Inexact Newton regularization in Banach spaces based on two-point gradient method with uniformly convex penalty terms

Abstract

In this paper, we propose and analyze a new inexact Newton regularization method for solving nonlinear inverse problems in Banach spaces. The method consists of an outer Newton iteration and an inner scheme which provides increments by applying a regularization technique to the local linearized equation around the current iterate. In order to accelerate the convergence, we employ a two-point gradient method as inner regularization scheme, which is based on the Landweber iteration and an extrapolation strategy. The penalty term is allowed to be non-smooth, including L1-like and total variation-like penalty functionals, to detect special features of solutions such as sparsity and piecewise constancy. We present, under certain assumptions, the detailed analysis of convergence and regularization properties of the method. Finally, some numerical experiments on elliptic parameter identification and Robin coefficient reconstruction problems are provided to illustrate the effectiveness of reconstructing the properties of sought solutions and the acceleration effect of the method.