Landweber-Kaczmarz method in Banach spaces with inexact inner solvers. (5th August 2016)

• Record Type: Journal Article
• Authors: Jin, Qinian
• Is Part Of: Inverse problems. Volume 32:Number 10(2016:Oct.)
• Publication Date: 2016-08-05
• Subjects: Landweber-Kaczmarz method -- inexact inner solver -- regularization property -- convex penalty functions; Inverse problems (Differential equations) -- Periodicals; 515.357
• DOI: 10.1088/0266-5611/32/10/104005
• Languages: English
• ISSNs: 0266-5611

Abstract

Abstract: In recent years the 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 the Landweber-Kaczmarz method in Banach spaces in which the minimization problem involved in each iteration step is solved inexactly. Based on the ε -subdifferential calculus we give a convergence analysis of our method. Furthermore, using Nesterov's strategy, we propose a possible accelerated version of the 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.