Title: Radial Basis Function Finite Difference Method for Solving PDEs and Eigenvalue Problems on Riemannian Manifolds

Speaker: Jiang Shixiao (ShanghaiTech University)

Time: June 27, 2023, 14:00-16:00

Location: 204, Building 2, Hainayuan

Abstract: 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.

Brief introduction to the speaker: Jiang Shixiao, Ph. D., currently an assistant professor/researcher of ShanghaiTech University, graduated from Shanghai Jiaotong University, and later engaged in postdoctoral research in Pennsylvania State University. Currently, his research interests include meshless methods on Differentiable manifold, numerical partial differential equations, manifold learning, model dimension reduction, parameter estimation of differential equations, etc., in CPAM, J. Comput Phys, J. Scientific Compute, J. Fluid Mech, Published papers in journals such as New J. Phys.