Inexact Newton methods for the stable solution of nonlinear ill-posed problems are considered. The corresponding inner scheme can be chosen to be any linear regularization with a sufficient modulus of convergence. The regularization property of these Newton-type algorithms is verified, that is, the iterates converge to a solution of the nonlinear problem with exact data when the noise level tends to zero. Moreover, convergence rates are given. Finally, implementation issues are discussed and the algorithm is applied to a parameter identification problem for an elliptic PDE. The numerical results reproduce nicely theoretical predictions and show the efficiency of the proposed method.

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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Summary. REGINN is an algorithm of inexact Newton type for the regularization of nonlinear ill-posed problems [Inverse Problems 15 (1999), pp. 309-327]. In the present article convergence is shown under weak smoothness assumptions (source conditions). Moreover, convergence rates are established. Some computational illustrations support the theoretical results. Classification (1991): 65J15, 65J20 Mathematics Subject

Abstract. It is common knowledge -mainly based on experience -that parameter identification problems in partial differential equations are ill-posed. Yet, a mathematical sound argumentation is missing, except for some special cases. We present a general theory for inverse problems related to abstract evolution equations which explains not only their local ill-posedness but also provides the Fréchet derivative of the corresponding parameter-to-solution map which is needed, e.g., in Newton-like solvers. Our abstract results are applied to inverse problems related to the following first order hyperbolic systems: Maxwell's equation (electromagnetic scattering in conducting media) and elastic wave equation (seismic imaging).