The Averaged Kaczmarz Iteration for Solving Inverse Problems

TL;DR

The paper introduces the averaged Kaczmarz (AVEK) iterative regularization method for inverse problems, combining stability and efficiency, with convergence analysis and numerical validation in photoacoustic tomography.

Contribution

It proposes the AVEK method, a hybrid of Landweber and Kaczmarz, with convergence proof and superior performance in tomographic image reconstruction.

Findings

AVEK improves stability over traditional methods.

Numerical results show AVEK outperforms Landweber and Kaczmarz.

AVEK is effective for limited data photoacoustic tomography.

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.