Multivariate quadrature on adaptive sparse grids

Authors:
Hans-Joachim Bungartz;Stefan Dirnstorfer
Affiliations:
IPVS, Universität Stuttgart, Universitätsstraße 38, D-70569 Stuttgart, Germany;Nagler & Company, D-92263 Schnaittenbach, Germany
Venue:
Computing
Year:
2003

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Abstract

In this paper, we study the potential of adaptive sparse grids for multivariate numerical quadrature in the moderate or high dimensional case, i.e. for a number of dimensions beyond three and up to several hundreds. There, conventional methods typically suffer from the curse of dimension or are unsatisfactory with respect to accuracy. Our sparse grid approach, based upon a direct higher order discretization on the sparse grid, overcomes this dilemma to some extent, and introduces additional flexibility with respect to both the order of the 1 D quadrature rule applied (in the sense of Smolyak's tensor product decomposition) and the placement of grid points. The presented algorithm is applied to some test problems and compared with other existing methods.