Weighted Discrepancy and High-Dimensional Numerical Integration

Authors: Larcher G.¹; Pillichshammer F.²; Scheicher K.³

Source: Bit Numerical Mathematics, Volume 43, Number 1, March 2003 , pp. 123-137(15)

Publisher: Springer

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Abstract:

The concept of weighted discrepancy of sequences was introduced by Sloan and Wo zacute niakowski when they proved a general form of a Koksma–Hlawka inequality for the numerical integration of functions. This version takes imbalances in the “importance” of the projections of the integrand into account.

In this paper we give estimates for the weighted discrepancy of several important point sets. Further we carry out various (high-dimensional) numerical integration experiments and we compare the results with the error bounds provided by the generalized Koksma–Hlawka inequality and by the estimates for the weighted discrepancy. Finally we discuss various consequences.

Keywords: Weighted discrepancy; numerical integration; Koksma–Hlawka inequality; Faure–Niederreiter net; Sobol–Niederreiter net

Language: English

Document Type: Research article

Affiliations: 1: Institut für Analysis, Universität Linz, Altenbergerstraße 69, A-4040 Linz, Austria. email: gerhard.larcher@jku.at 2: Institut für Analysis, Universität Linz, Altenbergerstraße 69, A-4040 Linz, Austria. email: friedrich.pillichshammer@jku.at 3: Institut für Analysis, Universität Linz, Altenbergerstraße 69, A-4040 Linz, Austria. email: klaus.scheicher@jku.at

Publication date: 2003-03-01