Efficient HSS-based preconditioners for generalized saddle point problems
- PDF / 734,116 Bytes
- 19 Pages / 439.37 x 666.142 pts Page_size
- 30 Downloads / 225 Views
Efficient HSS-based preconditioners for generalized saddle point problems Ke Zhang1 · Lin-Na Wang1 Received: 10 April 2019 / Revised: 3 April 2020 / Accepted: 29 April 2020 © SBMAC - Sociedade Brasileira de Matemática Aplicada e Computacional 2020
Abstract A fast iteration method based on HSS is proposed for solving the nonsymmetric generalized saddle point problem. It converges to the unique solution of the generalized saddle point problem unconditionally. We devise a new preconditioner induced by the new iteration method. We analyze the spectrum of the preconditioned coefficient matrix, and reveal the relation between the theoretically required number of iteration steps and the dimension of the preconditioned Krylov subspace. Furthermore, some practical inexact variants of the new preconditioner have been developed to reduce the computational overhead. Numerical experiments validate the effectiveness of the proposed preconditioners. Keywords Generalized saddle point problem · Preconditioner · Hermitian and skew-Hermitian splitting · Krylov subspace method Mathematics Subject Classification 65F10 · 65N22
1 Introduction We consider the solution of the generalized saddle point linear system f A BT x = , or Au = b, y −g −B C
(1)
where A is an n × n symmetric and positive definite matrix, B is an m × n full-rank matrix, C is an m × m symmetric and positive semidefinite matrix and m ≤ n. With such assumptions, the existence and uniqueness of the solution of (1) can be assured. The block linear system (1) stems from many real-world applications, including the constrained optimal control (Betts 2001), the computational fluid dynamics (Saad 2003) and the mixed finite element of elliptic PDEs (Benzi et al. 2005), etc.
Communicated by Natasa Krejic.
B 1
Ke Zhang [email protected] Department of Mathematics, Shanghai Maritime University, Shanghai 201306, People’s Republic of China 0123456789().: V,-vol
123
154
Page 2 of 19
K. Zhang, L.-N. Wang
Recent years have witnessed vigorous developments in the numerical algorithms for solving (1). Due to the large size and sparsity of the coefficient matrix A, iterative methods are usually favored. In addition to those ad hoc ones, existing approaches include the stationary iteration schemes (Bai et al. 2005b; Zhang and Shang 2010; Yun 2013; Zhang et al. 2019), the matrix-splitting methods (Bai et al. 2008; Cao et al. 2016; Bai and Golub 2007; Li and Wu 2015; Liang and Zhang 2016; Shen 2014; Zeng and Ma 2016), the more involved Krylov subspace methods (Bai 2015; Saad 2003) and their hybrid variants (Cao and Yi 2016). Among all methods, the restarted GMRES method (Saad and Schultz 1986) has received much attention and been used extensively in view of their effectiveness. However, it is known that GMRES(k) often suffers from slow convergence or even stagnation. Therefore, a reasonable preconditioner must be chosen to accelerate GMRES(k). By far, some kinds of preconditioners pertaining to Krylov subspace methods have been proposed, including the HSS-type preconditi
Data Loading...
