报告题目 Title：GMLS and RBF-FD Methods for Solving PDEs on Manifolds

报告人 Speaker：蒋诗晓

报告人所在单位 Affiliation：上海科技大学

时间 Time：2026-09-24 15:00-16:00

地点 Venue：Room 1513, Guanghua Eastern Main Tower, Fudan University (Handan Campus)

报告摘要 Abstract：Solving partial differential equations (PDEs) on manifolds is a challenging problem in scientific computing and has broad applications in various fields. In this talk, we introduce two meshfree approaches, generalized moving least-squares (GMLS) and radial basis function-generated finite difference (RBF-FD), for solving PDEs on manifolds with or without boundaries, identified by randomly sampled point cloud data. For RBF-FD, we develop a novel two-step generalized RBF-FD (gRBF-FD) method based on a PHS+Poly interpolant defined over the tangent space in a Monge coordinate system, where PHS stands for polyharmonic spline. Our gRBF-FD method shares the same interpolant form as the standard RBF-FD but differs in how the interpolation coefficients are computed. To enhance stability and reduce the solution error, we employ a specific weight function in both GMLS and gRBF-FD. We establish an error bound for the operator approximation in terms of the so-called local stencil diameter as well as the number of data points. We further demonstrate the accuracy of the two approaches on three tasks, including manifold approximation, scalar-valued PDEs, and vector-valued PDEs, across various manifolds.

个人简介 Bio：蒋诗晓，博士，现任上海科技大学研究员/助理教授/博士生导师，博士毕业于上海交通大学，博士期间在纽约科朗所访问，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SISC, CPAM, JCP, ACHA, JMLR, JSC, JFM, IP, New J. Phys.等期刊发表论文。

海报 Poster:蒋诗晓 学术报告.jpg