Inexact Newton regularization combined with gradient methods in Banach spaces

In this paper we investigate convergence and stability properties of an adaptation to Banach spaces of the algorithm REGINN (Rieder 1999 Inverse Problems 15 309-27). This inexact Newton method solves nonlinear inverse problems by means of linearizing the equation around the current iterate and subsequently applying a regularization technique in the so-called inner iteration in order to obtain a stable approximation of the resulting linearized system. The current iterate is then updated by adding this approximate solution. Using a generic gradient-like method as inner iteration, we provide a whole converge analysis of this Newton-type method, proving, under reasonable assumptions, strong convergence of the generated sequence to a solution of the inverse problem in the noiseless situation and the regularization property in the noisy-data case. Some numerical experiments are performed at the end of the paper for providing the necessary support to the theoretical results