Ghost Point Diffusion Maps for Solving Elliptic PDEs on Manifolds with Classical Boundary Conditions

doi:10.1002/cpa.22035

	Ghost Point Diffusion Maps for Solving Elliptic PDEs on Manifolds with Classical Boundary Conditions
	Jiang, Shixiao Willing1 ; Harlim, John 2
	2021-12
发表期刊	COMMUNICATIONS ON PURE AND APPLIED MATHEMATICS
ISSN	0010-3640
EISSN	1097-0312
发表状态	已发表
DOI	10.1002/cpa.22035
摘要	In this paper, we extend the class of kernel methods, the so-called diffusion maps (DM) and its local kernel variants to approximate second-order differential operators defined on smooth manifolds with boundaries that naturally arise in elliptic PDE models. To achieve this goal, we introduce the ghost point diffusion maps (GPDM) estimator on an extended manifold, identified by the set of point clouds on the unknown original manifold together with a set of ghost points, specified along the estimated tangential direction at the sampled points on the boundary. The resulting GPDM estimator restricts the standard DM matrix to a set of extrapolation equations that estimates the function values at the ghost points. This adjustment is analogous to the classical ghost point method in a finite-difference scheme for solving PDEs on flat domains. As opposed to the classical DM, which diverges near the boundary, the proposed GPDM estimator converges pointwise even near the boundary. Applying the consistent GPDM estimator to solve well-posed elliptic PDEs with classical boundary conditions (Dirichlet, Neumann, and Robin), we establish the convergence of the approximate solution under appropriate smoothness assumptions. We numerically validate the proposed mesh-free PDE solver on various problems defined on simple submanifolds embedded in Euclidean spaces as well as on an unknown manifold. Numerically, we also found that the GPDM is more accurate compared to DM in solving elliptic eigenvalue problems on bounded smooth manifolds. (c) 2021 Wiley Periodicals LLC.
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收录类别	SCI ; SCIE ; SCOPUS
语种	英语
资助项目	National Science Foundation[DMS-1854299]
WOS研究方向	Mathematics
WOS类目	Mathematics, Applied ; Mathematics
WOS记录号	WOS:000733178900001
出版者	WILEY
Scopus 记录号	2-s2.0-85121552110
来源库	Scopus
引用统计
文献类型	期刊论文
条目标识符	https://kms.shanghaitech.edu.cn/handle/2MSLDSTB/145762
专题	数学科学研究所_PI研究组(P)_蒋诗晓组
通讯作者	Jiang, Shixiao Willing
作者单位	1.ShanghaiTech Univ, Inst Math Sci, Shanghai 201210, Peoples R China 2.Penn State Univ, Dept Math, Dept Meteorol & Atmospher Sci, Inst Computat & Data Sci, University Pk, PA 16802 USA
第一作者单位	数学科学研究所
通讯作者单位	数学科学研究所
第一作者的第一单位	数学科学研究所
推荐引用方式 GB/T 7714	Jiang, Shixiao Willing,Harlim, John. Ghost Point Diffusion Maps for Solving Elliptic PDEs on Manifolds with Classical Boundary Conditions[J]. COMMUNICATIONS ON PURE AND APPLIED MATHEMATICS,2021.
APA	Jiang, Shixiao Willing,&Harlim, John.(2021).Ghost Point Diffusion Maps for Solving Elliptic PDEs on Manifolds with Classical Boundary Conditions.COMMUNICATIONS ON PURE AND APPLIED MATHEMATICS.
MLA	Jiang, Shixiao Willing,et al."Ghost Point Diffusion Maps for Solving Elliptic PDEs on Manifolds with Classical Boundary Conditions".COMMUNICATIONS ON PURE AND APPLIED MATHEMATICS (2021).