Two-point Landweber-type method with convex penalty terms for nonsmooth nonlinear inverse problems

Abstract

Abstract In this work, we propose a two-point Landweber-type method with general convex penalty terms for solving nonsmooth nonlinear inverse problems. The design of our method can cope with nonsmooth nonlinear inverse problems or the nonlinear inverse problems whose data are contaminated by various types of noise. The method consists of the two-point acceleration strategy and inner solvers. Inner solvers are used to solve the minimization problems with respect to the penalty terms in each steps. If the minimization problems can be solved explicitly, the inner solvers will be chosen to be the exact solvers. Otherwise, we will use inexact solvers as inner solvers. Convergence results are given without utilizing the Gâteaux differentiability of the forward operator or the reflexivity of the image space. Numerical simulations are given to test the performance of the proposed method.