Volume 22, Issue 1
Point Integral Method for Solving Poisson-Type Equations on Manifolds from Point Clouds with Convergence Guarantees

Commun. Comput. Phys., 22 (2017), pp. 228-258.

Published online: 2019-10

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• Abstract

Partial differential equations (PDE) on manifolds arise in many areas, including mathematics and many applied fields. Due to the complicated geometrical structure of the manifold, it is difficult to get efficient numerical method to solve PDE on manifold. In the paper, we propose a method called point integral method (PIM) to solve the Poisson-type equations from point clouds. Among different kinds of PDEs, the Poisson-type equations including the standard Poisson equation and the related eigenproblem of the Laplace-Beltrami operator are one of the most important. In PIM, the key idea is to derive the integral equations which approximates the Poisson-type equations and contains no derivatives but only the values of the unknown function. This feature makes the integral equation easy to be discretized from point cloud. In the paper, we explain the derivation of the integral equations, describe the point integral method and its implementation, and present the numerical experiments to demonstrate the convergence of PIM.

• Keywords

Point integral method, point cloud, Laplace-Beltrami operator, convergence.

65N12, 65N25, 65N75

zlin12@mails.tsinghua.edu.cn (Zhen Li)

zqshi@mail.tsinghua.edu.cn (Zuoqiang Shi)

jsun@math.tsinghua.edu.cn (Jian Sun)

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@Article{CiCP-22-228, author = {Li , Zhen and Shi , Zuoqiang and Sun , Jian }, title = {Point Integral Method for Solving Poisson-Type Equations on Manifolds from Point Clouds with Convergence Guarantees}, journal = {Communications in Computational Physics}, year = {2019}, volume = {22}, number = {1}, pages = {228--258}, abstract = {

Partial differential equations (PDE) on manifolds arise in many areas, including mathematics and many applied fields. Due to the complicated geometrical structure of the manifold, it is difficult to get efficient numerical method to solve PDE on manifold. In the paper, we propose a method called point integral method (PIM) to solve the Poisson-type equations from point clouds. Among different kinds of PDEs, the Poisson-type equations including the standard Poisson equation and the related eigenproblem of the Laplace-Beltrami operator are one of the most important. In PIM, the key idea is to derive the integral equations which approximates the Poisson-type equations and contains no derivatives but only the values of the unknown function. This feature makes the integral equation easy to be discretized from point cloud. In the paper, we explain the derivation of the integral equations, describe the point integral method and its implementation, and present the numerical experiments to demonstrate the convergence of PIM.

}, issn = {1991-7120}, doi = {https://doi.org/10.4208/cicp.111015.250716a}, url = {http://global-sci.org/intro/article_detail/cicp/13354.html} }
TY - JOUR T1 - Point Integral Method for Solving Poisson-Type Equations on Manifolds from Point Clouds with Convergence Guarantees AU - Li , Zhen AU - Shi , Zuoqiang AU - Sun , Jian JO - Communications in Computational Physics VL - 1 SP - 228 EP - 258 PY - 2019 DA - 2019/10 SN - 22 DO - http://dor.org/10.4208/cicp.111015.250716a UR - https://global-sci.org/intro/article_detail/cicp/13354.html KW - Point integral method, point cloud, Laplace-Beltrami operator, convergence. AB -

Partial differential equations (PDE) on manifolds arise in many areas, including mathematics and many applied fields. Due to the complicated geometrical structure of the manifold, it is difficult to get efficient numerical method to solve PDE on manifold. In the paper, we propose a method called point integral method (PIM) to solve the Poisson-type equations from point clouds. Among different kinds of PDEs, the Poisson-type equations including the standard Poisson equation and the related eigenproblem of the Laplace-Beltrami operator are one of the most important. In PIM, the key idea is to derive the integral equations which approximates the Poisson-type equations and contains no derivatives but only the values of the unknown function. This feature makes the integral equation easy to be discretized from point cloud. In the paper, we explain the derivation of the integral equations, describe the point integral method and its implementation, and present the numerical experiments to demonstrate the convergence of PIM.

Zhen Li, Zuoqiang Shi & Jian Sun. (2019). Point Integral Method for Solving Poisson-Type Equations on Manifolds from Point Clouds with Convergence Guarantees. Communications in Computational Physics. 22 (1). 228-258. doi:10.4208/cicp.111015.250716a
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