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

TL;DR

This paper introduces an efficient multilevel Monte Carlo method to compute the fractional Laplacian's field and first eigenvalue, leveraging hierarchical meshes and stable Levy process sampling for improved accuracy and computational complexity.

Contribution

It extends the walk-outside-spheres algorithm to approximate the entire fractional Laplacian field and eigenvalues using multilevel Monte Carlo and hierarchical meshes.

Findings

Accurate field approximation demonstrated on test problems.

Complexity bounds derived based on fractional parameter alpha.

Coupling with Arnoldi iteration effectively computes the smallest 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).