A Surrogate Hyperplane Bregman–Kaczmarz Method for Solving Linear Inverse Problems

Linear inverse problems arise in many practical applications. In the present work, we propose a residual-based surrogate hyperplane Bregman-Kaczmarz method (RSHBK) for solving this kind of problems. The convergence theory of the proposed method is investigated detailedly. When the data is contaminated by the independent noise, which means the observed measurement at each new iteration in the algorithm is refreshed with noise which is new and independent of that in the previous iterations, an adaptive version of our RSHBK method is developed. An adaptive relaxation parameter is derived for optimizing the bound on the expectation error. We demonstrate the efficiency of our proposed methods for both noise-free and independent noise problems by comparing with other state-of-the-art Kaczmarz methods in terms of computation time and convergence rate through synthetic experiments and real-world applications. © The Author(s), under exclusive licence to Springer Science+Business Media, LLC, part of Springer Nature 2024.