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
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