Speaker：Changtao Sheng（Shanghai University of Finance and Economics）

Time：2022-10-27, 10:00

Location：Tencent Meeting ID：236-388-485（No Password）

Abstract:

In this talk, we introduce the Monte Carlo methods for solving PDEs involving an integral fractional Laplacian (IFL) in multiple dimensions. We first construct a new Feynman-Kac representation based on the Green function for the fractional Laplacian operator on the unit ball in arbitrary dimensions. Inspired by the ``walk-on-spheres" algorithm proposed in [Kyprianou, Osojnik, and Shardlow, IMA J. Numer. Anal.(2018)], we extend our algorithm for solving fractional Poisson equations in the complex domain. Then, we can compute the expectation of a multi-dimensional random variable with a known density function to obtain the numerical solution efficiently. The proposed algorithm finds it remarkably efficient in solving fractional PDEs: it only needs to evaluate the integrals of expectation form over a series of inside ball tangent boundaries with the known Green function. Moreover, we carry out the error estimates of the proposed method for the d-dimensional unit ball. Ample numerical results are presented to demonstrate the robustness and effectiveness of the proposed method. Finally, we extended the proposed algorithm to solve space-fractional diffusion equations in high dimensions.