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

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).