Inexact Newton regularizations with uniformly convex stability terms: a unified convergence analysis

Abstract We present a unified convergence analysis of inexact Newton regularizations with general uniformly convex penalty terms for nonlinear ill-posed problems in Banach spaces. These schemes consist of an outer (Newton) iteration and an inner iteration, which provides the update of the current outer iterate. To this end the nonlinear problem is linearized about the current iterate and the resulting linear system is inexactly solved by an inner regularization method. In our analysis we only rely on generic assumptions of the inner methods, and we show that a variety of regularization techniques satisfy these assumptions. For instance, gradient-type and iterated-Tikhonov methods are covered. Not only the technique of proof is novel, but also some results, because for the first time uniformly convex penalty terms stabilize the inner scheme in full generality. Numerical experiments based on the inverse problem of electrical impedance tomography illustrate the impact of different uniformly convex penalty terms.