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

Christoph Aistleitner*, Dmitriy Bilyk, Aleksandar Nikolov

*Korrespondierende/r Autor/-in für diese Arbeit

Publikation: Beitrag in Buch/Bericht/KonferenzbandBeitrag in einem KonferenzbandBegutachtung

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.

Originalspracheenglisch
TitelMonte Carlo and Quasi-Monte Carlo Methods - MCQMC 2016
Herausgeber (Verlag)Springer New York LLC
Seiten169-180
Seitenumfang12
ISBN (Print)9783319914350
DOIs
PublikationsstatusVeröffentlicht - 2018
Veranstaltung12th International Conference on Monte Carlo and Quasi-Monte Carlo Methods in Scientific Computing: MCQMC 2016 - Stanford, USA / Vereinigte Staaten
Dauer: 14 Aug. 201619 Aug. 2016

Publikationsreihe

NameSpringer Proceedings in Mathematics & Statistics
Band241

Konferenz

Konferenz12th International Conference on Monte Carlo and Quasi-Monte Carlo Methods in Scientific Computing
KurztitelMCQMC 2016
Land/GebietUSA / Vereinigte Staaten
OrtStanford
Zeitraum14/08/1619/08/16

ASJC Scopus subject areas

  • Mathematik (insg.)

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