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

The solution of the exterior-value problem for the fractional Laplacian can be computed by a Walk Outside Spheres algorithm. This involves sampling α-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 α 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).