# TSG TASMANIAN Sparse Grid Library

TSG MATLAB examples and an interface library which demonstrate the use of the TasmanianSparseGrid (TSG) package, which implements routines for working with sparse grids, to efficiently estimate integrals or compute interpolants of scalar functions of multidimensional arguments.

TSG can create three kinds of rules:

• Global grids, for globally smooth functions;
• Local polynomial grids, a piecewise polynomial approach for non-smooth functions
• Wavelet grids, also for non-smooth functions, but this option can sometimes achieve better accuracy with fewer points.

TSG is a family of algorithms for constructing multidimensional quadrature and interpolation rules from tensor products of 1-dimensional versions of such rules.

TSG includes a large variety of 1-dimensional rules, which are generally defined, by default, over the interval [-1,+1]. In that case, the multidimensional quadrature and interpolation rules will be defined over the hypercube [-1,+1]^d, but a simple linear transformation can be carried out by TSG to redefine the linearly redefine the region.

The one-dimensional rules that can be used include:

• Chebyshev;
• Clenshaw Curtis;
• Fejer Type 2;
• Gauss Chebyshev Type 1;
• Gauss Chebyshev Type 2;
• Gauss Gegenbauer;
• Gauss Hermite;
• Gauss Jacobi;
• Gauss Laguerre;
• Gauss Legendre;
• Local polynomial;
• Local polynomial zero;
• Local wavelet.

The MATLAB programs must actually work with the compiled C++ version of the TasmanianSparseGrid library. They do this by interfacing with a compiled C++ program called tasg, communicating via the MATLAB system() command, with data passed by writing and reading files from a workfile directory whose location is specified by the user.

### Languages:

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

### Related Programs:

SGMGA, a MATLAB library which creates sparse grids based on a mixture of 1D quadrature rules, allowing anisotropic weights for each dimension.

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

SPINTERP, a MATLAB library which carries out piecewise multilinear hierarchical sparse grid interpolation; an earlier version of this software is ACM TOMS Algorithm 847, by Andreas Klimke;

SPQUAD, a MATLAB library which computes the points and weights of a sparse grid quadrature rule for an M-dimensional integral, based on the Clenshaw-Curtis quadrature rule, by Greg von Winckel.

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;

### Author:

Miroslav Stoyanov, Oak Ridge National Laboratory, mkstoyanov@gmail.com.

### References:

1. Miro Stoyanov,
User Manual: TASMANIAN Sparse Grids,
ORNL Report,
Oak Ridge National Laboratory, August 2013.

### Examples and Tests:

• tsg_test.m, the calling program;
• tsg_test01.m, estimates an integral in 2D using a Clenshaw-Curtis sparse grid.
• tsg_test02.m, interpolates a function in 2D using a Clenshaw-Curtis sparse grid.
• tsg_test03.m, uses a Gauss-Legendre sparse grid to integrate a function in 2D.
• tsg_test04.m, uses an anisotropic Gauss-Gegenbauer sparse grid for integration.
• tsg_test06.m, generates 2D Clenshaw Curtis grids for various levels, and compares the sizes of "hyperbolic", "level", and "basis" growth options.
• timestamp.m, prints the YMDHMS date as a timestamp.
• tsg_test_output.txt, the output file.

### Source Code:

• tsgDeleteGrid.m clears all data from the workfile directory.
• tsgDeleteGridByName.m clears data associated with a particular grid from the workfile directory.
• tsgEvaluate.m reads a grid from a file, and reads a list of evaluation points from a matrix file. The interpolant is evaluated at the evaluation points, and the results are written to a matrix file.
• tsgExample.m, contains five example calculations.
• tsgGetInterpolationWeights.m reads a grid from a file that was created previously, and returns the interpolation weights.
• tsgGetNeededPoints.m reads a grid from a file and extracts the grid abscissas that are not yet associated with values of the interpolated function.
• tsgGetPaths.m stores the location of the tsg() executable and the user's TSG workfile directory.
• tsgGetPoints.m reads a grid from a file that was created previously, and returns the abscissas of the associated quadrature rule.
• tsgGetQuadrature.m reads a grid from a file that was created previously, and computes the associated quadrature rule.
• tsgIntegrate.m reads a grid from a file and integrates the interpolant over the domain, returning the value of the integral.
• tsgListGridsByName.m lists the grids for which data is currently stored in the workfile directory.
• tsgLoadValues.m updates a grid from a file that was created previously, by reading the values of the interpolated function from a matrix file, and adding them to the grid file.
• tsgMakeFilenames.m, is given the paths defined by tsgGetPaths() and returns the names of all the files associated with a given grid.
• tsgMakeGrid.m creates information defining a particular grid.
• tsgMakeQuadrature.m returns the abscissas and weights of a particular sparse grid rule.
• tsgPlotPoints2D.m plots a set of 2D points.
• tsgPrintStats.m reads a grid from a file and prints out a short summary of grid statistics.