PDE-CONSTRAINED OPTIMAL CONTROL APPROACH FOR THE APPROXIMATION OF AN INVERSE CAUCHY PROBLEM

doi:10.3934/ipi.2015.9.791

CORC > 数学与系统科学研究院 > 中国科学院数学与系统科学研究院 > 计算数学与科学工程计算研究所

	PDE-CONSTRAINED OPTIMAL CONTROL APPROACH FOR THE APPROXIMATION OF AN INVERSE CAUCHY PROBLEM
	Chang, Lili 1; Gong, Wei2 ; Sun, Guiquan 1; Yan, Ningning3
刊名	INVERSE PROBLEMS AND IMAGING
	2015-08-01
卷号	9 期号:3 页码:791-814
关键词	Elliptic equation Cauchy problem Tikhonov regularization PDE-constrained optimal control finite element method
ISSN号	1930-8337
DOI	10.3934/ipi.2015.9.791
英文摘要	This paper concerns the approximation of a Cauchy problem for the elliptic equation. The inverse problem is transformed into a PDE-constrained optimal control problem and these two problems are equivalent under some assumptions. Different from the existing literature which is also based on the optimal control theory, we consider the state equation in the sense of very weak solution defined by the transposition technique. In this way, it does not need to impose any regularity requirement on the given data. Moreover, this method can yield theoretical analysis simply and numerical computation conveniently. To deal with the ill-posedness of the control problem, Tikhonov regularization term is introduced. The regularized problem is well-posed and its solution converges to the non-regularized counterpart as the regularization parameter approaches zero. We establish the finite element approximation to the regularized control problem and the convergence of the discrete problem is also investigated. Then we discuss the first order optimality condition of the control problem further and obtain an efficient numerical scheme for the Cauchy problem via the adjoint state equation. The paper is ended with numerical experiments.
资助项目	National Basic Research Program of China[2010CB731505] ; National Basic Research Program of China[2012CB821204] ; National Nature Science Foundation of China[11171337] ; National Nature Science Foundation of China[11201464]
WOS研究方向	Mathematics ; Physics
语种	英语
出版者	AMER INST MATHEMATICAL SCIENCES
WOS记录号	WOS:000360672300008
内容类型	期刊论文
源URL	[http://ir.amss.ac.cn/handle/2S8OKBNM/20710]
专题	计算数学与科学工程计算研究所系统科学研究所
通讯作者	Chang, Lili
作者单位	1.Shanxi Univ, Complex Syst Res Ctr, Taiyuan 030006, Shanxi, Peoples R China 2.Chinese Acad Sci, Acad Math & Syst Sci, Inst Computat Math, LSEC, Beijing 100190, Peoples R China 3.Chinese Acad Sci, Acad Math & Syst Sci, Inst Syst Sci, LSEC,NCMIS, Beijing 100190, Peoples R China
推荐引用方式 GB/T 7714	Chang, Lili,Gong, Wei,Sun, Guiquan,et al. PDE-CONSTRAINED OPTIMAL CONTROL APPROACH FOR THE APPROXIMATION OF AN INVERSE CAUCHY PROBLEM[J]. INVERSE PROBLEMS AND IMAGING,2015,9(3):791-814.
APA	Chang, Lili,Gong, Wei,Sun, Guiquan,&Yan, Ningning.(2015).PDE-CONSTRAINED OPTIMAL CONTROL APPROACH FOR THE APPROXIMATION OF AN INVERSE CAUCHY PROBLEM.INVERSE PROBLEMS AND IMAGING,9(3),791-814.
MLA	Chang, Lili,et al."PDE-CONSTRAINED OPTIMAL CONTROL APPROACH FOR THE APPROXIMATION OF AN INVERSE CAUCHY PROBLEM".INVERSE PROBLEMS AND IMAGING 9.3(2015):791-814.