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Multilevel Monte Carlo for the Feynman–Kac formula for the Laplace equation

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Abstract

Since its formulation in the late 1940s, the Feynman–Kac formula has proven to be an effective tool for both theoretical reformulations and practical simulations of differential equations. The link it establishes between such equations and stochastic processes can be exploited to develop Monte Carlo sampling methods that are effective, especially in high dimensions. There exist many techniques of improving standard Monte Carlo sampling methods, a relatively new development being the so-called Multilevel Monte Carlo method. This paper investigates the applicability of multilevel ideas to the stochastic representation of partial differential equations by the Feynman–Kac formula, using the Walk on Spheres algorithm to generate the required random paths. We focus on the Laplace equation, the simplest elliptic PDE, while mentioning some extension possibilities.

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Acknowledgments

We thank the anonymous reviewers for their valuable comments and suggestions that improved the quality of the paper.

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Author notes

  1. Robert Nicholas Gantner

    Present address: Seminar for Applied Mathematics, ETH Zurich, Zurich, Switzerland

Authors and Affiliations

  1. Computer Science Department and Seminar for Applied Mathematics, ETH Zurich, Zurich, Switzerland

    Stefan Pauli

  2. Computer Science Department, ETH Zurich, Zurich, Switzerland

    Robert Nicholas Gantner & Peter Arbenz

  3. Large Research Facilities (GFA), Paul Scherrer Institute (PSI), Villigen, Switzerland

    Andreas Adelmann

Authors
  1. Stefan Pauli
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  2. Robert Nicholas Gantner
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  3. Peter Arbenz
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  4. Andreas Adelmann
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Corresponding author

Correspondence to Stefan Pauli.

Additional information

Communicated by Desmond Higham.

The work of Stefan Pauli has been funded by the ETH interdisciplinary research Grant CH1-03 10-1.

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Pauli, S., Gantner, R.N., Arbenz, P. et al. Multilevel Monte Carlo for the Feynman–Kac formula for the Laplace equation. Bit Numer Math 55, 1125–1143 (2015). https://doi.org/10.1007/s10543-014-0543-8

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  • DOI: https://doi.org/10.1007/s10543-014-0543-8

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