A walk outside spheres for the fractional Laplacian: fields and first eigenvalue
For researchers in computational fractional PDEs, this provides an efficient field approximation and eigenvalue solver, though it is an incremental extension of existing algorithms.
This paper extends a walk-outside-spheres algorithm for the fractional Laplacian to efficiently approximate the whole field in L_2(D) using multilevel Monte Carlo on triangular meshes, and couples it with Arnoldi iteration to compute the smallest eigenvalue. The method achieves accurate results on test problems and matches analytical eigenvalue results.
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).