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.

Homotopy perturbation iteration is an effective and fast method for solving nonlinear ill-posed problems. It only needs approximately half the computation time of Landweber iteration to reach the similar recovery precision. In this paper, a Nesterov-type accelerated sequential subspace optimization method based on homotopy perturbation iteration is proposed for solving nonlinear inverse problems. The convergence analysis is provided under the general assumptions for iterative regularization methods. The numerical experiments on inverse potential problem indicate that the proposed method dramatically reduces the total number of iterations and time consumption to obtain satisfying approximations, especially for the problems with costly solution of forward calculation.

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