Generalized inexact Newton regularization for nonlinear ill-posed problems in Banach spaces

Abstract

In this paper, we generalize inexact Newton regularization methods to solve nonlinear inverse problems from a reflexive Banach space to a Banach space. The image space is not necessarily reflexive so that the method can be used to deal with various types of noise such as the Gaussian noise and the impulsive noise. The method consists of an outer Newton iteration and an inner scheme which provides increments by applying the regularization technique to the local linearized equations. Under some assumptions, in particular, the reflexivity of the image space is not required, we present a novel convergence analysis of the inexact Newton regularization method with inner scheme defined by Landweber iteration. Furthermore, by employing a two-point gradient method as inner regularization scheme to accelerate the convergence, we propose an accelerated version of inexact Newton–Landweber method and present the detailed convergence analysis. The numerical simulations are provided to demonstrate the effectiveness of the proposed methods in handling different kinds of noise and the fast convergence of the accelerated method.