A range-relaxed criteria for choosing the Lagrange multipliers in the iterated Tikhonov Kaczmarz method for solving systems of linear ill-posed equations

Abstract

In this article we propose and analyze a nonstationary iterated Tikhonov Kaczmarz (iTK) type method for obtaining stable approximate solutions to systems of ill-posed equations modeled by linear operators acting between Hilbert spaces. We generalize for the iTK iteration the criteria proposed in [] for the iterated Tikhonov method. The goal is to devise an efficient strategy for choosing the Lagrange multipliers in this method. Convergence analysis for the resulting iTK method is provided, including convergence for exact data, stability and semi-convergence. Numerical experiments are presented for two distinct applications, namely: an image deblurring problem and a 2D elliptic parameter identification problem (the inverse potential problem). The obtained numerical results validate the efficiency of the proposed method.