Where are the Logs?

Art B. Owen () and Zexin Pan ()
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Art B. Owen: Stanford University
Zexin Pan: Stanford University

A chapter in Advances in Modeling and Simulation, 2022, pp 381-400 from Springer

Abstract: Abstract The commonly quoted error rates for QMC integration with an infinite low discrepancy sequence is $$O(n^{-1}\log (n)^r)$$ O ( n - 1 log ( n ) r ) with $$r=d$$ r = d for extensible sequences and $$r=d-1$$ r = d - 1 otherwise. Such rates hold uniformly over all d dimensional integrands of Hardy-Krause variation one when using n evaluation points. Implicit in those bounds is that for any sequence of QMC points, the integrand can be chosen to depend on n. In this paper we show that rates with any $$r 1$$ r > 1 is needed. An example with $$d=3$$ d = 3 and n up to $$2^{100}$$ 2 100 might possibly require $$r>1$$ r > 1 .

Keywords: Discrepancy; Koksma-Hlawka inequality; Quasi-Monte Carlo; Trojan’s theorem (search for similar items in EconPapers)
Date: 2022
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Persistent link: https://EconPapers.repec.org/RePEc:spr:sprchp:978-3-031-10193-9_19

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DOI: 10.1007/978-3-031-10193-9_19

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