A simplified HSS preconditioner for generalized saddle point problems
type: Item
instance of: scholarly article
title: A simplified HSS preconditioner for generalized saddle point problems
zbMATH Open document ID: 1347.65056
author: Yang Cao
author: Zhi-Ru Ren
author: Quan Shi
published in: BIT
publication date: 2016-06-10T00:00:00Z
review text: This paper develops the SHSS preconditioner for generalized saddle point problems \[ \begin{bmatrix} A & B^T\\ -B & C \end{bmatrix}\, \begin{bmatrix} x\\y\end{bmatrix} = \begin{bmatrix} f\\ -g \end{bmatrix} \] that is built on and simplified from the classical HSS preconditioner of \textit{Z.-Z. Bai} et al. [SIAM J. Matrix Anal. Appl. 24, No. 3, 603--626 (2003; Zbl 1036.65032)]. The idea is to leave off parts of the HSS preconditioner \(P_{HSS} = \dfrac{1}{\alpha}\) \[ \begin{bmatrix} \alpha I + A& 0\\ 0 & \alpha I + C \end{bmatrix} \begin{bmatrix} \alpha I & B^T\\ -B & \alpha I\end{bmatrix} \] in the new preconditioner \(P_{SHSS} = \dfrac{1}{\alpha}\) \[ \begin{bmatrix} A& 0\\ 0 & \alpha I \end{bmatrix} \begin{bmatrix} \alpha I & B^T\\ -B & \alpha I\end{bmatrix}. \] The simpler preconditioner \(P_SHSS\) differs from the saddle point matrix \[ \begin{bmatrix} A & B^T\\ -B & C \end{bmatrix} \] only in the upper triangular block. This allows the thus preconditioned algorithm to solve only two linear systems where the more complex HSS methods need to solve three. Moreover, the spectral distribution of the preconditioned matrix improves significantly, leading to much shorter iteration sequences. A detailed spectral analysis of SHSS is included. Numerical experiments deal with preconditioned GMRES for the Stokes equation in 2D. Extensive numerical tests show the advantages (fewer iterations, CPU speedup by factors above \(5\), same accuracy) of SHSS over HSS as well as choices for a near optimal parameter \(\alpha\).
reviewed by: Frank Uhlig
Mathematics Subject Classification ID: 65F08
Mathematics Subject Classification ID: 65F10
Mathematics Subject Classification ID: 65F50
zbMATH DE Number: 6591883
zbMATH Keywords: preconditioning
zbMATH Keywords: Krylov subspace method
zbMATH Keywords: saddle point problem
zbMATH Keywords: SHSS preconditioner
zbMATH Keywords: Stokes problem
zbMATH Keywords: algorithm
zbMATH Keywords: numerical experiment
MaRDI profile type: Publication
full work available at URL: s10543-015-0588-3
OpenAlex ID: W2274312341
cites work: Structured preconditioners for nonsingular matrices of block two-by-two structures
cites work: Optimal parameters in the HSS-like methods for saddle-point problems
cites work: Block alternating splitting implicit iteration methods for saddle-point problems from time-harmonic eddy current models.
cites work: Eigenvalue estimates for saddle point matrices of Hermitian and indefinite leading blocks
cites work: Accelerated Hermitian and skew-Hermitian splitting iteration methods for saddle-point problems
cites work: Optimal Parameter in Hermitian and Skew-Hermitian Splitting Method for Certain Two-by-Two Block Matrices
cites work: Convergence properties of preconditioned Hermitian and skew-Hermitian splitting methods for non-Hermitian positive semidefinite matrices
cites work: Hermitian and Skew-Hermitian Splitting Methods for Non-Hermitian Positive Definite Linear Systems
cites work: Preconditioned Hermitian and skew-Hermitian splitting methods for non-Hemitian positive semidefinite linear systems
cites work: On Inexact Preconditioners for Nonsymmetric Matrices
cites work: Constraint Preconditioners for Symmetric Indefinite Matrices
cites work: Element‐free Galerkin methods
cites work: Optimization of the Hermitian and skew-Hermitian splitting iteration for saddle-point problems
cites work: A Preconditioner for Generalized Saddle Point Problems
cites work: Numerical solution of saddle point problems
cites work: A dimensional split preconditioner for Stokes and linearized Navier-Stokes equations
cites work: A relaxed dimensional factorization preconditioner for the incompressible Navier-Stokes equations
cites work: Mixed and Hybrid Finite Element Methods
cites work: A relaxed deteriorated PSS preconditioner for nonsymmetric saddle point problems from the steady Navier-Stokes equation
cites work: A splitting preconditioner for saddle point problems.
cites work: A modified dimensional split preconditioner for generalized saddle point problems
cites work: A Relaxed HSS Preconditioner for Saddle Point Problems from Meshfree Discretization
cites work: A weighted nodal-radial point interpolation meshless method for 2D solid problems
cites work: Q5319020
cites work: Finite elements and fast iterative solvers. With applications in incompressible fluid dynamics
cites work: On Solving Block-Structured Indefinite Linear Systems
cites work: The spectral properties of the Hermitian and skew-Hermitian splitting preconditioner for generalized saddle point problems
cites work: A practical formula for computing optimal parameters in the HSS iteration methods
cites work: On parameterized block triangular preconditioners for generalized saddle point problems
cites work: Inequalities: theory of majorization and its applications
cites work: Numerical Optimization
cites work: New preconditioners for saddle point problems
cites work: An alternating positive-semidefinite splitting preconditioner for saddle point problems from time-harmonic eddy current models
cites work: Q4414854
cites work: Stabilised bilinear-constant velocity-pressure finite elements for the conjugate gradient solution of the Stokes problem
cites work: Spectral Properties of the Hermitian and Skew-Hermitian Splitting Preconditioner for Saddle Point Problems
cites work: On HSS-based constraint preconditioners for generalized saddle-point problems
DOI: 10.1007/S10543-015-0588-3
description: scientific article; zbMATH DE number 6591883