WIAS Preprint No. 1116, (2006)

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



Authors

  • Eppler, Karsten

2010 Mathematics Subject Classification

  • 49Q10 49M15 65N38 65K10 49K20 65T60

Keywords

  • electrical impedance tomography, shape calculus, boundary integral equations, ill-posed problems, two norm discrepancy

Abstract

In the paper [17], 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.

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