A shape calculus analysis for tracking type formulations in electrical impedance tomography
- Eppler, Karsten
2010 Mathematics Subject Classification
- 49Q10 49M15 65N38 65K10 49K20 65T60
- electrical impedance tomography, shape calculus, boundary integral equations, ill-posed problems, two norm discrepancy
In the paper , 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.