Unbiased `walk-on-spheres' Monte Carlo methods for the fractional Laplacian

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Abstract

We consider Monte Carlo methods for simulating solutions to the analogue of the Dirichlet boundary value problem in which the Laplacian is replaced by the fractional Laplacian and boundary conditions are replaced by conditions on the exterior of the domain. Specifically, we consider the analogue of the so-called ewalk-on-spheresf algorithm. In the diffusive setting, this entails sampling the path of Brownian motion as it uniformly exits a sequence of spheres maximally inscribed in the domain. As this algorithm would otherwise never end, it is truncated when the ewalk-on-spheresf comes within ϵ 0 of the boundary. In the setting of the fractional Laplacian, the role of Brownian motion is replaced by an isotropic α-stable process with α ∈ (0, 2). A significant difference to the Brownian setting is that the stable processes will exit spheres by a jump rather than hitting their boundary. This difference ensures that disconnected domains may be considered and that, unlike the diffusive setting, the algorithm ends after an almost surely finite number of steps.

Original languageEnglish
Pages (from-to)1550-1578
Number of pages29
JournalIMA Journal of Numerical Analysis
Volume38
Issue number3
Early online date21 Aug 2017
DOIs
Publication statusPublished - 1 Jul 2018

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Keywords

  • Lévy processes
  • Monte Carlo methods
  • Walk-on-spheres
  • fractional Laplacian

ASJC Scopus subject areas

  • Mathematics(all)
  • Computational Mathematics
  • Applied Mathematics

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