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On Dropping the First Sobol’ Point

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Monte Carlo and Quasi-Monte Carlo Methods (MCQMC 2020)

Part of the book series: Springer Proceedings in Mathematics & Statistics ((PROMS,volume 387))

Abstract

Quasi-Monte Carlo (QMC) points are a substitute for plain Monte Carlo (MC) points that greatly improve integration accuracy under mild assumptions on the problem. Because QMC can give errors that are o(1/n) as \(n\rightarrow \infty \), and randomized versions can attain root mean squared errors that are o(1/n), changing even one point can change the estimate by an amount much larger than the error would have been and worsen the convergence rate. As a result, certain practices that fit quite naturally and intuitively with MC points can be very detrimental to QMC performance. These include thinning, burn-in, and taking sample sizes such as powers of 10, when the QMC points were designed for different sample sizes. This article looks at the effects of a common practice in which one skips the first point of a Sobol’ sequence. The retained points ordinarily fail to be a digital net and when scrambling is applied, skipping over the first point can increase the numerical error by a factor proportional to \(\sqrt{n}\) where n is the number of function evaluations used.

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Acknowledgements

This work was supported by the NSF under grant IIS-1837931, and a grant from Hitachi, Ltd. I thank Fred Hickernell, Pierre L’Ecuyer, Alex Keller, Max Balandat, Michael McCourt, Pamphile Roy and Sergei Kucherenko for stimulating discussions. I think Sifan Liu for catching an error in some code. Thanks to Mike Giles, Arnaud Doucet, Alex Keller and the whole team at ICMS for making MCQMC 2020 happen despite all the pandemic disruption. This paper benefited from comments of two anonymous reviewers. I thank Alex Keller for handling the proceedings volume.

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    Art B. Owen

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    Alexander Keller

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Owen, A.B. (2022). On Dropping the First Sobol’ Point. In: Keller, A. (eds) Monte Carlo and Quasi-Monte Carlo Methods. MCQMC 2020. Springer Proceedings in Mathematics & Statistics, vol 387. Springer, Cham. https://doi.org/10.1007/978-3-030-98319-2_4

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