Randomized iterated Tikhonov regularization for linear ill-posed problems in Hilbert spaces

Abstract In this paper we introduce a mini-batch randomized iterated Tikhonov regularization method for solving ill-posed inverse problems governed by linear systems. To capture the features of the sought solutions we incorporate convex regularization terms into our algorithm. Our approach combines the advantages of Newton-type methods with stochastic optimization techniques, enabling efficient handling of large-scale problems while mitigating oscillations and semiconvergence phenomena typically induced by noise. We propose various step-size selection rules, particularly emphasizing one based on the discrepancy principle, which ensures almost sure termination within a finite number of iterations. Under reasonable conditions we establish the convergence and the convergence rate of the method. Numerical simulations validate the promising performance of the proposed method.