Numerical integration

• R. Cools
• A. Haegemans
• B. Maerten

The approximation of multivariate integrals is part of many large scale computer applications, e.g., mathematical models in finance, physics and chemistry, computer graphics, finite element applications, parameter estimation using Bayesian statistics.

The project CUBPACK aims to develop a package with cubature software. The goal of the project is to develop software for the automatic computation of multi-dimensional integrals over several regions. Subproblems that require attention are, e.g., the construction of integration rules (cubature formulae) and realistic error estimators, subdivision strategies, detection of special behaviour of the integrand and methods to deal with this and the development of alternative strategies for parallel architectures.

Recently we

• researched alternative subdivision strategies for n-dimensional simplices,
• studied the behaviour of integration rules constructed using Smolyak's technique starting from a variety of 1-dimensional quadrature rules,
• implemented extrapolation techniques for multidimensional regions,
• implemented an interface to Cubpack that does no longer require a user to compile his integrand function.

At the moment the Fortran 90 version (joint work with Prof. A. Genz, Washington State University) concentrates on n-dimensional simplices and hyperrectangles. The C++ version (joint work with Prof. D. Laurie, University of Potchefstroom) concentrates on all types of 2-dimensional regions.