A class of modified DPSS preconditioners for generalized saddle-point linear systems
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(2019) 38:84
A class of modified DPSS preconditioners for generalized saddle-point linear systems Zhao-Zheng Liang1 · Guo-Feng Zhang1 Received: 14 October 2018 / Revised: 16 March 2019 / Accepted: 18 March 2019 © SBMAC - Sociedade Brasileira de Matemática Aplicada e Computacional 2019
Abstract In this paper, a class of new preconditioners based on matrix splitting are presented for generalized saddle-point linear systems, which improve some recently published preconditioners in view of spectral distributions and numerical performances. Moreover, we widen the scope of the new preconditioners to solve more general but rarely considered saddle-point linear systems with singular leading blocks and rank-deficient off-diagonal blocks. The new variants can result in much better convergence properties and spectrum distributions than the original existing preconditioners. Numerical experiments are given to illustrate the efficiency of the new proposed preconditioners. Keywords Saddle-point linear systems · Matrix splitting · Preconditioning · Convergence analysis · Spectral analysis Mathematics Subject Classification 65F10 · 65F08 · 65F15 · 65F50
1 Introduction We consider the following generalized saddle-point linear system: x f A BT Au ≡ = ≡ c, y g −B C
(1)
where A ∈ Rn×n is positive semidefinite, in the sense that it has positive semidefinite symmetric part, C ∈ Rm×m is symmetric positive semidefinite. We further assume that B ∈ Rm×n satisfies
Communicated by Ernesto G. Birgin. This work was supported by the National Natural Science Foundation of China (Nos. 11801242 and 11771193) and the Fundamental Research Funds for the Central Universities (No. lzujbky-2018-31).
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Zhao-Zheng Liang [email protected] Guo-Feng Zhang [email protected]
1
School of Mathematics and Statistics, Lanzhou University, Lanzhou 730000, People’s Republic of China
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Z.-Z. Liang, G.-F. Zhang
null(B T ) ∩ null(C) = {0} and null(H ) ∩ null(B) = {0}
(2)
1 2 (A
+ A T ) to guarantee the nonsingularity of the saddle-point matrix A in (1); with H = see Lemma 1 for detailed proofs. Here, (·)T and null(·) denote the transpose and the null space of a matrix, respectively. Linear systems of the form (1) are ubiquitous in science and engineering and rightfully a large amount of solution methods have been proposed in the literature; see, for example, (Benzi et al. 2005; Wathen 2015) for a comprehensive list of applications and numerical solution methods. Compared with the direct methods, iterative methods are more recommended due to the large, sparse structure of the linear system (1). However, well-established iterative methods such as the Krylov subspace methods Saad (2003), if without preconditioning, will converge slowly or even fail to converge. As a general rule, the preconditioning should reduce the number of iterations required; meanwhile, it should not significantly increase the amount of computation required at each iteration. Efficient preconditioning techniques for saddle-point linear system (1) inclu
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