Convergence Analysis of the Nonlinear Kaczmarz Method for Systems of Nonlinear Equations with Componentwise Convex Mappings and Applications to Image Reconstruction in Multispectral CT

Motivated by a class of nonlinear imaging inverse problems, for instance, multispectral computed tomography (MSCT), this paper studies the convergence theory of the nonlinear Kaczmarz method (NKM) for solving the system of nonlinear equations with componentwise convex mapping, namely, the function corresponding to each equation being convex. Such kind of nonlinear mapping may not satisfy the commonly used componentwise tangential cone condition (TCC). For this purpose, we propose a novel condition named relative gradient discrepancy condition (RGDC) and make use of it to prove the convergence and even the convergence rate of the NKM with several general index selection strategies, where these strategies include the cyclic strategy and the maximum residual strategy. Particularly, we investigate the application of the NKM for solving nonlinear systems in MSCT image reconstruction. We prove that the nonlinear mappings in this context fulfill the proposed RGDC rather than the componentwise TCC and provide a global convergence of the NKM based on the previously obtained results. Numerical experiments further illustrate the numerical convergence of the NKM for MSCT image reconstruction.