Two new variants of nonlinear inexact Uzawa algorithms for saddle-point problems

Summary. In this paper, we consider some nonlinear inexact Uzawa methods for iteratively solving linear saddle-point problems. By means of a new technique, we first give an essential improvement on the convergence results of Bramble-Paschiak-Vassilev for a known nonlinear inexact Uzawa algorithm. Then we propose two new algorithms, which can be viewed as a combination of the known nonlinear inexact Uzawa method with the classical steepest descent method and conjugate gradient method respectively. The two new algorithms converge under very practical conditions and do not require any apriori estimates on the minimal and maximal eigenvalues of the preconditioned systems involved, including the preconditioned Schur complement. Numerical results of the algorithms applied for the Stokes problem and a purely linear system of algebraic equations are presented to show the efficiency of the algorithms. Received December 8, 1999 / Revised version received September 8, 2001 / Published online March 8, 2002 RID="*" ID="*" The work of this author was partially supported by a grant from The Institute of Mathematical Sciences, CUHK RID="**" ID="**" The work of this author was partially supported by Hong Kong RGC Grants CUHK 4292/00P and CUHK 4244/01P     

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.     

A generalization of parameterized inexact Uzawa method for generalized saddle point problems

Recently, a class of parameterized inexact Uzawa methods has been proposed for generalized saddle point problems by Bai and Wang [Z.-Z. Bai, Z.-Q. Wang, On parameterized inexact Uzawa methods for generalized saddle point problems, Linear Algebra Appl. 428 (2008) 2900–2932], and a generalization of the inexact parameterized Uzawa method has been studied for augmented linear systems by Chen and Jiang [F. Chen, Y.-L. Jiang, A generalization of the inexact parameterized Uzawa methods for saddle point problems, Appl. Math. Comput. (2008)]. This paper is concerned about a generalization of the parameterized inexact Uzawa method for solving the generalized saddle point problems with nonzero (2, 2) blocks. Some new iterative methods are presented and their convergence are studied in depth. By choosing different parameter matrices, we derive a series of existing and new iterative methods, including the preconditioned Uzawa method, the inexact Uzawa method, the SOR-like method, the GSOR method, the GIAOR method, the PIU method, the APIU method and so on. Numerical experiments are used to demonstrate the feasibility and effectiveness of the generalized parameterized inexact Uzawa methods.     

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 sufficient condition for the convergence of the inexact Uzawa algorithm for saddle point problems

In this paper, the convergence property of the inexact Uzawa algorithm for solving symmetric indefinite linear systems is studied. A simple sufficient condition for the convergence of the inexact Uzawa algorithm is obtained. Two examples and numerical experiments illustrating the conclusion are provided.     

Inexact subgradient methods for quasi-convex optimization problems

In this paper, we consider a generic inexact subgradient algorithm to solve a nondifferentiable quasi-convex constrained optimization problem. The inexactness stems from computation errors and noise, which come from practical considerations and applications. Assuming that the computational errors and noise are deterministic and bounded, we study the effect of the inexactness on the subgradient method when the constraint set is compact or the objective function has a set of generalized weak sharp minima. In both cases, using the constant and diminishing stepsize rules, we describe convergence results in both objective values and iterates, and finite convergence to approximate optimality. We also investigate efficiency estimates of iterates and apply the inexact subgradient algorithm to solve the Cobb–Douglas production efficiency problem. The numerical results verify our theoretical analysis and show the high efficiency of our proposed algorithm, especially for the large-scale problems.     

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.     

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.     

Convergence analysis of generalized nonlinear inexact Uzawa algorithm for stabilized saddle point problems

This paper deals with a modified nonlinear inexact Uzawa (MNIU) method for solving the stabilized saddle point problem. The modified Uzawa method is an inexact inner-outer iteration with a variable relaxation parameter and has been discussed in the literature for uniform inner accuracy. This paper focuses on the general case when the accuracy of inner iteration can be variable and the convergence of MNIU with variable inner accuracy, based on a simple energy norm. Sufficient conditions for the convergence of MNIU are proposed. The convergence analysis not only greatly improves the existing convergence results for uniform inner accuracy in the literature, but also extends the convergence to the variable inner accuracy that has not been touched in literature. Numerical experiments are given to show the efficiency of the MNIU algorithm.     

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.     

具有滑动边界条件Stokes问题的自适应Uzawa块松弛算法

对一类具有非线性滑动边界条件的Stokes问题,得到了求其数值解的自适应Uzawa块松弛算法（SUBRM）.通过该问题导出的变分问题,引入辅助变量将原问题转化为一个基于增广Lagrange函数表示的鞍点问题,并采用Uzawa块松弛算法（UBRM）求解.为了提高算法性能,提出利用迭代函数自动选取合适罚参数的自适应法则.该算法的优点是每次迭代只需计算一个线性问题,同时显式计算辅助变量.对算法的收敛性进行了理论分析,最后用数值结果验证了该算法的可行性和有效性.     

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.

     

Convergence of a generalized Newton and an inexact generalized Newton algorithms for solving nonlinear equations with nondifferentiable terms

In this paper, we consider two versions of the Newton-type method for solving a nonlinear equations with nondifferentiable terms, which uses as iteration matrices, any matrix from B-differential of semismooth terms. Local and global convergence theorems for the generalized Newton and inexact generalized Newton method are proved. Linear convergence of the algorithms is obtained under very mild assumptions. The superlinear convergence holds under some conditions imposed on both terms of equation. Some numerical results indicate that both algorithms works quite well in practice.      

Uzawa iteration method for stokes type variational inequality of the second kind

In this paper,the Uzawa iteration algorithm is applied to the Stokes problem with nonlinear slip boundary conditions whose variational formulation is the variational inequality of the second kind.Firstly, the multiplier in a convex set is introduced such that the variational inequality is equivalent to the variational identity.Moreover,the solution of the variational identity satisfies the saddle-point problem of the Lagrangian functional ?.Subsequently,the Uzawa algorithm is proposed to solve the solution of the saddle-point problem. We show the convergence of the algorithm and obtain the convergence rate.Finally,we give the numerical results to verify the feasibility of the Uzawa algorithm.     

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.     

自由边界问题的自适应Uzawa块松弛算法

利用增广Lagrange乘子法和自适应法则，得到求解单侧障碍自由边界问题的自适应Uzawa块松弛法.单侧障碍自由边界问题离散为有限维线性互补问题，等价于一个用辅助变量和增广Lagrange函数表示的鞍点问题.采用Uzawa块松弛算法求解该问题得到一个两步迭代法，主要的子问题为一个线性问题，同时能显式求解辅助变量.由于Uzawa块松弛算法的收敛速度显著依赖于罚参数，而且对具体问题很难选择合适的罚参数.为提高算法的性能，提出了自适应法则，该方法自动调整每次迭代所需的罚参数.数值结果验证了该算法的理论分析.     

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.     

Modified parameterized inexact Uzawa method for singular saddle-point problems

As well known, each of the consistent singular saddle-point (CSSP) problems has more than one solutions, and most of the iteration methods can only be proved to converge to one of the solutions of the CSSP problem. However, we do not know which solution it is and whether this solution depends on the initial iteration guesses. In this work, we introduce a new iteration method by slightly modifying the parameterized inexact Uzawa (PIU) iteration scheme. Theoretical analysis shows that, under suitable restrictions on the involved iteration parameters, the iteration sequence produced by the new method converges to the solution \(\mathcal {A}^{\dag }b\) for any initial guess, no matter the singular saddle-point system \(\mathcal {A}~x=b\) is consistent or inconsistent, where \(\mathcal {A}^{\dag }\) denotes the Moore-Penrose inverse of matrix \(\mathcal {A}\). In addition, the quasi-optimal iteration parameters and the corresponding quasi-optimal convergence factor are determined. Numerical examples are given to verify the correctness of the theoretical results and the effectiveness of our new method.     

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.     

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