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Numerische Mathematik

, Volume 130, Issue 4, pp 579–613 | Cite as

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

  • Claudio Bierig
  • Alexey ChernovEmail author
Article

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.

Mathematics Subject Classification

65N30 65C05 65C30 65N55 65K15 

Notes

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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Copyright information

© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  1. 1.Department of Mathematics and StatisticsUniversity of ReadingReadingUK

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