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A Brief Discussion of the Discrepancy Bounds

  • Gunther Leobacher
  • Friedrich Pillichshammer
Chapter
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Part of the Compact Textbooks in Mathematics book series (CTM)

Abstract

In many applications the dimension s can be rather large. In this case, the asymptotically almost optimal bounds on the discrepancy which we obtained, e.g., for the Hammersley point set or for (t, m, s)-nets soon become useless for a modest number N of points. For example, assume that for every \(s,N \in \mathbb{N}\) we have a point set \(\mathcal{P}_{s,N}\) in the s-dimensional unit cube of cardinality N with star discrepancy of at most
$$\displaystyle\begin{array}{rcl} D_{N}^{{\ast}}(\mathcal{P}_{ s,N}) \leq c_{s}\frac{(\log N)^{s-1}} {N},& &{}\end{array}$$
(6.1)
with some c s  > 0 that is independent of N.

Keywords

Variable Boundary Variable Stars Cardinality Polynomial Tractability Aistleitner 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer International Publishing Switzerland 2014

Authors and Affiliations

  • Gunther Leobacher
    • 1
  • Friedrich Pillichshammer
    • 2
  1. 1.Institute of Financial MathematicsUniversity of LinzLinzAustria
  2. 2.University of LinzLinzAustria

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