A walk outside spheres for the fractional Laplacian

fields and first eigenvalue

Research output: Contribution to journalArticle

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Abstract

The Feynman-Kac formula for the exterior-value problem for the fractional Laplacian leads to a walk-outside-spheres algorithm via sampling alpha-stable Levy processes on their exit from maximally inscribed balls and sampling their occupation distribution. Kyprianou, Osojnik, and Shardlow (2017) 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 alpha 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
Early online date14 Mar 2019
DOIs
Publication statusE-pub ahead of print - 14 Mar 2019

Keywords

  • math.NA

Cite this

A walk outside spheres for the fractional Laplacian : fields and first eigenvalue. / Shardlow, Tony.

In: Mathematics of Computation (MCOM), Vol. 88, No. 320, 31.07.2030, p. 2767-2792.

Research output: Contribution to journalArticle

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