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

doi:10.3934/ipi.2015.9.791

	PDE-CONSTRAINED OPTIMAL CONTROL APPROACH FOR THE APPROXIMATION OF AN INVERSE CAUCHY PROBLEM
	Chang, Lili 1; Gong, Wei2 ; Sun, Guiquan 1; Yan, Ningning3
	2015-08-01
发表期刊	INVERSE PROBLEMS AND IMAGING
ISSN	1930-8337
卷号	9 期号:3 页码:791-814
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.
关键词	Elliptic equation Cauchy problem Tikhonov regularization PDE-constrained optimal control finite element method
DOI	10.3934/ipi.2015.9.791
语种	英语
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
WOS类目	Mathematics, Applied ; Physics, Mathematical
WOS记录号	WOS:000360672300008
出版者	AMER INST MATHEMATICAL SCIENCES
引用统计
文献类型	期刊论文
条目标识符	http://ir.amss.ac.cn/handle/2S8OKBNM/20711
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.