NDLI: Efficient Preconditioners for Optimality Systems Arising in Connection with Inverse Problems

Content Provider	Society for Industrial and Applied Mathematics (SIAM)
Author	Mardal, Kent-Andre Nielsen, Bjrn Fredrik
Copyright Year	2010
Abstract	This paper is devoted to the numerical treatment of linear optimality systems (OS) that arise in connection with inverse problems for partial differential equations. If such inverse problems are regularized by Tikhonov regularization, then it follows from standard theory that the associated OS is well-posed, provided that the regularization parameter $\alpha$ is positive and that the involved state equation satisfies suitable assumptions. We explain and analyze how certain mapping properties of the operators appearing in the OS can be employed to define efficient preconditioners for finite element (FE) approximations of such systems. The key feature of the scheme is that the number of iterations needed to solve the preconditioned problem by the minimal residual method is bounded independently of the mesh parameter h, used in the FE discretization, and increases only moderately as $\alpha\rightarrow0$. More specifically, if the stopping criterion for the iteration process is defined in terms of the associated energy norm, then the number of iterations required (in the severely ill-posed case) cannot grow faster than $O((\ln(\alpha))^2)$. Our analysis is based on a careful study of the operators involved, which yields the distribution of the eigenvalues of the preconditioned OS. Finally, the theoretical results are illuminated by a number of numerical experiments addressing both a model problem studied by Borzi, Kunisch, and Kwak [SIAM J. Control Optim., 41 (2003), pp. 14771497] and an inverse problem arising in connection with electrocardiography [Nielsen, Cai, and Lysaker, Math. Biosci., 210 (2007), pp. 523553].
Starting Page	5143
Ending Page	5177
Page Count	35
File Format	PDF
ISSN	03630129
DOI	10.1137/080719960
e-ISSN	10957138
Issue Number	8
Volume Number	48
Language	English
Publisher	Society for Industrial and Applied Mathematics
Publisher Date	2010-10-26
Access Restriction	Subscribed
Subject Keyword	isolated eigenvalues optimality systems Inverse problems minimal residual method inverse problems Eigenvalues, eigenvectors Tikhonov regularization Preconditioners for iterative methods Problems involving partial differential equations all-at-once methods preconditioning electrocardiography
Content Type	Text
Resource Type	Article
Subject	Control and Optimization Applied Mathematics

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