A Fast Data-Driven Iteratively Regularized Method with Convex Penalty for Solving Ill-Posed Problems

Abstract

We propose a new iterative regularization method for solving inverse problems in Hilbert spaces. The iterative process of the proposed method combines classical iterative regularization format and Data-Driven approach. Data-Driven technique is based on the idea of deep learning to estimate the interior of a black box through a training set, so as to solve problems better and faster in some cases. In order to capture the special feature of solutions, convex functions are utilized to be penalty terms. Algorithmically, the two-point gradient acceleration strategy based on homotopy perturbation method is applied to the iterative scheme, which makes the method have satisfactory acceleration effect. We provide convergence analysis of the method under standard assumptions for iterative regularization methods. Finally, several numerical experiments are presented to show the effectiveness and acceleration effect of our method.