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A Walk Outside Spheres for the fractional Laplacian: Fields and first eigenvalue

Author: Tony Shardlow
Journal: Math. Comp. 88 (2019), 2767-2792
MSC (2010): Primary 65C05, 34A08, 60J75, 34B09
Published electronically: March 14, 2019
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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 $ \alpha $-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 $ \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).

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Tony Shardlow
Affiliation: Department of Mathematical Sciences, University of Bath, Bath BA2 7AY, United Kingdom

Keywords: Fractional Laplacian, walk on spheres, Levy processes, Arnoldi algorithm, exterior-value problems, multilevel Monte Carlo, numerical solution of PDEs, eigenvalue problems
Received by editor(s): April 25, 2018
Received by editor(s) in revised form: November 28, 2018
Published electronically: March 14, 2019
Article copyright: © Copyright 2019 American Mathematical Society