Radial Basis Function-Finite Difference Method for Solving PDEs and Eigenvalue Problems on Riemannian Manifolds
题目：Radial Basis Function-Finite Difference Method for Solving PDEs and Eigenvalue Problems on Riemannian Manifolds
地点：海纳苑2幢204摘要：In this talk, we will first discuss the Radial Basis Function (RBF) approximation to differential operators on smooth tensor fields defined on Riemannian submanifolds of Euclidean space, identified by randomly sampled point cloud data. The formulation in this work leverages a fundamental fact that the covariant derivative on a submanifold is the projection of the directional derivative in the ambient Euclidean space onto the tangent space of the submanifold. To differentiate a test function (or vector field) on the submanifold with respect to the Euclidean metric, the RBF interpolation is applied to extend the function (or vector field) in the ambient Euclidean space. Theoretically, we establish the convergence of the eigenpairs of both the Laplace-Beltrami operator and Bochner Laplacian in the limit of large data with convergence rates. Numerically, we provide supporting examples for approximations of the Laplace-Beltrami operator and various vector Laplacians, including the Bochner, Hodge, and Lichnerowicz Laplacians. Next, since the RBF Laplacian matrix is dense, the standard RBF approach for solving PDEs and eigenvalue problems is often computational expensive. To overcome this issue, we will talk about a RBF finite-difference type method for approximating the Laplacian operators using sparse matrices. Finally, we will solve PDEs involving the Laplace-Beltrami operator and supporting numerical examples are provided.
报告人简介：蒋诗晓，博士，现任上海科技大学助理教授/研究员，博士毕业于上海交通大学，后在美国宾州州立大学从事博士后研究，目前研究兴趣包括微分流形上的无网格方法、数值偏微分方程、流形学习，模型降维、微分方程参数估计等，在CPAM, J. Comput. Phys., J. Scientific Comput., J. Fluid Mech., New J. Phys.等期刊发表论文。