Updated preconditioned Hermitian and skew-Hermitian splitting-type iteration methods for solving saddle-point problems
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Updated preconditioned Hermitian and skew-Hermitian splitting-type iteration methods for solving saddle-point problems Fang Chen1
· Tian-Yi Li1 · Kang-Ya Lu1
Received: 12 February 2020 / Revised: 21 April 2020 / Accepted: 6 May 2020 © SBMAC - Sociedade Brasileira de Matemática Aplicada e Computacional 2020
Abstract For the saddle-point problems, we discuss and analyze the updated preconditioned Hermitian and skew-Hermitian splitting (UPHSS) iteration method in detail. On this basis, we introduce a two-stage iteration method for the UPHSS iteration method. Theoretical analysis shows that the UPHSS and two-stage UPHSS iteration methods are convergent to the unique solution of the saddle-point linear system when the parameter is suitably chosen. Numerical examples show the correctness of the theory and the effectiveness of these methods. Keywords Saddle-point linear problem · Hermitian and skew-Hermitian splitting · Matrix splitting iteration · Convergence Mathematics Subject Classification 65F10 · 65F15
1 Introduction We consider the saddle-point problem B E y f Ax ≡ = ≡ b, ∗ −E 0 z g
(1.1)
where B ∈ C p× p is Hermitian positive definite, and E ∈ C p×q is of full column rank ( p ≥ q), with n = p + q. Therefore, A ∈ Cn×n is nonsingular, and the solution of the saddle-point problem (1.1) exists and is unique. This kind of problem arises in many scientific
Communicated by Zhong-Zhi Bai. Supported by The National Natural Science Foundation (no. 11501038), and The Science and Technology Planning Projects of Beijing Municipal Education Commission (no. KM201911232010, no. KM202011232019 and no. KM201811232020), People’s Republic of China.
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Fang Chen [email protected] School of Applied Science, Beijing Information Science and Technology University, Beijing 100192, People’s Republic of China 0123456789().: V,-vol
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and engineering applications. A large variety of methods for solving linear systems of the form (1.1) can be found in the literature, such as Uzawa-type schemes (Elman and Golub 1994; Bramble et al. 1997), matrix splitting methods (Bai and Benzi 2017), and Cholesky factorization method (Zhao 1998), etc. In accordance with the Hermitian and skew-Hermitian splitting (HSS) method (Bai et al. 2003), many iteration methods have been proposed to solve linear systems of the form (1.1) (Bai 2018; Bai and Benzi 2017; Bai and Golub 2007; Bai et al. 2004; Benzi and Golub 2004; Cao et al. 2016; Huang et al. 2016, 2018; Wang et al. 2016; Yang et al. 2010). For the more recent results, we mention the regulated HSS iteration method in Bai and Benzi (2017) which is based on the Hermitian and skew-Hermitian splitting. Numerical experiments show that these HSS methods and its variants are very effective for solving the saddle-point problem (1.1). In general, the matrix splitting iteration methods of the form M x (k+1) = N x (k) + b can be used to solve the saddle-point problem (1.1), where M is a nonsingular matrix such that A = M − N . This class of iteration methods
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