Paper Info
Title
Solve Partial Differential Equations on Manifold 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 mainly 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 interpoint distance matrix with possible missing values. This paper discusses a framework to discretize PDEs on manifolds represented as incomplete interpoint distance information. Without conducting a time-consuming global coordinates reconstruction, we propose a more efficient strategy by discretizing differential operators only based on pointwisely 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) eigensystem. 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.
Year
DOI
Venue
2017
10.1137/17M1111176
SIAM JOURNAL ON SCIENTIFIC COMPUTING
Keywords
DocType
Volume
manifolds,Laplace-Beltrami eigenproblem,eikonal equation,low-rank matrix completion
Journal
39
Issue
ISSN
Citations 
5
1064-8275
0
PageRank 
References 
Authors
0.34
0
2
Name
Order
Citations
PageRank
Rongjie Lai123919.84
Jia Li200.34