Landweber-Kaczmarz method in Banach spaces with inexact inner solvers

Abstract: In recent years Landweber(-Kaczmarz) method has been proposed for solving nonlinear ill-posed inverse problems in Banach spaces using general convex penalty functions. The implementation of this method involves solving a (nonsmooth) convex minimization problem at each iteration step and the existing theory requires its exact resolution which in general is impossible in practical applications. In this paper we propose a version of Landweber-Kaczmarz method in Banach spaces in which the minimization problem involved in each iteration step is solved inexactly. Based on the $\varepsilon$-subdifferential calculus we give a convergence analysis of our method. Furthermore, using Nesterov's strategy, we propose a possible accelerated version of Landweber-Kaczmarz method. Numerical results on computed tomography and parameter identification in partial differential equations are provided to support our theoretical results and to demonstrate our accelerated method.