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 component-wise convex mapping, namely, the function corresponding to each equation being convex. However, such kind of nonlinear mapping may not satisfy the commonly used component-wise 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 cyclic strategy and maximum residual strategy. Particularly, we investigate the application of the NKM for solving nonlinear systems in MSCT image reconstruction. We prove that the nonlinear mapping in this context fulfills the proposed RGDC rather than the component-wise 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.

翻译：受一类非线性成像逆问题（例如多光谱计算机断层扫描（MSCT））的启发，本文研究了非线性Kaczmarz方法（NKM）在求解具有分量凸映射（即每个方程对应的函数为凸函数）的非线性方程组时的收敛理论。然而，此类非线性映射可能不满足常用的分量切向锥条件（TCC）。为此，本文提出一种名为相对梯度偏差条件（RGDC）的新条件，并利用该条件证明了NKM在若干通用指标选取策略（包括循环策略和最大残差策略）下的收敛性乃至收敛速率。特别地，本文研究了NKM在MSCT图像重建非线性方程组求解中的应用。我们证明了该场景下的非线性映射满足所提出的RGDC而非分量TCC，并基于先前结果给出了NKM的全局收敛性。数值实验进一步展示了NKM在MSCT图像重建中的数值收敛性。