scholarly journals Numerical solution of saddle point problems

Acta Numerica ◽  
2005 ◽  
Vol 14 ◽  
pp. 1-137 ◽  
Author(s):  
Michele Benzi ◽  
Gene H. Golub ◽  
Jörg Liesen
Keyword(s):  
Saddle Point ◽  
Linear Systems ◽  
Saddle Point Problems ◽  
Science And Engineering ◽  
Solution Methods ◽  
Computational Science And Engineering ◽  
Sparse Problems ◽  
Large Linear Systems ◽  
Point Form ◽  
Selection Of

Large linear systems of saddle point type arise in a wide variety of applications throughout computational science and engineering. Due to their indefiniteness and often poor spectral properties, such linear systems represent a significant challenge for solver developers. In recent years there has been a surge of interest in saddle point problems, and numerous solution techniques have been proposed for this type of system. The aim of this paper is to present and discuss a large selection of solution methods for linear systems in saddle point form, with an emphasis on iterative methods for large and sparse problems.

scholarly journals New Block Triangular Preconditioners for Saddle Point Linear Systems with Highly Singular (1,1) Blocks

2009 ◽  
Vol 2009 ◽  
pp. 1-13 ◽  
Author(s):  
TingZhu Huang ◽  
GuangHui Cheng ◽  
Liang Li
Keyword(s):  
Saddle Point ◽  
Linear Systems ◽  
Numerical Experiments ◽  
Optimal Parameter ◽  
Spectral Characteristics ◽  
Saddle Point Problems

We establish two types of block triangular preconditioners applied to the linear saddle point problems with the singular (1,1) block. These preconditioners are based on the results presented in the paper of Rees and Greif (2007). We study the spectral characteristics of the preconditioners and show that all eigenvalues of the preconditioned matrices are strongly clustered. The choice of the parameter is involved. Furthermore, we give the optimal parameter in practical. Finally, numerical experiments are also reported for illustrating the efficiency of the presented preconditioners.

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

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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