New choices of preconditioning matrices for generalized inexact parameterized iterative methods

For large sparse saddle point problems, Chen and Jiang recently studied a class of generalized inexact parameterized iterative methods (see [F. Chen, Y.-L. Jiang, A generalization of the inexact parameterized Uzawa methods for saddle point problems, Appl. Math. Comput. 206 (2008) 765-771]). In this paper, the methods are modified and some choices of preconditioning matrices are given. These preconditioning matrices have advantages in solving large sparse linear system. Numerical experiments of a model Stokes problem are presented.     

Block diagonally preconditioned PIU methods of saddle point problem

For large sparse saddle point problems, we firstly introduce the block diagonally preconditioned Gauss-Seidl method (PBGS) which reduces to the GSOR method [Z.-Z. Bai, B.N. Parlett, Z.-Q. Wang, On generalized successive overrelaxation methods for augmented linear systems, Numer. Math. 102 (2005) 1-38] and PIU method [Z.-Z. Bai, Z.-Q. Wang, On parameterized inexact Uzawa methods for generalized saddle point problems, Linear Algebra Appl. 428 (2008) 2900-2932] when the preconditioners equal to different matrices, respectively. Then we generalize the PBGS method to the PPIU method and discuss the sufficient conditions such that the spectral radius of the PPIU method is much less than one. Furthermore, some rules are considered for choices of the preconditioners including the splitting method of the (1, 1) block matrix in the PIU method and numerical examples are given to show the superiority of the new method to the PIU method.     

A class of Uzawa-SOR methods for saddle point problems

In this paper, we consider a class of Uzawa-SOR methods for saddle point problems, and prove the convergence of the proposed methods. We solve a lower triangular system per iteration in the proposed methods, instead of solving a linear equation Az=b. Actually, the new methods can be considered as an inexact iteration method with the Uzawa as the outer iteration and the SOR as the inner iteration. Although the proposed methods cannot achieve the same convergence rate as the GSOR methods proposed by Bai et al. [Z.-Z. Bai, B.N. Parlett, Z.-Q. Wang, On generalized successive overrelaxation methods for augmented linear systems, Numer. Math. 102 (2005) 1-38], but our proposed methods have less workloads per iteration step. Experimental results show that our proposed methods are feasible and effective.     

A convergence analysis of the nonlinear Uzawa algorithm for saddle point problems

In this paper we discuss the convergence behavior of the nonlinear inexact Uzawa algorithm for solving saddle point problems presented in a recent paper by Cao [Z.H. Cao, Fast Uzawa algorithm for generalized saddle point problems, Appl. Numer. Math. 46 (2003) 157–171]. We show that this algorithm converges under a condition weaker than that stated in this paper.     

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.     

Uzawa methods for a class of block three‐by‐three saddle‐point problems

In this work, we consider numerical methods for solving a class of block three‐by‐three saddle‐point problems, which arise from finite element methods for solving time‐dependent Maxwell equations and some other applications. The direct extension of the Uzawa method for solving this block three‐by‐three saddle‐point problem requires the exact solution of a symmetric indefinite system of linear equations at each step. To avoid heavy computations at each step, we propose an inexact Uzawa method, which solves the symmetric indefinite linear system in some inexact way. Under suitable assumptions, we show that the inexact Uzawa method converges to the unique solution of the saddle‐point problem within the approximation level. Two special algorithms are customized for the inexact Uzawa method combining the splitting iteration method and a preconditioning technique, respectively. Numerical experiments are presented, which demonstrated the usefulness of the inexact Uzawa method and the two customized algorithms.     

Improved PHSS iterative methods for solving saddle point problems

An improvement on a generalized preconditioned Hermitian and skew-Hermitian splitting method (GPHSS), originally presented by Pan and Wang (J. Numer. Methods Comput. Appl. 32, 174–182, 2011), for saddle point problems, is proposed in this paper and referred to as IGPHSS for simplicity. After adding a matrix to the coefficient matrix on two sides of first equation of the GPHSS iterative scheme, both the number of required iterations for convergence and the computational time are significantly decreased. The convergence analysis is provided here. As saddle point problems are indefinite systems, the Conjugate Gradient method is unsuitable for them. The IGPHSS is compared with Gauss-Seidel, which requires partial pivoting due to some zero diagonal entries, Uzawa and GPHSS methods. The numerical experiments show that the IGPHSS method is better than the original GPHSS and the other two relevant methods.     

Nonlinear uzawa methods for solving nonsymmetric saddle point problems

In [A new nonlinear Uzawa algorithm for generalized saddle point problems, Appl. Math. Comput., 175(2006), 1432–1454], a nonlinear Uzawa algorithm for solving symmetric saddle point problems iteratively, which was defined by two nonlinear approximate inverses, was considered. In this paper, we extend it to the nonsymmetric case. For the nonsymmetric case, its convergence result is deduced. Moreover, we compare the convergence rates of three nonlinear Uzawa methods and show that our method is more efficient than other nonlinear Uzawa methods in some cases. The results of numerical experiments are presented when we apply them to Navier-Stokes equations discretized by mixed finite elements.     

Preconditioned Uzawa Algorithm for Elastic Contact Problems

After finite element discretization of the elastic contact problems with friction, we obtain a big sparse non–symmetric and nonlinear systems of equations, and in many cases ill–conditioned. Solving these systems by direct methods or classical iterative methods are non efficient and with bad convergence properties. One way to overcome these difficulties is to use the preconditioned Uzawa–type algorithms. On this paper we focus on the transformation of the generalized Signorini elastic contact problems into a saddle point problem of some augmented Lagrangian functional and give a preconditioning technique for Uzawa algorithm. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Optimization of the parameterized Uzawa preconditioners for saddle point matrices

The parameterized Uzawa preconditioners for saddle point problems are studied in this paper. The eigenvalues of the preconditioned matrix are located in (0, 2) by choosing the suitable parameters. Furthermore, we give two strategies to optimize the rate of convergence by finding the suitable values of parameters. Numerical computations show that the parameterized Uzawa preconditioners can lead to practical and effective preconditioned GMRES methods for solving the saddle point problems.     

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.     

Analysis of iterative methods for saddle point problems: a unified approach

In this paper two classes of iterative methods for saddle point problems are considered: inexact Uzawa algorithms and a class of methods with symmetric preconditioners. In both cases the iteration matrix can be transformed to a symmetric matrix by block diagonal matrices, a simple but essential observation which allows one to estimate the convergence rate of both classes by studying associated eigenvalue problems. The obtained estimates apply for a wider range of situations and are partially sharper than the known estimates in literature. A few numerical tests are given which confirm the sharpness of the estimates.

     

An adaptive finite element method for singular parabolic equations

Summary. In this paper we consider additive Schwarz-type iteration methods for saddle point problems as smoothers in a multigrid method. Each iteration step of the additive Schwarz method requires the solutions of several small local saddle point problems. This method can be viewed as an additive version of a (multiplicative) Vanka-type iteration, well-known as a smoother for multigrid methods in computational fluid dynamics. It is shown that, under suitable conditions, the iteration can be interpreted as a symmetric inexact Uzawa method. In the case of symmetric saddle point problems the smoothing property, an important part in a multigrid convergence proof, is analyzed for symmetric inexact Uzawa methods including the special case of the additive Schwarz-type iterations. As an example the theory is applied to the Crouzeix-Raviart mixed finite element for the Stokes equations and some numerical experiments are presented. Mathematics Subject Classification (1991):65N22, 65F10, 65N30Supported by the Austrian Science Foundation (FWF) under the grant SFB F013}\and Walter Zulehner     

Schur complements on Hilbert spaces and saddle point systems

For any continuous bilinear form defined on a pair of Hilbert spaces satisfying the compatibility Ladyshenskaya–Babušca–Brezzi condition, symmetric Schur complement operators can be defined on each of the two Hilbert spaces. In this paper, we find bounds for the spectrum of the Schur operators only in terms of the compatibility and continuity constants. In light of the new spectral results for the Schur complements, we review the classical Babušca–Brezzi theory, find sharp stability estimates, and improve a convergence result for the inexact Uzawa algorithm. We prove that for any symmetric saddle point problem, the inexact Uzawa algorithm converges, provided that the inexact process for inverting the residual at each step has the relative error smaller than 1/3. As a consequence, we provide a new type of algorithm for discretizing saddle point problems, which combines the inexact Uzawa iterations with standard a posteriori error analysis and does not require the discrete stability conditions.     

Variable parameter Uzawa method for solving a class of block three-by-three saddle point problems

In this work, we consider numerical methods for solving a class of block three-by-three saddle point problems, which arise from finite element methods for solving time-dependent Maxwell equations and a class of quadratic programs. We present a variant of Uzawa method with two variable parameters for the saddle point problems. These two parameters can be updated easily in each iteration, similar to the evaluation of the two iteration parameters in the conjugate gradient method. We show that the new iterative method converges to the unique solution of the saddle point problems under a reasonable condition. Numerical experiments highlighting the performance of the proposed method for problems are presented.

     

一类求解鞍点问题的广义不精确Uzawa方法

本文提出了一类求解大型稀疏鞍点问题的新的广义不精确Uzawa算法.该方法不仅可以包含 前人的方法, 而且可以拓展出很多新方法. 理论分析给出该方法收敛的条件, 并详细的分析了其收敛性质和参数矩阵的选取方法. 通过对有限元离散的Stokes问题的数值实验表明, 新方法是行之有效的, 其收敛速度明显优于原来的算法.     

A comparative study of efficient iterative solvers for generalized Stokes equations

We consider a generalized Stokes equation with problem parameters ξ?0 (size of the reaction term) and ν>0 (size of the diffusion term). We apply a standard finite element method for discretization. The main topic of the paper is a study of efficient iterative solvers for the resulting discrete saddle point problem. We investigate a coupled multigrid method with Braess–Sarazin and Vanka‐type smoothers, a preconditioned MINRES method and an inexact Uzawa method. We present a comparative study of these methods. An important issue is the dependence of the rate of convergence of these methods on the mesh size parameter and on the problem parameters ξ and ν. We give an overview of the main theoretical convergence results known for these methods. For a three‐dimensional problem, discretized by the Hood–Taylor ??₂–??₁ pair, we give results of numerical experiments. Copyright © 2007 John Wiley & Sons, Ltd.     

A generalization of the HSS-based sequential two-stage method for solving non-Hermitian saddle point problems

For non-Hermitian saddle point problems with non-Hermitian positive definite (1,1)-block, Zhu et al. studied the HSS-based sequential two-stage method (see Zhu et al. Appl. Math. Comput. 242, 907–916 19). However, this approach may not work when the (1,1)-block of the saddle point problems is weakly Hermitian or skew-Hermitian dominant. By introducing a new preconditioning matrix, a generalization of the HSS-based sequential two-stage method is proposed for solving non-Hermitian saddle-point problems with non-Hermitian positive definite and Hermitian or skew-Hermitian dominant (1,1)-block. Theoretical analysis shows that the proposed iterative method is convergent. Numerical experiments are provided to confirm the theoretical results, which demonstrate that the generalized method is effective and feasible for solving saddle point problems with non-Hermitian positive definite and Hermitian or skew-Hermitian dominant (1,1)-block.     

On the optimal convergence factor of the accelerated parameterized inexact Uzawa method with three parameters for augmented systems

Under the assumption that all eigenvalues of the preconditioned Schur complement are real, we present an analytical proof for obtaining the optimal convergence factor of the real accelerated parameterized inexact Uzawa (APIU) method when P=A. It is proved that the optimal convergence factor is the same as that of the generalized successive overrelaxation method, which was published at the same time, and that it can be attained only at the unique optimum point of parameters, regardless of whether m>n or m=n. In addition, we generalize the APIU method and analyze the relationship between the APIU method and 10 additional Uzawa‐like methods.     

The alternating-direction iterative method for saddle point problems

In the paper, a new alternating-direction iterative method is proposed based on matrix splittings for solving saddle point problems. The convergence analysis for the new method is given. When the better values of parameters are employed, the proposed method has faster convergence rate and less time cost than the Uzawa algorithm with the optimal parameter and the Hermitian and skew-Hermitian splitting iterative method. Numerical examples further show the effectiveness of the method.