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Multilevel Monte Carlo method for parabolic stochastic partial differential equations

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Abstract

We analyze the convergence and complexity of multilevel Monte Carlo discretizations of a class of abstract stochastic, parabolic equations driven by square integrable martingales. We show under low regularity assumptions on the solution that the judicious combination of low order Galerkin discretizations in space and an Euler–Maruyama discretization in time yields mean square convergence of order one in space and of order 1/2 in time to the expected value of the mild solution. The complexity of the multilevel estimator is shown to scale log-linearly with respect to the corresponding work to generate a single path of the solution on the finest mesh, resp. of the corresponding deterministic parabolic problem on the finest mesh.

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Acknowledgements

This research was supported in part by the European Research Council under grant ERC AdG 247277.

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Authors and Affiliations

  1. Seminar für Angewandte Mathematik, ETH, Rämistrasse 101, 8092, Zürich, Switzerland

    Andrea Barth, Annika Lang & Christoph Schwab

Authors
  1. Andrea Barth
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  2. Annika Lang
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  3. Christoph Schwab
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Corresponding author

Correspondence to Annika Lang.

Additional information

Communicated by Desmond Higham.

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Barth, A., Lang, A. & Schwab, C. Multilevel Monte Carlo method for parabolic stochastic partial differential equations. Bit Numer Math 53, 3–27 (2013). https://doi.org/10.1007/s10543-012-0401-5

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  • DOI: https://doi.org/10.1007/s10543-012-0401-5

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