Non-linear integral equations for the complete electrode model in inverse impedance tomography

Abstract

We consider the two-dimensional inverse electrical impedance problem for piecewise constant conductivities with the data given in terms of the complete electrode model. Our approach is based on a system of non-linear integral equations arising from Green's representation formula from which the unknown conductivities and the unknown shapes of the interfaces are obtained iteratively via linearization. The method is an extension of our previous work for the case of classical data in terms of full Cauchy data on ∂D. This in turn originated from a method that has been suggested by Kress and Rundell for the case of a perfectly conducting inclusion. For the choice of the regularization parameters occurring in the algorithm, we propose an evolutionary algorithm and the initial guess for the iterations is obtained through employing a Newton-type finite element method. We describe the method in detail and illustrate its feasibility by numerical examples.

Keywords:

• impedance tomography
• non-linear integral equations
• evolutionary algorithm

Acknowledgements

The research of H.E. was supported by the German Research Foundation DFG through the Graduiertenkolleg Identification in Mathematical Models and by the German Federal Ministry of Education and Research BMBF through the joint research project Regularization in Electrical Impedance Tomography.