Convergence and Semi-Convergence of a Class of Constrained Block Iterative Methods. (26th October 2021)

• Record Type: Journal Article
• Authors: Mirzapour, Mahdi; Cegielski, Andrzej; Elfving, Tommy
• Is Part Of: Numerical functional analysis and optimization. Volume 42:Number 14(2021)
• Page Start: 1718
• Page End: 1746
• Publication Date: 2021-10-26
• Subjects: Block iterative method -- constraints -- Landweber type iteration -- relaxation parameters -- split feasibility problem -- semi-convergence -- tomographic imaging; 68Q25 -- 68R10 -- 68U05; Functional analysis -- Periodicals; Numerical analysis -- Periodicals; Mathematical optimization -- Periodicals; Numerical Analysis, Computer-Assisted; 515.705
• DOI: 10.1080/01630563.2021.2001822
• Languages: English
• ISSNs: 0163-0563

Abstract

Abstract: In this paper, we analyze the 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.