A projective two-point gradient Kaczmarz iteration for nonlinear ill-posed problems

Abstract

In this paper, we propose and analyze a fast Kaczmarz type method for solving multiple nonlinear ill-posed problems in Hilbert spaces. The method is the combination of the projective two-point gradient method and the Kaczmarz method. The key idea, in contrast to the standard two-point gradient method, is to use modified discrete backtracking search algorithm in each iteration in combination with metric projection of the step size to reduce the total number of performed steps and the computation time. Under reasonable conditions used in this work, we establish the strong convergence result of the method in the noise-free case. Moreover, we present the stability and regularity of the proposed method terminated by the discrepancy principle for the case of noisy data. Finally, some numerical experiments on a nonlinear parameter identification problem are presented, which exhibit that the effectiveness of reconstruction results and the acceleration effect of the method.