Tusnády’s problem, the transference principle, and non-uniform QMC sampling

Christoph Aistleitner*, Dmitriy Bilyk, Aleksandar Nikolov

*Corresponding author for this work

Research output: Chapter in Book/Report/Conference proceedingConference paperpeer-review

Abstract

It is well-known that for every N≥ 1 and d≥ 1 there exist point sets x1, ⋯, xN∈ [0, 1]d whose discrepancy with respect to the Lebesgue measure is of order at most (log N)d - 1N-1. In a more general setting, the first author proved together with Josef Dick that for any normalized measure μ on [0, 1 ]d there exist points x1, ⋯, xN whose discrepancy with respect to μ is of order at most (log N)( 3 d + 1 ) / 2N- 1. The proof used methods from combinatorial mathematics, and in particular a result of Banaszczyk on balancings of vectors. In the present note we use a version of the so-called transference principle together with recent results on the discrepancy of red-blue colorings to show that for any μ there even exist points having discrepancy of order at most (logN)d-12N-1, which is almost as good as the discrepancy bound in the case of the Lebesgue measure.

Original languageEnglish
Title of host publicationMonte Carlo and Quasi-Monte Carlo Methods - MCQMC 2016
PublisherSpringer New York LLC
Pages169-180
Number of pages12
Volume241
ISBN (Print)9783319914350
DOIs
Publication statusPublished - 1 Jan 2018
Event12th International Conference on Monte Carlo and Quasi-Monte Carlo Methods in Scientific Computing, MCQMC 2016 - Stanford, United States
Duration: 14 Aug 201619 Aug 2016

Conference

Conference12th International Conference on Monte Carlo and Quasi-Monte Carlo Methods in Scientific Computing, MCQMC 2016
Country/TerritoryUnited States
CityStanford
Period14/08/1619/08/16

Keywords

  • combinatorial discrepancy
  • Gates of Hell
  • Low-discrepancy sequences
  • Non-uniform sampling
  • Tusnády’s problem

ASJC Scopus subject areas

  • Mathematics(all)

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