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Weighted Discrepancy and High-Dimensional Numerical Integration

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Abstract

The concept of weighted discrepancy of sequences was introduced by Sloan and Woźniakowski 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.

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Authors and Affiliations

  1. Institut für Analysis, Universität Linz, Altenbergerstraße 69, A-4040, Linz, Austria

    G. Larcher, F. Pillichshammer & K. Scheicher

Authors
  1. G. Larcher
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  2. F. Pillichshammer
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  3. K. Scheicher
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Larcher, G., Pillichshammer, F. & Scheicher, K. Weighted Discrepancy and High-Dimensional Numerical Integration. BIT Numerical Mathematics 43, 123–137 (2003). https://doi.org/10.1023/A:1023605123264

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