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.
盛长滔，上海财经大学助理研究员，研究方向为偏微分方程数值方法，主要包括谱方法及其应用。2018 年于厦门大学获得博士学位，之后在新加坡南洋理工大学从事博士后研究。主持国家自然青年基金和上海市浦江人才计划，并获郭本瑜青年学者优秀论文奖。目前为止，在SIAM J. Numer. Anal., Math.Comp., ESAIM M2AN.等知名国内外期刊上发表SCI 论文10 余篇。