Spectral methods for solving elliptic PDEs on unknown manifolds

doi:10.1016/j.jcp.2023.112132

KMS > 数学科学研究所

	Spectral methods for solving elliptic PDEs on unknown manifolds
	Yan, Qile 1; Jiang, Shixiao Willing2 ; Harlim, John 3
	2023-08-01
发表期刊	JOURNAL OF COMPUTATIONAL PHYSICS
ISSN	0021-9991
EISSN	1090-2716
卷号	486
发表状态	已发表
DOI	10.1016/j.jcp.2023.112132
摘要	In this paper, we propose a mesh-free numerical method for solving elliptic PDEs on unknown manifolds, identified with randomly sampled point cloud data. The PDE solver is formulated as a spectral method where the test function space is the span of the leading eigenfunctions of the Laplacian operator, which are approximated from the point cloud data. While the framework is flexible for any test functional space, we will consider the eigensolutions of a weighted Laplacian obtained from a symmetric Radial Basis Function (RBF) method induced by a weak approximation of a weighted Laplacian on an appropriate Hilbert space. In this paper, we consider a test function space that encodes the geometry of the data yet does not require us to identify and use the sampling density of the point cloud. To attain a more accurate approximation of the expansion coefficients, we adopt a second -order tangent space estimation method to improve the RBF interpolation accuracy in estimating the tangential derivatives. This spectral framework allows us to efficiently solve the PDE many times subjected to different parameters, which reduces the computational cost in the related inverse problem applications. In a well-posed elliptic PDE setting with randomly sampled point cloud data, we provide a theoretical analysis to demonstrate the convergence of the proposed solver as the sample size increases. We also report some numerical studies that show the convergence of the spectral solver on simple manifolds and unknown, rough surfaces. Our numerical results suggest that the proposed method is more accurate than a graph Laplacian-based solver on smooth manifolds. On rough manifolds, these two approaches are comparable. Due to the flexibility of the framework, we empirically found improved accuracies in both smoothed and unsmoothed Stanford bunny domains by blending the graph Laplacian eigensolutions and RBF interpolator. (c) 2023 Elsevier Inc. All rights reserved.
关键词	Radial basis function Galerkin approximation Elliptic PDEs solver Point cloud data
URL	查看原文
收录类别	SCI ; EI
语种	英语
资助项目	NSF["DMS-1854299","DMS-2207328","DMS-2229435"] ; ONR[N00014-22-1-2193] ; NSFC[12101408]
WOS研究方向	Computer Science ; Physics
WOS类目	Computer Science, Interdisciplinary Applications ; Physics, Mathematical
WOS记录号	WOS:000984517400001
出版者	ACADEMIC PRESS INC ELSEVIER SCIENCE
EI入藏号	20231613896756
EI主题词	Inverse problems
EI分类号	802.3 Chemical Operations ; 921 Mathematics ; 921.3 Mathematical Transformations ; 921.6 Numerical Methods
原始文献类型	Journal article (JA)
引用统计
文献类型	期刊论文
条目标识符	https://kms.shanghaitech.edu.cn/handle/2MSLDSTB/305029
专题	数学科学研究所数学科学研究所_PI研究组(P)_蒋诗晓组
通讯作者	Yan, Qile
作者单位	1.Penn State Univ, Dept Math, University Pk, PA 16802 USA 2.ShanghaiTech Univ, Inst Math Sci, Shanghai 201210, Peoples R China 3.Penn State Univ, Inst Computat & Data Sci, Dept Math, Dept Meteorol & Atmospher Sci, University Pk, PA 16802 USA
推荐引用方式 GB/T 7714	Yan, Qile,Jiang, Shixiao Willing,Harlim, John. Spectral methods for solving elliptic PDEs on unknown manifolds[J]. JOURNAL OF COMPUTATIONAL PHYSICS,2023,486.
APA	Yan, Qile,Jiang, Shixiao Willing,&Harlim, John.(2023).Spectral methods for solving elliptic PDEs on unknown manifolds.JOURNAL OF COMPUTATIONAL PHYSICS,486.
MLA	Yan, Qile,et al."Spectral methods for solving elliptic PDEs on unknown manifolds".JOURNAL OF COMPUTATIONAL PHYSICS 486(2023).