A range-relaxed criteria for choosing the Lagrange multipliers in the Levenberg–Marquardt–Kaczmarz method for solving systems of non-linear ill-posed equations: Application to EIT-CEM with real data

Abstract

Abstract In this article we propose and analyze a Levenberg–Marquardt–Kaczmarz-type (LMK) method for obtaining stable approximate solutions to systems of ill-posed equations modeled by non-linear operators acting between Hilbert spaces. We extend to the LMK iteration the strategy proposed in [A. Leitão, F. Margotti and B. F. Svaiter, Range-relaxed criteria for choosing the Lagrange multipliers in the Levenberg–Marquardt method, IMA J. Numer. Anal. 41 2021, 4, 2962–2989] for choosing the Lagrange multipliers in the Levenberg–Marquardt (LM) method. Our main goal is to devise a simple (and easy to implement) strategy for computing the multiplier in each iterative step, such that the resulting LMK iteration is both stable and numerically efficient. Convergence analysis for the proposed LMK type method is provided, including convergence for exact data, stability and semi-convergence. Numerical experiments using real data are presented for a 2D parameter identification problem, namely the Electrical Impedance Tomography (EIT) problem. The mathematical model known as complete electrode model (EIT-CEM) is considered. The obtained numerical results validate the efficiency of the proposed LMK-type method.