Convergence and Semi-convergence of a class of constrained block iterative methods

TL;DR

This paper investigates the convergence and semi-convergence properties of projected block iterative methods for solving large, noisy, and ill-conditioned linear systems, providing new strategies for parameter selection to enhance image reconstruction.

Contribution

It introduces a detailed analysis of convergence for P-BIM, derives noise error bounds, and proposes a noise-dependent relaxation parameter strategy to improve reconstruction quality.

Findings

P-BIM converges linearly for polyhedral convex sets in finite dimensions.

An upper bound for noise error in P-BIM is established.

A new relaxation parameter strategy accelerates convergence and enhances image quality.

Abstract

In this paper, we analyze the convergence %semi-convergence properties of projected non-stationary block iterative methods (P-BIM) aiming to find a constrained solution to large linear, usually both noisy and ill-conditioned, systems of equations. We split the error of the $k$th iterate into noise error and iteration error, and consider each error separately. The iteration error is treated for a more general algorithm, also suited for solving split feasibility problems in Hilbert space. The results for P-BIM come out as a special case. The algorithmic step involves projecting onto closed convex sets. When these sets are polyhedral, and of finite dimension, it is shown that the algorithm converges linearly. We further derive an upper bound for the noise error of P-BIM. Based on this bound, we suggest a new strategy for choosing relaxation parameters, which assist in speeding up the reconstruction process and improving the quality of obtained images. The relaxation parameters may depend on the noise. The performance of the suggested strategy is shown by examples taken from the field of image reconstruction from projections.