BIT Numerical Mathematics

, Volume 55, Issue 4, pp 1125–1143 | Cite as

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

  • Stefan PauliEmail author
  • Robert Nicholas Gantner
  • Peter Arbenz
  • Andreas Adelmann


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.


Multilevel Monte Carlo Feynman–Kac Walk on Spheres  Laplace equation 

Mathematics Subject Classification

60H30 65C05 65N99 



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


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Copyright information

© Springer Science+Business Media Dordrecht 2015

Authors and Affiliations

  • Stefan Pauli
    • 1
    Email author
  • Robert Nicholas Gantner
    • 2
    • 3
  • Peter Arbenz
    • 2
  • Andreas Adelmann
    • 4
  1. 1.Computer Science Department and Seminar for Applied MathematicsETH ZurichZurichSwitzerland
  2. 2.Computer Science DepartmentETH ZurichZurichSwitzerland
  3. 3.Seminar for Applied MathematicsETH ZurichZurichSwitzerland
  4. 4.Large Research Facilities (GFA), Paul Scherrer Institute (PSI)VilligenSwitzerland

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