Abstract
The commonly quoted error rates for QMC integration with an infinite low discrepancy sequence is \(O(n^{-1}\log (n)^r)\) with \(r=d\) for extensible sequences and \(r=d-1\) 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 \(r<(d-1)/2\) can hold when f is held fixed as \(n\rightarrow \infty \). This is accomplished following a suggestion of Erich Novak to use some unpublished results of Trojan from the 1980s as given in the information based complexity monograph of Traub, Wasilkowski and Woźniakowski. The proof is made by applying a technique of Roth with the theorem of Trojan. The proof is non constructive and we do not know of any integrand of bounded variation in the sense of Hardy and Krause for which the QMC error exceeds \((\log n)^{1+\epsilon }/n\) for infinitely many n when using a digital sequence such as one of Sobol’s. An empirical search when \(d=2\) for integrands designed to exploit known weaknesses in certain point sets showed no evidence that \(r>1\) is needed. An example with \(d=3\) and n up to \(2^{100}\) might possibly require \(r>1\).
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Acknowledgements
This paper is dedicated to Pierre L’Ecuyer on the occasion of his 70th birthday. Pierre has made singular contributions to Monte Carlo and quasi-Monte Carlo. His influence goes well beyond establishing theoretical results by also providing computational tools, definitive survey articles, and papers with exemplary applications, on top of service to those fields through editorial work and conference organization.
This work was supported by the U.S. NSF under grant IIS-1837931. Thanks to Fred Hickernell and Erich Novak for discussions related to this problem. We are also grateful to Traub, Wasilkowski and Woźniakowski for ensuring that Trojan’s work was not lost. Finally we thank two reviewers for their helpful comments. Finally, we thank the festschrift organizers, Bruno Tuffin, Christiane Lemieux, Alex Keller and Zdravko Botev for organizing this project.
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Owen, A.B., Pan, Z. (2022). Where are the Logs?. In: Botev, Z., Keller, A., Lemieux, C., Tuffin, B. (eds) Advances in Modeling and Simulation. Springer, Cham. https://doi.org/10.1007/978-3-031-10193-9_19
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