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 systems of nonlinear equations with component-wise convex mapping, namely, the function corresponding to each equation being convex. Although the tangential cone condition (TCC) is often used to prove the convergence of NKM, it may be impossible or difficult to verify/satisfy this condition for such kind of nonlinear systems. 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 NKM with several general index selection strategies, where these strategies include the cyclic strategy and maximum residual strategy. Particularly, we investigate the application of NKM for solving nonlinear systems in MSCT image reconstruction. We prove that the nonlinear mapping of interest fulfills the proposed RGDC rather than the component-wise local TCC, and provide the global convergence of NKM based on the previously obtained results. Numerical experiments further illustrate the numerical convergence of NKM for MSCT image reconstruction.

翻译：受一类非线性成像逆问题（例如多谱计算机断层扫描（MSCT））的启发，本文研究非线性Kaczmarz方法（NKM）在求解具有分量凸映射的非线性方程组系统（即每个方程对应的函数为凸函数）时的收敛性理论。尽管切向锥条件（TCC）常被用于证明NKM的收敛性，但对于此类非线性系统，该条件可能无法满足或难以验证/满足。我们提出一种名为相对梯度差异条件（RGDC）的新条件，并利用它证明了在多种通用索引选择策略（包括循环策略和最大残差策略）下NKM的收敛性乃至收敛速率。特别地，我们研究了NKM在MSCT图像重建中求解非线性系统的应用。我们证明所关注的非线性映射满足所提出的RGDC而非分量局部TCC，并基于先前获得的结果给出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。