A note on the block alternating splitting implicit iteration method for complex saddle-point problems

2018 ◽  
Vol 25 (6) ◽  
pp. e2209
Author(s):  
Zeng-Qi Wang
Keyword(s):  
Saddle Point ◽  
Iteration Method ◽  
Saddle Point Problems ◽  
Implicit Iteration

A New Uzawa-Type Iteration Method for Non-Hermitian Saddle-Point Problems

2017 ◽  
Vol 7 (1) ◽  
pp. 211-226
Author(s):  
Yan Dou ◽  
Ai-Li Yang ◽  
Yu-Jiang Wu
Keyword(s):  
Saddle Point ◽  
Iteration Method ◽  
Spectral Properties ◽  
Numerical Results ◽  
Saddle Point Problems ◽  
Convergence Properties ◽  
Saddle Point Matrix

AbstractBased on a preconditioned shift-splitting of the (1,1)-block of non-Hermitian saddle-point matrix and the Uzawa iteration method, we establish a new Uzawa-type iteration method. The convergence properties of this iteration method are analyzed. In addition, based on this iteration method, a preconditioner is proposed. The spectral properties of the preconditioned saddle-point matrix are also analyzed. Numerical results are presented to verify the robustness and the efficiency of the new iteration method and the preconditioner.

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 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.

Regularized HSS iteration methods for stabilized saddle-point problems

2018 ◽  
Vol 39 (4) ◽  
pp. 1888-1923 ◽  
Author(s):  
Zhong-Zhi Bai
Keyword(s):  
Saddle Point ◽  
Iteration Method ◽  
Krylov Subspace ◽  
Krylov Subspace Methods ◽  
Iterative Solvers ◽  
Convergence Theory ◽  
Unconditional Convergence ◽  
Saddle Point Problems ◽  
Subspace Methods ◽  
Iteration Methods

Abstract We extend the regularized Hermitian and skew-Hermitian splitting (RHSS) iteration methods for standard saddle-point problems to stabilized saddle-point problems and establish the corresponding unconditional convergence theory for the resulting methods. Besides being used as stationary iterative solvers, this class of RHSS methods can also be used as preconditioners for Krylov subspace methods. It is shown that the eigenvalues of the corresponding preconditioned matrix are clustered at a small number of points in the interval $(0, \, 2)$ when the iteration parameter is close to $0$ and, furthermore, they can be clustered near $0$ and $2$ when the regularization matrix is appropriately chosen. Numerical results on stabilized saddle-point problems arising from finite element discretizations of an optimal boundary control problem and of a Cahn–Hilliard image inpainting problem, as well as from the Gauss–Newton linearization of a nonlinear image restoration problem, show that the RHSS iteration method significantly outperforms the Hermitian and skew-Hermitian splitting iteration method in iteration counts and computing times when they are used either as linear iterative solvers or as matrix splitting preconditioners for Krylov subspace methods, and optimal convergence behavior can be achieved when using inexact variants of the proposed RHSS preconditioners.

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