Efficient Monte Carlo Methods for fractional PDEs in High Dimensions

In this talk, we introduce the Monte Carlo methods for solving PDEs involving an integral fractional Laplacian (IFL) in high dimensions. We first construct the Feynman-Kac representation based on the Green function for the IFL 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.

报告人介绍： 盛长滔，上海财经大学数学学院助理研究员。2018年于厦门大学获得理学博士学位，之后在新加坡南洋理工大学从事博士后研究。主要研究方向为谱方法和谱元法以其应用、奇性问题的高精度数值方法、高维偏微分方程的随机算法等。目前，主持国家自然科学青年基金和上海市浦江人才计划，并荣获“2021年郭本瑜青年学者优秀论文 奖”一等奖。