The Averaged Kaczmarz Iteration for Solving Inverse Problems
For researchers in inverse problems and image reconstruction, AVEK offers a more stable and efficient iterative regularization method, though it is an incremental improvement over existing techniques.
The paper introduces the averaged Kaczmarz (AVEK) method for solving inverse problems, combining aspects of Landweber and Kaczmarz iterations. In photoacoustic tomography, AVEK achieves faster convergence and improved reconstruction quality compared to standard methods, with up to 50% reduction in iterations for similar accuracy.
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.