Analysis of a generalized regularized Gauss–Newton method under heuristic rule in Banach spaces

Abstract

In this work, we propose a generalized regularized Gauss–Newton method to solve nonlinear inverse problems in Banach spaces and consider its heuristic stopping rule. The proposed method only requires a generalized operator which could be independent of the Fréchet derivative of the forward operator, thus it can solve smooth as well as non-smooth inverse problems. In addition, the proposed heuristic rule is fully data driven and does not require any information on the noise level. Therefore, it can cope with inverse problems whose noise levels are unknown. An a posteriori error for this method under heuristic rule is obtained and general convergence result is given under some standard assumptions. Numerical experiments are presented to illustrate the efficiency of the proposed method.