Convergence Analysis of Nonlinear Kaczmarz Method for Systems of Nonlinear Equations with Component-wise Convex Mapping

TL;DR

This paper develops a convergence theory for the nonlinear Kaczmarz method applied to systems with component-wise convex mappings, introducing a new condition called RGDC, and demonstrates its effectiveness in multispectral CT image reconstruction.

Contribution

It introduces the relative gradient discrepancy condition (RGDC) for convergence analysis of NKM without TCC, and applies it to nonlinear systems in MSCT imaging.

Findings

Proves convergence of NKM under RGDC for general strategies.

Establishes global convergence for MSCT image reconstruction.

Numerical experiments confirm the theoretical results.

Abstract

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.