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 The ULT-HSS hybrid iteration method for symmetric saddle point problems

2021 ◽  
pp. 128-128
Author(s):  
Jun-Feng Lu
Keyword(s):  
Saddle Point ◽  
Iteration Method ◽  
Numerical Experiments ◽  
Optimal Parameter ◽  
Sufficient Conditions ◽  
Saddle Point Problem ◽  
Point Problem ◽  
Saddle Point Problems ◽  
Necessary And Sufficient ◽  
Direction Iteration

This paper proposes a hybrid iteration method for solving symmetric saddle point problem arising in computational fluid dynamics. It is an implicit alternative direction iteration method and named as the ULT-HSS method. The convergence analysis is provided, and the necessary and sufficient conditions are given for the convergence of the method. Some practical approaches are formulated for setting the optimal parameter of the method. Numerical experiments are given to show its efficiency.

scholarly journals New Preconditioners for Nonsymmetric Saddle Point Systems with Singular (1,1) Block

2013 ◽  
Vol 2013 ◽  
pp. 1-8
Author(s):  
Qingbing Liu
Keyword(s):  
Saddle Point ◽  
Krylov Subspace ◽  
Optimal Parameter ◽  
Spectral Characteristics ◽  
Ill Conditioning ◽  
Preconditioned Iterative ◽  
Preconditioned Iterative Methods ◽  
Saddle Point Matrix ◽  
Large Linear Systems ◽  
Point Type

We investigate the solution of large linear systems of saddle point type with singular (1,1) block by preconditioned iterative methods and consider two parameterized block triangular preconditioners used with Krylov subspace methods which have the attractive property of improved eigenvalue clustering with increased ill-conditioning of the (1,1) block of the saddle point matrix, including the choice of the parameter. Meanwhile, we analyze the spectral characteristics of two preconditioners and give the optimal parameter in practice. Numerical experiments that validate the analysis are presented.

A Fast Shift-Splitting Iteration Method for Nonsymmetric Saddle Point Problems

2017 ◽  
Vol 7 (1) ◽  
pp. 172-191 ◽  
Author(s):  
Quan-Yu Dou ◽  
Jun-Feng Yin ◽  
Ze-Yu Liao
Keyword(s):  
Saddle Point ◽  
Iteration Method ◽  
Numerical Experiments ◽  
Saddle Point Problems ◽  
Splitting Technique ◽  
Splitting Iteration Method

AbstractBased on the shift-splitting technique and the idea of Hermitian and skew-Hermitian splitting, a fast shift-splitting iteration method is proposed for solving nonsingular and singular nonsymmetric saddle point problems in this paper. Convergence and semi-convergence of the proposed iteration method for nonsingular and singular cases are carefully studied, respectively. Numerical experiments are implemented to demonstrate the feasibility and effectiveness of the proposed method.

scholarly journals New Preconditioning Techniques for Saddle Point Problems Arising from the Time-Harmonic Maxwell Equations

2013 ◽  
Vol 2013 ◽  
pp. 1-8 ◽  
Author(s):  
Qingbing Liu
Keyword(s):  
Saddle Point ◽  
Numerical Experiments ◽  
Maxwell Equations ◽  
Saddle Point Problems ◽  
Finite Element Discretization ◽  
Preconditioning Techniques ◽  
Element Discretization ◽  
Electromagnetic Problems ◽  
Time Harmonic ◽  
Saddle Point Matrices

We study two parameterized preconditioners for iteratively solving the saddle point linear systems arising from finite element discretization of the mixed formulation of the time-harmonic Maxwell equations in electromagnetic problems. We establish some spectral properties of the preconditioned saddle point matrices, involving the choice of the parameter. Meanwhile, we provide some results of numerical experiments to show the effectiveness of the proposed parameterized preconditioners.

scholarly journals ANALYSIS OF THE INEXACT UZAWA ALGORITHMS FOR NONLINEAR SADDLE-POINT PROBLEMS

2010 ◽  
Vol 51 (3) ◽  
pp. 369-382 ◽  
Author(s):  
JIAN-LEI LI ◽  
TING-ZHU HUANG ◽  
LIANG LI
Keyword(s):  
Saddle Point ◽  
Numerical Experiments ◽  
Sufficient Condition ◽  
Saddle Point Problems ◽  
Simple Sufficient Condition ◽  
Inexact Uzawa Algorithms

AbstractInexact Uzawa algorithms for solving nonlinear saddle-point problems are proposed. A simple sufficient condition for the convergence of the inexact Uzawa algorithms is obtained. Numerical experiments show that the inexact Uzawa algorithms are convergent.

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.

A relaxed splitting preconditioner for saddle point problems

2015 ◽  
Vol 23 (4) ◽  
Author(s):  
Cui-Xia Li ◽  
Shi-Liang Wu ◽  
Ai-Qun Huang
Keyword(s):  
Saddle Point ◽  
Spectral Properties ◽  
Numerical Experiments ◽  
Saddle Point Problems ◽  
Preconditioned Matrix ◽  
Appl Math

AbstractIn this paper, we introduce a relaxed splitting preconditioner for saddle point problems. Spectral properties of the preconditioned matrix are analyzed and compared with the closely related preconditioner in recent paper [New preconditioners for saddle point problems, Appl. Math. Comput. 172 (2006), 762-771] by Pan et al. Numerical experiments are given to illustrate the efficiency of the proposed precoditioner.

Generalized Accelerated Hermitian and Skew-Hermitian Splitting Methods for Saddle-Point Problems

2017 ◽  
Vol 10 (1) ◽  
pp. 167-185 ◽  
Author(s):  
H. Noormohammadi Pour ◽  
H. Sadeghi Goughery
Keyword(s):  
Saddle Point ◽  
Unique Solution ◽  
Numerical Experiments ◽  
Saddle Point Problem ◽  
Splitting Methods ◽  
Point Problem ◽  
Saddle Point Problems ◽  
Iteration Methods ◽  
Hss Iteration ◽  
Iteration Parameters

AbstractWe generalize the accelerated Hermitian and skew-Hermitian splitting (AHSS) iteration methods for large sparse saddle-point problems. These methods involve four iteration parameters whose special choices can recover the preconditioned HSS and accelerated HSS iteration methods. Also a new efficient case is introduced and we theoretically prove that this new method converges to the unique solution of the saddle-point problem. Numerical experiments are used to further examine the effectiveness and robustness of iterations.

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