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PDE-CONSTRAINED OPTIMAL CONTROL APPROACH FOR THE APPROXIMATION OF AN INVERSE CAUCHY PROBLEM
Chang, Lili1; Gong, Wei2; Sun, Guiquan1; Yan, Ningning3
2015-08-01
Source PublicationINVERSE PROBLEMS AND IMAGING
ISSN1930-8337
Volume9Issue:3Pages:791-814
AbstractThis 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.
KeywordElliptic equation Cauchy problem Tikhonov regularization PDE-constrained optimal control finite element method
DOI10.3934/ipi.2015.9.791
Language英语
Funding ProjectNational 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 Research AreaMathematics ; Physics
WOS SubjectMathematics, Applied ; Physics, Mathematical
WOS IDWOS:000360672300008
PublisherAMER INST MATHEMATICAL SCIENCES
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Cited Times:2[WOS]   [WOS Record]     [Related Records in WOS]
Document Type期刊论文
Identifierhttp://ir.amss.ac.cn/handle/2S8OKBNM/20711
Collection计算数学与科学工程计算研究所
系统科学研究所
Affiliation1.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
Recommended Citation
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.
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