Range-relaxed Levenberg-Marquardt Kaczmarz method for nonlinear ill-posed problems with strongly convex penalty

Abstract In this manuscript, we propose and analyze a Levenberg-Marquardt-Kaczmarz-$\Theta$ (LMK-$\Theta$) method for obtaining stable reconstructions for systems of nonlinear ill-posed problems between Hilbert spaces. A strongly convex penalty term is incorporated to reconstruct the special features of the sought solution, such as sparsity and discontinuity. The so-called range-relaxed criterion is utilized to determine the appropriate regularization parameter. Convergence analysis for the proposed LMK-$\Theta$ method is provided, including convergence for exact data, stability, and strong convergence for noisy data. Current density impedance imaging (CDII) is presented as a numerical example. We test a geometric phantom and a brain phantom, and the numerical results demonstrate the effectiveness of the proposed LMK-$\Theta$ method. The reconstructions not only achieve good overall accuracy but also successfully recover important structural features of the conductivity, including its discontinuities.