitting the endpoints, of orders 1, 3, 7, 15, 31, 63, and in general (2^(I+1))-1.
Another 1D rule is denoted by the authors as the "M" or "maximum norm" rule. This rule is the same as the CC rule, except that it starts with the rule of order 3. This seemingly minor difference forces this rule to use many more points than the other rules in the multidimensional case. The difference is evident in 2 dimensions, and quickly overwhelming even in dimensions as low as 4!
Andreas Klimke,
Universitaet Stuttgart,
Stuttgart, Germany.
SPARSE GRID INTERPOLATION TOOLBOX - LICENSE
Copyright (c) 2006 W. Andreas Klimke, Universitaet Stuttgart. Copyright (c) 2007-2008 W. A. Klimke. All Rights Reserved. All Rights Reserved.
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RBF_INTERP, a MATLAB library which defines and evaluates radial basis interpolants to multidimensional data.
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_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_INTERPOLANT a MATLAB library which can be used to define a sparse interpolant to a function f(x) of a multidimensional argument.
SPINTERP_EXAMPLES, a MATLAB library which demonstrates some simple uses of the spinterp program, which uses sparse grids for interpolation, optimization, and quadrature in higher dimensions.
TEST_INTERP_FUN_2D, a MATLAB library which defines a number of test problems for interpolation in 2D, provided as functions v = f(x,y).
TOMS847, a MATLAB library which carries out piecewise multilinear hierarchical sparse grid interpolation; this library is commonly called SPINTERP (version 2.1); this is ACM TOMS Algorithm 847, by Andreas Klimke;
You can go up one level to the MATLAB source codes.