# Solving Partial Differential Equations on Manifolds From Incomplete Inter-Point Distance

## TL;DR

This paper introduces an efficient method for solving PDEs on manifolds using incomplete inter-point distance data, leveraging local reconstruction and low-rank matrix completion without full manifold reconstruction.

## Contribution

It proposes a novel framework that discretizes PDEs directly from incomplete distance data using local reconstruction and low-rank matrix completion, avoiding global coordinate reconstruction.

## Key findings

- Effective PDE solutions on incomplete distance data.
- Successful manifold reconstruction from partial distances.
- Numerical experiments validate the approach.

## 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.