A generalized SHSS preconditioner for generalized saddle point problem
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A generalized SHSS preconditioner for generalized saddle point problem Jun Li1 · Shu-Xin Miao1 Received: 15 January 2020 / Revised: 30 June 2020 / Accepted: 8 July 2020 © SBMAC - Sociedade Brasileira de Matemática Aplicada e Computacional 2020
Abstract In this paper, based on the simplified Hermitian and skew-Hermitian (SHSS) preconditioner, a new preconditioner, called the generalized SHSS (GSHSS) preconditioner, is considered to solve the generalized saddle point problem. We prove that the GSHSS iteration method is convergent if the iteration parameters satisfy appropriate conditions. In addition, it is proved that all eigenvalues of the GSHSS preconditioned matrix are real and non-unit eigenvalues are located in a positive interval. Numerical experiment is provided to show the effectiveness of the GSHSS preconditioner. Keywords Generalized saddle point problems · GSHSS preconditioner · Preconditioning · Spectral properties Mathematics Subject Classification 65F10 · 65F50
1 Introduction We consider the iteration solution of large generalized saddle point problem of the following form: x p A BT = ≡ b, (1) Ax≡ y q −B C where A ∈ Rn×n is symmetric positive definite (SPD) matrix, B ∈ Rm×n has full row rank, C ∈ Rm×m is symmetric positive semi-definite (SPSD) matrix, p ∈ Rn and q ∈ Rm are
Communicated by Andreas Fischer. This work is supported by National Natural Science Foundation of China (nos. 11701458, 11861059).
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Jun Li [email protected] Shu-Xin Miao [email protected]
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College of Mathematics and Statistics, Northwest Normal University, Lanzhou 730070, People’s Republic of China 0123456789().: V,-vol
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J. Li, S.-X. Miao
given vectors, and B T denotes the transpose of the matrix B. It is easy to demonstrate that the solution of (1) exists and is unique under the assumptions above; see Bai (2009), Bai et al. (2006) and Bai and Wang (2008). Generalized saddle point problem of the form (1) arises in many scientific computing and engineering applications such as constrained optimization, computational fluid dynamics, mixed finite element methods for solving elliptic partial differential equations and Stokes problems, constrained least-squares problems, structure analysis and so on; see Benzi and Golub (2004). Many effective iteration methods have been studied for generalized saddle point problem, such as Hermitian and skew-Hermitian (HSS) iteration methods (Bai 2009; Bai et al. 2003, 2004, 2007), Uzawa-type methods (Yang and Wu 2014; Miao 2017), conjugate gradient methods (Bai and Wang 2006) and so on. These iteration methods can not only be used as stationary iteration solvers, but they can also be served as preconditioners for Krylov subspace methods (Salkuyeh and Masoudi 2016; Cao et al. 2019), leading to a more efficacious class of solvers. Due to the elegant mathematical properties, HSS method has attracted much attention and many researchers have finished many works on the basis of it. We know that Bai et al. (2004) discussed the following HSS preconditioner:
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