Quasi-Monte Carlo algorithms for unbounded, weighted integration problems

Authors:
Jürgen Hartinger;Reinhold F. Kainhofer;Robert F. Tichy
Affiliations:
Department of Mathematics, Graz University of Technology, Steyrergasse 30, Graz A-8010, Austria;Department of Mathematics, Graz University of Technology, Steyrergasse 30, Graz A-8010, Austria;Department of Mathematics, Graz University of Technology, Steyrergasse 30, Graz A-8010, Austria
Venue:
Journal of Complexity
Year:
2004

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Abstract

In this article we investigate quasi-Monte Carlo (QMC) methods for multidimensional improper integrals with respect to a measure other than the uniform distribution. Additionally, the integrand is allowed to be unbounded at the lower boundary of the integration domain. We establish convergence of the QMC estimator to the value of the improper integral under conditions involving both the integrand and the sequence used. Furthermore, we suggest a modification of an approach proposed by Hlawka and Mück for the creation of low-discrepancy sequences with regard to a given density, which are suited for singular integrands.