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Convergence analysis of multilevel Monte Carlo variance estimators and application for random obstacle problems

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Abstract

We develop a novel convergence theory for the multilevel sample variance estimators in the framework of the multilevel Monte Carlo methods. We prove that, dependent on the regularity of the quantity of interest, the multilevel sample variance estimator may achieve the same asymptotic cost/error relation as the multilevel sample mean, which is superior to the standard Monte Carlo method. Weaker regularity assumptions result in reduced convergence rates, quantified in our analysis. The general convergence theory is applied to a class of scalar elliptic obstacle problems with rough random obstacle profiles, which is a simple model of contact between a deformable body with a rough uncertain substrate. Numerical experiments confirm theoretical convergence proofs.

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Acknowledgments

The authors thank Prof. Christoph Schwab, Dr. Annika Lang and Jonas Šukys (ETH Zürich) for the discussion on unbiased variance estimators, and Prof. Rolf Krause (USI, Lugano) for mentioning the reference [23].

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Authors and Affiliations

  1. Department of Mathematics and Statistics, University of Reading, Whiteknights Campus, PO Box 220, Reading, RG6 6AX, UK

    Claudio Bierig & Alexey Chernov

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  1. Claudio Bierig
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  2. Alexey Chernov
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Correspondence to Alexey Chernov.

Additional information

The major part of the research presented in this paper has been carried out at the Hausdorff Center of Mathematics, University of Bonn, Germany. The authors acknowledge support by HCM and by the University of Reading.

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Bierig, C., Chernov, A. Convergence analysis of multilevel Monte Carlo variance estimators and application for random obstacle problems. Numer. Math. 130, 579–613 (2015). https://doi.org/10.1007/s00211-014-0676-3

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  • DOI: https://doi.org/10.1007/s00211-014-0676-3

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