The Averaged Kaczmarz Iteration for Solving Inverse Problems

Abstract

We introduce a new iterative regularization method for solving inverse problems that can be written as systems of linear or non-linear equations in Hilbert spaces. The proposed averaged Kaczmarz (AVEK) method can be seen as a hybrid method between the Landweber and the Kaczmarz method. As the Kaczmarz method, the proposed method only requires evaluation of one direct and one adjoint sub-problem per iterative update. On the other, similar to the Landweber iteration, it uses an average over previous auxiliary iterates which increases stability. We present a convergence analysis of the AVEK iteration. Further, detailed numerical studies are presented for a tomographic image reconstruction problem, namely the limited data problem in photoacoustic tomography. Thereby, the AVEK is compared with other iterative regularization methods including standard Landweber and Kaczmarz iterations, as well as recently proposed accelerated versions based on error minimizing relaxation strategies.

References in corpus (4)

• Regularization of systems of nonlinear ill-posed equations: I. Convergence Analysis
• Regularization of systems of nonlinear ill-posed equations: II. Applications
• On Levenberg-Marquardt-Kaczmarz iterative methods for solving systems of nonlinear ill-posed equations
• On projective Landweber-Kaczmarz methods for solving systems of nonlinear ill-posed equations

Cited by in corpus (2)

• Stochastic Proximal Gradient Algorithms for Multi-Source Quantitative Photoacoustic Tomography
• Convergence Analysis of Nonlinear Kaczmarz Method for Systems of Nonlinear Equations with Component-wise Convex Mapping