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

Yu Gao, Chong Chen

TL;DR

This work introduces the relative gradient discrepancy condition (RGDC) as a non-TCC nonlinearity framework to establish convergence and linear-rate convergence for the nonlinear Kaczmarz method (NKM) when solving systems with component-wise convex mappings. It proves global convergence under RGDC for several index-selection strategies, and demonstrates the framework on MSCT image reconstruction in geometrically consistent and inconsistent settings. The analysis shows that MSCT mappings satisfy RGDC and yields linear convergence rates, complemented by numerical experiments validating rapid convergence and accurate reconstructions. The results offer a practical convergence theory for a broad class of nonlinear inverse problems where component-wise convexity precludes standard TCC-based analysis, with MSCT as a key motivating and validating application.

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.

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

TL;DR

This work introduces the relative gradient discrepancy condition (RGDC) as a non-TCC nonlinearity framework to establish convergence and linear-rate convergence for the nonlinear Kaczmarz method (NKM) when solving systems with component-wise convex mappings. It proves global convergence under RGDC for several index-selection strategies, and demonstrates the framework on MSCT image reconstruction in geometrically consistent and inconsistent settings. The analysis shows that MSCT mappings satisfy RGDC and yields linear convergence rates, complemented by numerical experiments validating rapid convergence and accurate reconstructions. The results offer a practical convergence theory for a broad class of nonlinear inverse problems where component-wise convexity precludes standard TCC-based analysis, with MSCT as a key motivating and validating application.

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.
Paper Structure (27 sections, 17 theorems, 105 equations, 10 figures, 1 table, 3 algorithms)

This paper contains 27 sections, 17 theorems, 105 equations, 10 figures, 1 table, 3 algorithms.

Table of Contents

  1. Introduction
  2. Mathematical preliminaries
  3. Notation
  4. Preliminary results
  5. Convergence analysis
  6. A brief analysis for the component-wise local TCC
  7. The proposed condition
  8. Convergence analysis of the NKM
  9. Case 1: $F_{j_k}(\boldsymbol{x}^{k})\ge 0$.
  10. Case 2: $F_{j_k}(\boldsymbol{x}^{k})< 0$.
  11. Convergence of the NKM for MSCT reconstruction
  12. Discrete-to-discrete (DD)-data model of MSCT
  13. Geometrically-consistent MSCT
  14. Geometrically-inconsistent MSCT
  15. Numerical experiments
  16. ...and 12 more sections

Key Result

Proposition 2.1

(Scherzer08) Suppose that $\boldsymbol{F}$ satisfies local TCC eq:local_tcc_mapping in $\mathcal{B}_{\rho}(\boldsymbol{x}_0)$. Then for any $\bar{\boldsymbol{x}}\in\mathcal{B}_{\rho}(\boldsymbol{x}_0)$, and $Null(\boldsymbol{F}^{\prime}(\boldsymbol{x}))= Null(\boldsymbol{F}^{\prime}(\bar{\boldsymbol{x}}))$ for $\boldsymbol{x}\in \{\boldsymbol{x}\mid \boldsymbol{F}(\boldsymbol{x})=\boldsymbol{F}(\

Figures (10)

  • Figure 1: The diagram depicts an example which fulfills the \ref{['ass:relative_grad_discrepancy']}.
  • Figure 2: The figure depicts an example for the iterated procedure of the NKM using \ref{['stra:non_negative']}.
  • Figure 3: The figure depicts an example for the iterated procedure of the NKM satisfying strategies \ref{['stra:non_negative']} and \ref{['stra:infinite_often']}.
  • Figure 4: The figure depicts the two cases for the sign of $F_{j_k}(\boldsymbol{x}^k)$ in the NKM iteration \ref{['eq:non_kacz']}, as analyzed in \ref{['thm:nkm_converge_rate']}.
  • Figure 5: The used spectra pairs. Left: Spectra pair I, which contains 80-kV spectrum (red, solid) and 140-kV spectrum (blue, dashed). Right: Spectra pair II, which contains 80-kV spectrum (red, solid) and filtered 140-kV spectrum (blue, dashed dot).
  • ...and 5 more figures

Theorems & Definitions (44)

  • Definition 1
  • Proposition 2.1
  • Definition 2
  • Proposition 3.1
  • proof
  • Corollary 3.1
  • proof
  • Remark 3.1
  • Remark 3.2
  • Proposition 3.2
  • ...and 34 more