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A Preconditioned MINRES Method for Block Lower Triangular Toeplitz Systems

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Abstract

In this study, a novel preconditioner based on the absolute-value block \(\alpha \)-circulant matrix approximation is developed, specifically designed for nonsymmetric dense block lower triangular Toeplitz (BLTT) systems that emerge from the numerical discretization of evolutionary equations. Our preconditioner is constructed by taking an absolute-value of a block \(\alpha \)-circulant matrix approximation to the BLTT matrix. To apply our preconditioner, the original BLTT linear system is converted into a symmetric form by applying a time-reversing permutation transformation. Then, with our preconditioner, the preconditioned minimal residual method (MINRES) solver is employed to solve the symmetrized linear system. With properly chosen \(\alpha \), the eigenvalues of the preconditioned matrix are proven to be clustered around \(\pm 1\) without any significant outliers. With the clustered spectrum, we show that the preconditioned MINRES solver for the preconditioned system has a convergence rate independent of system size. The efficacy of the proposed preconditioner is corroborated by our numerical experiments, which reveal that it attains optimal convergence.

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The data of natural images and codes involved in this paper are available from the corresponding author on reasonable request.

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Funding

The work of Sean Hon was supported in part by the Hong Kong RGC under grant 22300921, a start-up grant from the Croucher Foundation, and a Tier 2 Start-up Grant from Hong Kong Baptist University. The work of Xuelei Lin was supported by research Grants: 2021M702281 from China Postdoctoral Science Foundation, 12301480 from NSFC, HA45001143 from Harbin Institute of Technology, Shenzhen, HA11409084 from Shenzhen. The work of Shu-Lin Wu was supported by research Grants: Fundamental Research Funds for the Central Universities (No. 2412022ZD035) and the Natural Science Foundation of Jilin Province (YDZJ202201ZYTS593).

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Authors and Affiliations

  1. School of Science, Harbin Institute of Technology, Shenzhen, 518055, China

    Congcong Li & Xuelei Lin

  2. Department of Mathematics, Hong Kong Baptist University, Kowloon Tong, Hong Kong SAR, China

    Congcong Li & Sean Hon

  3. School of Mathematics and Statistics, Northeast Normal University, Changchun, 130024, China

    Shu-Lin Wu

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  1. Congcong Li
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Correspondence to Xuelei Lin.

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Li, C., Lin, X., Hon, S. et al. A Preconditioned MINRES Method for Block Lower Triangular Toeplitz Systems. J Sci Comput 100, 63 (2024). https://doi.org/10.1007/s10915-024-02611-4

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