Title: Solving Partial Differential Equations on Manifolds From Incomplete Inter-Point Distance,
Abstract: Solutions of partial differential equations (PDEs) on manifolds have provided
important applications in different fields in science and engineering. Existing
methods are majorly based on discretization of manifolds as implicit functions,
triangle meshes, or point clouds, where the manifold structure is approximated
by either zero level set of an implicit function or a set of points. In many
applications, manifolds might be only provided as an inter-point distance
matrix with possible missing values. This paper discusses a framework to
discretize PDEs on manifolds represented as incomplete inter-point distance
information. Without conducting a time-consuming global coordinates
reconstruction, we propose a more efficient strategy by discretizing
differential operators only based on point-wisely local reconstruction. Our
local reconstruction model is based on the recent advances of low-rank matrix
completion theory, where only a very small random portion of distance
information is required. This method enables us to conduct analyses of
incomplete distance data using solutions of special designed PDEs such as the
Laplace-Beltrami (LB) eigen-system. As an application, we demonstrate a new way
of manifold reconstruction from an incomplete distance by stitching patches
using the spectrum of the LB operator. Intensive numerical experiments
demonstrate the effectiveness of the proposed methods.