Preconditioned AHSS iteration method for singular saddle point problems

In this paper, for solving the singular saddle point problems, we present a new preconditioned accelerated Hermitian and skew-Hermitian splitting (AHSS) iteration method. The semi-convergence of this method and the eigenvalue distribution of the preconditioned iteration matrix are studied. In addition, we prove that all eigenvalues of the iteration matrix are clustered for any positive iteration parameters α and β. Numerical experiments illustrate the theoretical results and examine the numerical effectiveness of the AHSS iteration method served either as a preconditioner or as a solver.     

On local Hermitian and skew-Hermitian splitting iteration methods for generalized saddle point problems

In this paper, we first present a local Hermitian and skew-Hermitian splitting (LHSS) iteration method for solving a class of generalized saddle point problems. The new method converges to the solution under suitable restrictions on the preconditioning matrix. Then we give a modified LHSS (MLHSS) iteration method, and further extend it to the generalized saddle point problems, obtaining the so-called generalized MLHSS (GMLHSS) iteration method. Numerical experiments for a model Navier-Stokes problem are given, and the results show that the new methods outperform the classical Uzawa method and the inexact parameterized Uzawa method.     

关于广义鞍点问题的HSS迭代方法的收缩因子(英文)

白中治等提出了解非埃尔米特正定线性方程组的埃尔米特和反埃尔米特分裂(HSS)迭代方法(Bai Z Z,Golub G H,Ng M K.Hermitian and skew-Hermitian splitting methodsfor non-Hermitian positive definite linear systems.SIAM J.Matrix Anal.Appl.,2003,24:603-626).本文精确地估计了用HSS迭代方法求解广义鞍点问题时在加权2-范数和2-范数下的收缩因子.在实际的计算中,正是这些收缩因子而不是迭代矩阵的谱半径,本质上控制着HSS迭代方法的实际收敛速度.根据文中的分析,求解广义鞍点问题的HSS迭代方法的收缩因子在加权2-范数下等于1,在2-范数下它会大于等于1,而在某种适当选取的范数之下,它则会小于1.最后,用数值算例说明了理论结果的正确性.     

On an iterative method for saddle point problems

Convergence behavior of a nested iterative scheme presented in a paper by Bank, Welfert and Yserentant is studied. It is shown that this scheme converges under conditions weaker than that stated in their paper. Received November 18, 1996 / Revised version received March 18, 1997     

Semi-convergence analysis of preconditioned deteriorated PSS iteration method for singular saddle point problems

In this paper, we propose a two-parameter preconditioned variant of the deteriorated PSS iteration method (J. Comput. Appl. Math., 273, 41–60 (2015)) for solving singular saddle point problems. Semi-convergence analysis shows that the new iteration method is convergent unconditionally. The new iteration method can also be regarded as a preconditioner to accelerate the convergence of Krylov subspace methods. Eigenvalue distribution of the corresponding preconditioned matrix is presented, which is instructive for the Krylov subspace acceleration. Note that, when the leading block of the saddle point matrix is symmetric, the new iteration method will reduce to the preconditioned accelerated HSS iteration method (Numer. Algor., 63 (3), 521–535 2013), the semi-convergence conditions of which can be simplified by the results in this paper. To further improve the effectiveness of the new iteration method, a relaxed variant is given, which has much better convergence and spectral properties. Numerical experiments are presented to investigate the performance of the new iteration methods for solving singular saddle point problems.     

On the UPSS method for non-Hermitian singular saddle point problems

Recently, a new Uzawa-type method, referred as the UPSS method, is proposed for solving the non-Hermitian nonsingular saddle point problems, see Dou, Yang and Wu (2017). In this paper, we give the semi-convergence analysis of the UPSS method when it is used to solve non-Hermitian singular saddle point problems. An example is given to verify the effectiveness of this method for solving non-Hermitian singular saddle point problems.     

鞍点问题的修正对称超松弛迭代算法

为了提高求解鞍点问题的迭代算法的速度,通过设置合适的加速变量,对修正超松弛迭代算法（简记作MSOR-like算法）和广义对称超松弛迭代算法（简记作GSSOR-like算法）进行了修正,给出了修正对称超松弛迭代算法,即MSSOR-like （modified symmetric successiveover-relaxation）算法,并研究了该算法收敛的充分必要条件.最后,通过数值例子表明,选择合适的参数后,新算法的迭代速度和迭代次数均优于MSOR-like （modified successive overrelaxation）和GSSOR-like （generalized symmetric successive over-relaxation）算法,因此,它是一种较好的解决鞍点问题的算法.     

On generalized stationary iterative method for solving the saddle point problems

In this paper, the generalized stationary iterative (GSI) method is studied for solving the saddle point problems. The convergence of this method is studied under suitable restrictions on iteration parameters. Numerical example for solving Stokes equation is presented to show the superiority of GSI method.     

The parameterized upper and lower triangular splitting methods for saddle point problems

Complex moment-based eigensolvers for solving interior eigenvalue problems have been studied because of their high parallel efficiency. Recently, we proposed the block Arnoldi-type complex moment-based eigensolver without a low-rank approximation. A low-rank approximation plays a very important role in reducing computational cost and stabilizing accuracy in complex moment-based eigensolvers. In this paper, we develop the method and propose block Krylov-type complex moment-based eigensolvers with a low-rank approximation. Numerical experiments indicate that the proposed methods have higher performance than the block SS–RR method, which is one of the most typical complex moment-based eigensolvers.     

On the optimal parameters of GMSSOR method for saddle point problems

For solving saddle point problems, SOR-type methods are investigated by many researchers in the literature. In this short note, we study the GMSSOR method for solving saddle point problems and obtain the optimal parameters which minimize the spectral (or pseudo-spectral) radii of the iteration matrices.     

On parameterized block triangular preconditioners for generalized saddle point problems

In this paper, we discuss two classes of parameterized block triangular preconditioners for the generalized saddle point problems. These preconditioners generalize the common block diagonal and triangular preconditioners. We will give distributions of the eigenvalues of the preconditioned matrix and provide estimates for the interval containing the real eigenvalues. Numerical experiments of a model Stokes problem are presented.     

On Schwarz-type smoothers for saddle point problems with applications to PDE-constrained optimization problems

In this paper we consider a (one-shot) multigrid strategy for solving the discretized optimality system (KKT system) of a PDE-constrained optimization problem. In particular, we discuss the construction of an additive Schwarz-type smoother for a certain class of optimal control problems. A rigorous multigrid convergence analysis is presented. Numerical experiments are shown which confirm the theoretical results. The work was supported by the Austrian Science Fund (FWF) under grant SFB 013/F1309.     

On the numerical analysis of nonlinear twofold saddle point problems

We provide a general abstract theory for the solvability andGalerkin approximation of nonlinear twofold saddle point problems.In particular, a Strang error estimate containing the consistencyterms arising from the approximation of the continuous operatorsinvolved is deduced. Then we apply these results to analysea fully discrete Galerkin scheme for a twofold saddle pointformulation of a nonlinear elliptic boundary value problem indivergence form. Some numerical results are also presented.     

An alternating preconditioner for saddle point problems

Based on matrix splittings, a new alternating preconditioner with two parameters is proposed for solving saddle point problems. Some theoretical analyses for the eigenvalues of the associated preconditioned matrix are given. The choice of the parameters is considered and the quasi-optimal parameters are obtained. The new preconditioner with these quasi-optimal parameters significantly improves the convergence rate of the generalized minimal residual (GMRES) iteration. Numerical experiments from the linearized Navier-Stokes equations demonstrate the efficiency of the new preconditioner, especially on the larger viscosity parameter ν. Further extensions of the preconditioner to generalized saddle point matrices are also checked.     

A splitting preconditioner for saddle point problems

For large sparse systems of linear equations iterative techniques are attractive. In this paper, we study a splitting method for an important class of symmetric and indefinite system. Theoretical analyses show that this method converges to the unique solution of the system of linear equations for all t>0 (t is the parameter). Moreover, all the eigenvalues of the iteration matrix are real and nonnegative and the spectral radius of the iteration matrix is decreasing with respect to the parameter t. Besides, a preconditioning strategy based on the splitting of the symmetric and indefinite coefficient matrices is proposed. The eigensolution of the preconditioned matrix is described and an upper bound of the degree of the minimal polynomials for the preconditioned matrix is obtained. Numerical experiments of a model Stokes problem and a least‐squares problem with linear constraints presented to illustrate the effectiveness of the method. Copyright © 2011 John Wiley & Sons, Ltd.     

On lower bounds on the complexity of solution of integral equations

We obtain lower bounds for the number of arithmetic operations required for the solution of Fredholm integral equations with accuracy ? under the assumption that the set of simplest operations is Φ = {arithmetic operations, evaluation of functionals}.     

Uzawa type algorithms for nonsymmetric saddle point problems

In this paper, we consider iterative algorithms of Uzawa type for solving linear nonsymmetric saddle point problems. Specifically, we consider systems, written as usual in block form, where the upper left block is an invertible linear operator with positive definite symmetric part. Such saddle point problems arise, for example, in certain finite element and finite difference discretizations of Navier-Stokes equations, Oseen equations, and mixed finite element discretization of second order convection-diffusion problems. We consider two algorithms, each of which utilizes a preconditioner for the operator in the upper left block. Convergence results for the algorithms are established in appropriate norms. The convergence of one of the algorithms is shown assuming only that the preconditioner is spectrally equivalent to the inverse of the symmetric part of the operator. The other algorithm is shown to converge provided that the preconditioner is a sufficiently accurate approximation of the inverse of the upper left block. Applications to the solution of steady-state Navier-Stokes equations are discussed, and, finally, the results of numerical experiments involving the algorithms are presented.