Newton regularizations for impedance tomography: convergence by local injectivity

Abstract

In [Inverse Problems 22(2006), pp. 1967-1987] we demonstrated experimentally that the Newton-like regularization method CG-REGINN is a competitive solver for the inverse problem of the complete electrode model in 2D-electrical impedance tomography. Here we establish rigorously the observed convergence of CG-REGINN (and related schemes). To this end we prove that the underlying nonlinear operator has an injective Frechét derivative whenever the number of electrodes is sufficiently large and the discretization step size is sufficiently small. Though injectivity of the Frechét derivative is an interesting new result on its own, it is only a secondary issue here. We namely rely on it to obtain a so-called tangential cone condition in the fully discrete setting which is the main ingredient in a well-developed convergence theory for Newton-like regularization schemes.

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