Landweber-Kaczmarz method in Banach spaces with inexact inner solvers

TL;DR

This paper introduces an inexact version of the Landweber-Kaczmarz method for Banach spaces, allowing practical implementation with convergence guarantees and an accelerated variant, supported by numerical experiments.

Contribution

It develops an inexact iterative scheme for nonlinear inverse problems in Banach spaces, including convergence analysis and an accelerated method using Nesterov's strategy.

Findings

Convergence of the inexact Landweber-Kaczmarz method is established.

Numerical experiments demonstrate the effectiveness of the accelerated method.

The method is applicable to computed tomography and PDE parameter identification.

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.