A walk outside spheres for the fractional Laplacian: fields and first eigenvalue

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Abstract

The solution of the exterior-value problem for the fractional Laplacian can be computed by a Walk Outside Spheres algorithm. This involves sampling α-stable Levy processes on their exit from maximally inscribed balls and sampling their occupation distribution. Kyprianou, Osojnik, and Shardlow (2018) developed this algorithm, providing a complexity analysis and an implementation, for approximating the solution at a single point in the domain. This paper shows how to efficiently sample the whole field by generating an approximation in L 2(D) for a domain D. The method takes advantage of a hierarchy of triangular meshes and uses the multilevel Monte Carlo method for Hilbert space-valued quantities of interest. We derive complexity bounds in terms of the fractional parameter α and demonstrate that the method gives accurate results for two problems with exact solutions. Finally, we show how to couple the method with the variable-accuracy Arnoldi iteration to compute the smallest eigenvalue of the fractional Laplacian. A criteria is derived for the variable accuracy and a comparison is given with analytical results of Dyda (2012).

Original languageEnglish
Pages (from-to)2767-2792
Number of pages26
JournalMathematics of Computation (MCOM)
Volume88
Issue number320
DOIs
Publication statusPublished - 14 Mar 2019

Keywords

  • math.NA
  • Arnoldi algorithm
  • Eigenvalue problems
  • Exterior-value problems
  • Levy processes
  • Multilevel Monte Carlo
  • Numerical solution of PDEs
  • Fractional Laplacian
  • Walk on spheres

ASJC Scopus subject areas

  • Computational Mathematics
  • Applied Mathematics
  • Algebra and Number Theory

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