# New Pair of Primal Dual Algorithms for Bregman Iterated Variational Regularization

## TL;DR

This paper introduces two new primal-dual algorithms for variational regularization using Bregman distances, providing convergence rates and stability analysis, and demonstrates their effectiveness on tomographic image reconstruction problems.

## Contribution

The work develops novel primal-dual iterative algorithms for Bregman-penalized variational regularization, with explicit parameter choices and convergence analysis under variational source conditions.

## Key findings

- Algorithms effectively reconstruct 3D atmospheric tomography images.
- Methods demonstrate stability and convergence in both continuous and iterative senses.
- Successful application to 2D and 3D tomographic problems confirms practical utility.

## Abstract

Primal-dual splitting involving proximity operators in order to be able to find some approximation to the minimizer for a general form of Tikhonov type functional is in the focus of this work. This approximation is produced by a pair of iterative variational regularization procedures. Under the assumption of some variational source condition (VSC), total error estimation both in the iterative sense and in the continuous sense has been analysed separately. Rates of convergence will be obtained in terms some concave and positive definite index function. Of the choice of the penalty term, we are interested in Bregman distance penalization associated with the non-smooth total variation (TV) functional. Furthermore, following up the lower and bounds defined for the regularization parameter, some deterministic choice of the regularization parameter is given explicitly. It is in the emphasis of this work that the regularization parameter obeys Morozov`s discrepancy principle (MDP) in order for the stability analysis of regularized solution. In the computerized environment, the algorithms are verified as iterative regularization methods by applying it to an atmospheric tomography problem named as GPS-Tomography. Apart from this 3-D tomographic inverse problem, we also apply the algorithms to some 2-D conventional tomographic image reconstruction problems in order to be able test algorithms` capability of capturing the details and observe that algorithms behave as iterative regularization procedures.