Solving PDEs on manifolds represented as point clouds and applications

Abstract

Abstract This paper serves as a review of our series work of solving partial differential equations (PDEs) on manifolds represented as point clouds and using their solutions to conduct geometric understanding of point clouds. We first review our two systemic methods of discretizing PDEs including the moving least square method and the local mesh method. These methods of approximating differential operator on manifold-structured point clouds are based only on local approximation using nearest neighbours and achieve high order numerical convergence of the desired equations including diffusion and nondiffusion types of equations. We further discuss extensions of these methods to approximate the committor function for understanding dynamic systems and to solve PDEs on manifolds represented as incomplete inter-point distance information by combining with low-rank matrix completion theory. As direct applications, we use solutions of PDEs on manifold-structured point clouds as bridges to link local and global information. With this strategy, we discuss a few key applications essential to geometric understanding for point clouds, including skeletons extraction from point clouds, the conformal structures construction from point clouds, and nonrigid intrinsic registration among point clouds.