Bounds and constructions for the star-discrepancy via δ-covers

  • Authors:

  • Benjamin Doerr;Michael Gnewuch;Anand Srivastav

  • Affiliations:

  • Mathematisches Seminar, Bereich II, Christian-Albrechts-Universität zu Kiel, Christian-Albrechts-Platz 4, 24098 Kiel, Germany;Mathematisches Seminar, Bereich II, Christian-Albrechts-Universität zu Kiel, Christian-Albrechts-Platz 4, 24098 Kiel, Germany;Mathematisches Seminar, Bereich II, Christian-Albrechts-Universität zu Kiel, Christian-Albrechts-Platz 4, 24098 Kiel, Germany

  • Venue:
  • Journal of Complexity
  • Year:
  • 2005

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Abstract

For numerical integration in higher dimensions, bounds for the star-discrepancy with polynomial dependence on the dimension d are desirable. Furthermore, it is still a great challenge to give construction methods for low-discrepancy point sets. In this paper, we give upper bounds for the star-discrepancy and its inverse for subsets of the d-dimensional unit cube. They improve known results. In particular, we determine the usually only implicitly given constants. The bounds are based on the construction of nearly optimal @d-covers of anchored boxes in the d-dimensional unit cube. We give an explicit construction of low-discrepancy points with a derandomized algorithm. The running time of the algorithm, which is exponentially in d, is discussed in detail and comparisons with other methods are given.