A new Kaczmarz-type method and its acceleration for nonlinear ill-posed problems

Abstract

In this paper, we first introduce a new Kaczmarz-type method for solving inverse problems that can be written as a system of a finite number of nonlinear equations. The proposed homotopy perturbation Kaczmarz (HPK) iteration is seen as a hybrid method between the homotopy perturbation iteration and the Kaczmarz strategy. Furthermore, an accelerated homotopy perturbation Kaczmarz (AHPK) method is presented based on the general case of Nesterov’s acceleration scheme. Under the classical assumptions for iterative regularization methods, we provide the corresponding convergence analysis for HPK and AHPK, respectively. The HPK iteration is shown to have faster calculation speed and less time consumption than the Landweber–Kaczmarz iteration through some numerical experiments on inverse potential problem. Besides, the significantly reduced computation cost and much better reconstruction quality indicate a remarkable acceleration effect for AHPK.