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BIT Numerical Mathematics

, Volume 53, Issue 1, pp 3–27 | Cite as

Multilevel Monte Carlo method for parabolic stochastic partial differential equations

  • Andrea Barth
  • Annika LangEmail author
  • Christoph Schwab
Article

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.

Keywords

Multilevel Monte Carlo Stochastic partial differential equations Stochastic Finite Element Methods Stochastic parabolic equation Multilevel approximations 

Mathematics Subject Classification (2010)

60H15 60H35 65C30 41A25 65C05 65N30 

Notes

Acknowledgements

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

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Copyright information

© Springer Science + Business Media B.V. 2012

Authors and Affiliations

  1. 1.Seminar für Angewandte MathematikETHZürichSwitzerland

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