Component-by-component construction of low-discrepancy point sets of small size

Doerr Benjamin, Gnewuch Michael, Kritzer Peter and Pillichshammer Friedrich
Additional contact information
Doerr Benjamin: Max-Planck-Institut für Informatik, Stuhlsatzenhausweg 85, 66123 Saarbrücken, Germany.
Gnewuch Michael: Department of Computer Science, Christian-Albrechts-Universität Kiel, Christian-Albrechts-Platz 4, 24098 Kiel, Germany. Email: mig@informatik.uni-kiel.de
Kritzer Peter: Department of Mathematics, University of Salzburg, Hellbrunnerstr. 34, 5020 Salzburg, Austria, and School of Mathematics and Statistics, University of New South Wales, Sydney, NSW, Australia. Email: p.kritzer@unsw.edu.au
Pillichshammer Friedrich: Institut für Finanzmathematik, Universität Linz, Altenbergerstr. 69, 4040 Linz, Austria. Email: friedrich.pillichshammer@jku.at

Monte Carlo Methods and Applications, 2008, vol. 14, issue 2, 129-149

Abstract: We investigate the problem of constructing small point sets with low star discrepancy in the s-dimensional unit cube. The size of the point set shall always be polynomial in the dimension s. Our particular focus is on extending the dimension of a given low-discrepancy point set.This results in a deterministic algorithm that constructs N-point sets with small discrepancy in a component-by-component fashion. The algorithm also provides the exact star discrepancy of the output set. Its run-time considerably improves on the run-times of the currently known deterministic algorithms that generate low-discrepancy point sets of comparable quality.We also study infinite sequences of points with infinitely many components such that all initial subsegments projected down to all finite dimensions have low discrepancy. To this end, we introduce the inverse of the star discrepancy of such a sequence, and derive upper bounds for it as well as for the star discrepancy of the projections of finite subsequences with explicitly given constants. In particular, we establish the existence of sequences whose inverse of the star discrepancy depends linearly on the dimension.

Keywords: Star discrepancy; multivariate integration; derandomization (search for similar items in EconPapers)
Date: 2008
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DOI: 10.1515/MCMA.2008.007

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