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Fast rotated BSOR method for block two-by-two linear systems with application to PDE-constrained optimal control problems

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Abstract

In this paper, we consider iterative solution of certain large scale block two-by-two linear systems arising from numerical solution process of some PDE-constrained optimal control problems. Based upon skillful rotating technique, a new fast and robust stationary iteration method is constructed from the idea of classical block successive over relaxation (BSOR) iteration. Equipped with a practical problem independent parameter choice strategy, the proposed method can result in a sharp parameter independent convergence rate close to 0.17. Moreover, a robust preconditioner is developed from an equivalent form of the new iteration method, which is suitable for inexact variable right preconditioning within Krylov subspace acceleration. Numerical examples from both distributed steady control problem and unsteady control problem which leads to complex Kronecker structured linear system are tested to show that the new solution methods are competitive to some existing ones.

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Authors and Affiliations

  1. School of Mathematics and Statistics, Lanzhou University, Lanzhou, 730000, People’s Republic of China

    Zhao-Zheng Liang

  2. School of Science, Lanzhou University of Technology, Lanzhou, People’s Republic of China

    Yan Dou

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  1. Zhao-Zheng Liang
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  2. Yan Dou
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Correspondence to Zhao-Zheng Liang.

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Communicated by Lothar Reichel.

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This work was supported by the National Natural Science Foundation of China (Nos. 11801242 and 11901267).

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Liang, ZZ., Dou, Y. Fast rotated BSOR method for block two-by-two linear systems with application to PDE-constrained optimal control problems. Bit Numer Math 62, 1175–1206 (2022). https://doi.org/10.1007/s10543-022-00908-0

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