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A dimension expanded preconditioning technique for saddle point problems

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Abstract

In this paper, we present a new block preconditioner for solving the saddle point linear systems. The proposed method is developed from an augmented reformulation of the saddle point problem into a new linear system with an almost block triangular coefficient matrix. Theoretical results are derived on the eigenvalue distribution of the preconditioned matrix, and an efficient algorithmic implementation is developed and presented. Several numerical examples are reported to support the theoretical findings and to illustrate the favourable convergence properties of the proposed preconditioner, also compared to other popular solvers for saddle point problems.

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Acknowledgements

The authors would like to thank Reviewers and Editor for the valuable comments and helpful suggestions. This research was supported by NSFC (11801463), the Applied Basic Research Project of Sichuan Province (2020YJ0007), the Guanghua Talent Project of Southwestern University of Finance and Economics and the Scientific Research Fund of Hunan Provincial Science and Technology Department (2022JJ30416). The third author is a member of Gruppo Nazionale per il Calcolo Scientifico (GNCS) of Istituto Nazionale di Alta Matematica (INdAM), and this work was partially supported by INdAM-GNCS Project CUP_E55F22000270001.

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Authors and Affiliations

  1. School of Mathematics and Physics, Hunan University of Arts and Science, Changde, 415000, Hunan, People’s Republic of China

    Wei-Hua Luo

  2. School of Mathematics, Southwestern University of Finance and Economics, Chengdu, 611130, Sichuan, People’s Republic of China

    Xian-Ming Gu

  3. Faculty of Computer Science, Free University of Bozen-Bolzano, Bolzano, Italy

    Bruno Carpentieri

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  1. Wei-Hua Luo
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  2. Xian-Ming Gu
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  3. Bruno Carpentieri
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Correspondence to Xian-Ming Gu.

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Communicated by Michiel E. Hochstenbach.

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The original online version of this article was revised: In Table 14, the second “Density(B)\(=\)0.01” should be “Density(B)\(=\)0.1”. In Table 15, the second “Density(B)\(=\)0.001” should be “Density(B)=0.005”.

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Luo, WH., Gu, XM. & Carpentieri, B. A dimension expanded preconditioning technique for saddle point problems. Bit Numer Math 62, 1983–2004 (2022). https://doi.org/10.1007/s10543-022-00938-8

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  • DOI: https://doi.org/10.1007/s10543-022-00938-8

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