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Journal of Inverse and Ill-posed Problems

Editor-in-Chief: Kabanikhin, Sergey I.


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A shape calculus analysis for tracking type formulations in electrical impedance tomography

K. Eppler1

1Institute of Numerical Mathematics, TU Dresden, Zellescher Weg 12–14, 01062 Dresden, Germany. Email:

Citation Information: Journal of Inverse and Ill-posed Problems. Volume 17, Issue 8, Pages 733–751, ISSN (Online) 1569-3945, ISSN (Print) 0928-0219, DOI: 10.1515/JIIP.2009.043, November 2009

Publication History

Received:
2008-11-11
Published Online:
2009-11-25

Abstract

In the paper [Eppler and Harbrecht, Control & Cybernetics 34: 203–225, 2005], the authors investigated the identification of an obstacle or void of perfectly conducting material in a two-dimensional domain by measurements of voltage and currents at the boundary. In particular, the reformulation of the given nonlinear identification problem was considered as a shape optimization problem using the Kohn and Vogelius criterion. The compactness of the complete shape Hessian at the optimal inclusion was proven, verifying strictly the ill-posedness of the identification problem. The aim of the paper is to present a similar analysis for the related least square tracking formulations. It turns out that the two-norm-discrepancy is of the same principal nature as for the Kohn and Vogelius objective. As a byproduct, the necessary first order optimality condition are shown to be satisfied if and only if the data are perfectly matching. Finally, we comment on possible consequences of the two-norm-discrepancy for the regularization issue.

Key words.: Electrical impedance tomography; shape calculus; boundary integral equations; ill-posed problems; two norm discrepancy

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[1]
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