2018/03/11 by Tony Shardlow, Shardlow, Tony
Mathematics · Physics and Astronomy · #Mathematical Approximation and Integration #Stochastic processes and statistical mechanics #Theoretical and Computational Physics
paper · pdf · doi:10.48550/arxiv.1803.03921
The Feynman-Kac formula for the exterior-value problem for the fractional\nLaplacian leads to a walk-outside-spheres algorithm via sampling alpha-stable\nLevy processes on their exit from maximally inscribed balls and sampling their\noccupation distribution. Kyprianou, Osojnik, and Shardlow (2017) developed this\nalgorithm, providing a complexity analysis and an implementation, for\napproximating the solution at a single point in the domain. This paper shows\nhow to efficiently sample the whole field by generating an approximation in\nL2(D), for a domain D . The method takes advantage of a hierarchy of\ntriangular meshes and uses the multilevel Monte Carlo method for Hilbert\nspace-valued quantities of interest. We derive complexity bounds in terms of\nthe fractional parameter alpha and demonstrate that the method gives accurate\nresults for two problems with exact solutions. Finally, we show how to couple\nthe method with the variable-accuracy Arnoldi iteration to compute the smallest\neigenvalue of the fractional Laplacian. A criteria is derived for the variable\naccuracy and a comparison is given with analytical results of Dyda (2012).\n