A new double-step splitting iteration method for certain block two-by-two linear systems
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A new double-step splitting iteration method for certain block two-by-two linear systems Zheng-Ge Huang1 Received: 10 January 2020 / Revised: 1 June 2020 / Accepted: 6 June 2020 © SBMAC - Sociedade Brasileira de Matemática Aplicada e Computacional 2020
Abstract We consider the iterative solution of certain block two-by-two linear systems and introduce a new double-step splitting (NDSS) iteration method. The proposed method is based on the transformed matrix iteration method proposed recently, and obtained by applying two-step and preconditioning techniques for the original linear system. We prove that the NDSS iteration method is convergent under mild conditions. Upper bounds on the spectral radius of the iteration matrix of the NDSS method are presented and the parameters which minimize these bounds are computed. We also consider the inexact NDSS iteration method. The proposed methods are compared theoretically and numerically with some existing ones, which shows the good performance of the NDSS iteration method and its inexact version. Keywords Block two-by-two linear system · New double-step splitting iteration method · Two-step technique · Preconditioning technique · Convergence properties · Quasi-optimal parameters · Inexact implementation Mathematics Subject Classification 65F10 · 65F50
1 Introduction Consider the iterative solution of block two-by-two systems of linear equations of the form u W −T u p = = , A v T W v q
(1)
Communicated by Zhong-Zhi Bai. This research was supported by the National Natural Science Foundation of China (No. 11901123), the Guangxi Natural Science Foundation (No. 2018JJB110062) and the Xiangsihu Young Scholars Innovative Research Team of Guangxi University for Nationalities (No. 2019RSCXSHQN03).
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Zheng-Ge Huang [email protected] Faculty of Science, Guangxi University for Nationalities, Nanning 530006, People’s Republic of China 0123456789().: V,-vol
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where W , T ∈ Rn×n are symmetric positive semi-definite matrices with at least one of them, e.g., W , being positive definite. This kind of linear systems frequently arises in many problems in many scientific and engineering applications, such as finite element discretizations of elliptic partial differential equation (PDE)-constrained optimization problems, wave propagation, numerical solution of Cahn–Hilliard phase filed modeling problems, real equivalent formulations of complex symmetric linear systems (Bai et al. 2013; Axelsson and Salkuyeh 2019; Axelsson and Kucherov 2000; Liang and Zhang 2019), and so forth. There are some illustrations: • (Liang and Zhang 2019) Consider the distributed control problem, which can be transformed into the linear systems √ y b √M − β K = , βK M u d using the discretize-then optimize method, where M is the mass matrix and K is the stiffness matrix. • (Bai et al. 2013) Consider large and sparse complex symmetric linear systems of the form Ax ≡ (W + i T )x = b.
(2)
Let x = u + iv, b = p + iq and u, v, p, q ∈ Rn , then (2) ca
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