19th GAMM-Seminar Leipzig on
High-dimensional problems - Numerical treatment and applications

Max-Planck-Institute for Mathematics in the Sciences
Inselstr. 22-26, D-04103 [O->]Leipzig
Phone: +49.341.9959.752, Fax: +49.341.9959.999

  19th GAMM-Seminar
January, 23th-25th, 2003
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  Abstract Thomas Gerstner, Fri, 17.00-17.25 Previous Contents Next  
  A dimension-adaptive approach to multivariate numerical integration
Thomas Gerstner (Uni Bonn)

We consider the numerical integration of multivariate functions defined over the unit hypercube. Here, we especially address the high-dimensional case, where in general the curse of dimension is encountered. Due to the concentration of measure phenomenon, such functions can often be well approximated by sums of lower-dimensional terms. The problem, however, is to find a good expansion given little knowledge of the integrand itself.

The dimension-adaptive quadrature method which is presented here aims to find such an expansion automatically. It is based on the sparse grid method which has shown to give good results for low- and moderate-dimensional problems. The dimension-adaptive quadrature method tries to find important dimensions and adaptively refines in this respect guided by suitable error estimators. This leads to an approach which is based on generalized sparse grid index sets. We propose efficient data structures for the storage and traversal of the index sets and discuss an efficient implementation of the algorithm.

The performance of the method is illustrated by several numerical examples from computational physics and finance where dimension reduction is obtained from the Brownian bridge discretization of the underlying stochastic process.

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[O->]Jens Burmeister (Uni Kiel), Kai Helms (MPI Leipzig)
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