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A Note on the Importance of Weak Convergence Rates for SPDE Approximations in Multilevel Monte Carlo Schemes

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Monte Carlo and Quasi-Monte Carlo Methods

Part of the book series: Springer Proceedings in Mathematics & Statistics ((PROMS,volume 163))

Abstract

It is a well-known rule of thumb that approximations of stochastic partial differential equations have essentially twice the order of weak convergence compared to the corresponding order of strong convergence. This is already known for many approximations of stochastic (ordinary) differential equations while it is recent research for stochastic partial differential equations. In this note it is shown how the availability of weak convergence results influences the number of samples in multilevel Monte Carlo schemes and therefore reduces the computational complexity of these schemes for a given accuracy of the approximations.

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Acknowledgments

This research was supported in part by the Knut and Alice Wallenberg foundation as well as the Swedish Research Council under Reg. No. 621-2014-3995. The author thanks Lukas Herrmann, Andreas Petersson, and two anonymous referees for helpful comments.

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Authors and Affiliations

  1. Department of Mathematical Sciences, Chalmers University of Technology and University of Gothenburg, SE-412 96, Gothenburg, Sweden

    Annika Lang

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  1. Annika Lang
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Correspondence to Annika Lang .

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Editors and Affiliations

  1. Department of Computer Science, KU Leuven, Heverlee, Belgium

    Ronald Cools

  2. Department of Computer Science, KU Leuven, Heverlee, Belgium

    Dirk Nuyens

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Lang, A. (2016). A Note on the Importance of Weak Convergence Rates for SPDE Approximations in Multilevel Monte Carlo Schemes. In: Cools, R., Nuyens, D. (eds) Monte Carlo and Quasi-Monte Carlo Methods. Springer Proceedings in Mathematics & Statistics, vol 163. Springer, Cham. https://doi.org/10.1007/978-3-319-33507-0_25

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