On parameterized block symmetric positive definite preconditioners for a class of block three-by-three saddle point problems

AbstractA lopsided alternating direction iteration (LADI) method and an induced block diagonal preconditioner for solving block two-by-two generalised saddle point problems are presented. The convergence of the LADI method is analysed, and the block diagonal preconditioner can accelerate the convergence rates of Krylov subspace iteration methods such as GMRES. Our new preconditioned method only requires a solver for two linear equation sub-systems with symmetric and positive definite coefficient matrices. Numerical experiments show that the GMRES with the new preconditioner is quite effective.

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An extension of the positive-definite and skew-Hermitian splitting method for preconditioning of generalized saddle point problems

Computers & Mathematics with Applications ◽

10.1016/j.camwa.2019.10.030 ◽

2020 ◽

Vol 79 (8) ◽

pp. 2304-2321

Author(s):

Mohsen Masoudi ◽

Davod Khojasteh Salkuyeh

Keyword(s):

Saddle Point ◽

Splitting Method ◽

Positive Definite ◽

Saddle Point Problems ◽

Generalized Saddle Point Problems

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Inexact modified positive-definite and skew-Hermitian splitting preconditioners for generalized saddle point problems

Advances in Mechanical Engineering ◽

10.1177/1687814018804092 ◽

2018 ◽

Vol 10 (10) ◽

pp. 168781401880409

Author(s):

Qin-Qin Shen ◽

Quan Shi

Keyword(s):

Saddle Point ◽

Minimal Polynomial ◽

Stokes Problem ◽

Positive Definite ◽

Navier Stokes ◽

Saddle Point Problems ◽

Numer Algor ◽

New Class ◽

Generalized Saddle Point Problems ◽

Theoretical Results

In this article, to better implement the modified positive-definite and skew-Hermitian splitting preconditioners studied recently (Numer. Algor., 72 (2016) 243–258) for generalized saddle point problems, a class of inexact modified positive-definite and skew-Hermitian splitting preconditioners is proposed with improved computing efficiency. Some spectral properties, including the eigenvalue distribution, the eigenvector distribution, and an upper bound of the degree of the minimal polynomial of the inexact modified positive-definite and skew-Hermitian splitting preconditioned matrices are studied. In addition, a theoretical optimal inexact modified positive-definite and skew-Hermitian splitting preconditioner is obtained. Numerical experiments arising from a model steady incompressible Navier–Stokes problem are used to validate the theoretical results and illustrate the effectiveness of this new class of proposed preconditioners.

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New modified shift-splitting preconditioners for non-symmetric saddle point problems

Arabian Journal of Mathematics ◽

10.1007/s40065-019-0256-6 ◽

2019 ◽

Vol 9 (2) ◽

pp. 245-257

Author(s):

Mahin Ardeshiry ◽

Hossein Sadeghi Goughery ◽

Hossein Noormohammadi Pour

Keyword(s):

Saddle Point ◽

Saddle Point Problem ◽

Positive Definite ◽

Point Problem ◽

Gmres Method ◽

Saddle Point Problems ◽

Singular Saddle Point Problems ◽

Performance Results ◽

Preconditioned Gmres ◽

Point Form

Abstract Zhou et al. and Huang et al. have proposed the modified shift-splitting (MSS) preconditioner and the generalized modified shift-splitting (GMSS) for non-symmetric saddle point problems, respectively. They have used symmetric positive definite and skew-symmetric splitting of the (1, 1)-block in a saddle point problem. In this paper, we use positive definite and skew-symmetric splitting instead and present new modified shift-splitting (NMSS) method for solving large sparse linear systems in saddle point form with a dominant positive definite part in (1, 1)-block. We investigate the convergence and semi-convergence properties of this method for nonsingular and singular saddle point problems. We also use the NMSS method as a preconditioner for GMRES method. The numerical results show that if the (1, 1)-block has a positive definite dominant part, the NMSS-preconditioned GMRES method can cause better performance results compared to other preconditioned GMRES methods such as GMSS, MSS, Uzawa-HSS and PU-STS. Meanwhile, the NMSS preconditioner is made for non-symmetric saddle point problems with symmetric and non-symmetric (1, 1)-blocks.

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