Convergence analysis for modified PAHSS-PU method with new parameter setting
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Convergence analysis for modified PAHSS-PU method with new parameter setting Bo Wu1 · Xing-Bao Gao1 Received: 17 September 2019 / Revised: 13 January 2020 / Accepted: 18 January 2020 © SBMAC - Sociedade Brasileira de Matemática Aplicada e Computacional 2020
Abstract Based on the modified preconditioned accelerated Hermitian and skew-Hermitian splitting (MPAHSS) and triangular splitting iterative (TSI) methods, this paper presents a new parameter setting to overcome the drawbacks of the MPAHSS-parameterized Uzawa (MPAHSS-PU) method proposed by Huang et al. (J Comput Appl Math 332:1–12, 2018) for sthe saddle point problems. A sufficient condition is provided to ensure the convergence of MPAHSSPU method with the new parameter setting, and a selection strategy for its parameters is also given. The new parameter setting not only lessens the parameter limitation of the MPAHSSPU method, but also improves its performance. The validity of the obtained results and the performance of MPAHSS-PU method with the new parameter setting are demonstrated by numerical examples. Keywords Saddle point problems · MPAHSS-PU method · New parameter setting · Convergence Mathematics Subject Classification 65F10 · 65F30 · 65F50
1 Introduction Consider the linear system with 2 × 2 block structure: Az ≡
A B −B T 0
x f = ≡ b, y −g
(1.1)
Communicated by Jinyun Yuan. This work is supported by the National Natural Science Foundation of China under Grant 61273311 and Grant 61502290.
B 1
Xing-Bao Gao [email protected] School of Mathematics and Information Science, Shaanxi Normal University, Xi’an 710062, Shaanxi, China 0123456789().: V,-vol
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B. Wu, X.-B. Gao
where A ∈ Rm×m is symmetric and positive definite, B ∈ Rm×n is of full column rank (n m), x, f ∈ Rm , y, g ∈ Rn , and B T is the transpose of B. Then, (1.1) has a unique solution (Benzi et al. 2005). The saddle point problem (1.1) arises extensively in the fields of scientific computing and engineering applications, including computational fluid dynamics, optimization, optimal control, and constrained and weighted least squares estimation (see Bai 2006; Betts 2001; Bjorck 1996; Elman 2002; Gould et al. 2001 and references therein). Although the sparse direct solvers are very competitive, they are less efficient for the large and sparse system (1.1) due to the requirement of computer storage. In contrast to the direct methods, iterative methods are effective for (1.1), and many promising iterative methods have been proposed to approximate its solution using the matrix splitting (see Bai 2006, 2018a, b; Bai and Benzi 2016; Bai and Golub 2007; Bai and Wang 2008; Bai et al. 2003, 2004, 2005, 2006; Cao 2018; Chen 2015; Elman and Golub 1994; Golub et al. 2001; Huang 2014; Huang et al. 2018a, 2019; Miao 2017; Njeru and Guo 2015; Shao et al. 2019; Shen et al. 2018; Wang and Zhang 2013; Yun 2013; Zheng and Ma 2016a). Among them, the Hermitian and skew-Hermitian splitting (HSS)-type methods (Bai 2009, 2018a; Bai and Benzi 2016; Bai and Golub 2007; Bai et al
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