Parameterized Inexact Uzawa Method Research Articles

Optimization of a parameterized inexact Uzawa method for saddle point problems

ABSTRACTFor large sparse saddle point problems, Cao et al. studied a modified generalized parameterized inexact Uzawa (MGPIU) method (see [Y. Cao, M.Q. Jiang, L.Q. Yao, New choices of preconditioning matrices for generalized inexact parameterized iterative methods, J. Comput. Appl. Math. 235 (1) (2010) 263–269]). For iterative methods of this type, the choice of the relaxation parameter is crucial for the methods to achieve their best performance. In this paper, for an example of 2D Stokes equations, we derive the optimal relaxation parameter for the continuous version of the MGPIU method, by minimizing the corresponding convergence factor that is obtained using Fourier analysis. In addition, we find that the MGPIU method is mesh parameter independent, however, it depends asymptotically linearly on the viscosity ν, which suggests that the numerical methods for Stokes equations should be investigated with the presence of the viscosity ν, though it can be scaled out from the equations in advance. We use numerical experiments to validate our theoretical findings.

A generalized preconditioned parameterized inexact Uzawa method for singular saddle point problems

In this short paper, we introduce and analyze a generalized preconditioned parameterized inexact Uzawa method for solving singular saddle point problems, this method is a generalization of the PPIU method (see, Gao and Kong, 2010), then we give the semi-convergent conditions of this method, which are more practical for the applications.

Variants of the accelerated parameterized inexact Uzawa method for saddle-point problems

In this paper, based on the SOR and SSOR splittings of the (1,1) part of saddle-point coefficient matrix, some variants of the accelerated parameterized inexact Uzawa (VAPIU) method are proposed for solving nonsingular and singular saddle-point problems. By choosing different parameter matrices, we derive some existing and new iterative methods. The corresponding convergence and semi-convergence of the VAPIU methods for solving nonsingular and singular saddle-point problems are studied in depth, respectively. The preconditioning strategies based on the VAPIU splittings of the coefficient matrices are presented. Numerical experiments are provided, which confirms that these new methods need less CPU times per iteration step comparing with some other methods for solving both nonsingular and singular saddle-point problems.

A generalization of preconditioned parameterized inexact Uzawa method for indefinite saddle point problems

The parameterized inexact Uzawa methods have been used to solve some of the symmetric saddle point problems. In this paper, a new preconditioned parameterized inexact Uzawa method is presented to solve indefinite saddle point problems. After preconditioning, theoretical analyses show that the iteration method converges under certain conditions. So we propose three new algorithms based on these conditions. Numerical experiments are provided to show the effectiveness of the proposed preconditioner and all these algorithms have fantastic convergence rates by choosing optimal parameters.

A new generalized parameterized inexact Uzawa method for solving saddle point problems

Recently, Bai, Parlett and Wang presented a class of parameterized inexact Uzawa (PIU) methods for solving saddle point problems (Bai et al., 2005). In this paper, we develop a new generalized PIU method for solving both nonsingular and singular saddle point problems. The necessary and sufficient conditions of the convergence (semi-convergence) for solving nonsingular (singular) saddle point problems are derived. Meanwhile, the characteristic of eigenvalues of the iteration matrix corresponding to the above iteration method is discussed. We further show that the generalized PIU-type method proposed in this paper has a wider convergence (semi-convergence) region than some classical Uzawa methods, such as the inexact Uzawa method, the SOR-like method, the GSOR method and so on. Finally, numerical examples are given to illustrate the feasibility and efficiency of this method.

Fast parameterized inexact Uzawa method for complex symmetric linear systems

In previous years, Bai and Wang presented a class of parameterized inexact Uzawa (PIU) methods for solving the generalized saddle point problems. In this paper, we consider the same method for iteratively solving the complex symmetric linear systems. Our main contribution is accelerating the convergence of the parameterized inexact Uzawa method by correction technique. First, the corrected model for the PIU method is established and the corrected PIU method is presented. Then we study the convergence property of the corrected PIU method. In fact, the corrected PIU method can converge faster than some Uzawa-type and HSS-like methods. Finally, numerical experiments on a few model problems are presented to illustrate the theoretical results and examine the numerical effectiveness of the new method.

On generalized parameterized inexact Uzawa methods for singular saddle-point problems

For a class of nonsingular saddle-point problems, Bai et al. in 2008 studied an efficient parameterized inexact Uzawa (PIU) method; see [Z.-Z. Bai, Z.-Q. Wang, On parameterized inexact Uzawa methods for generalized saddle-point problems, Linear Algebra Appl. 428 (2008) 2900---2932]. In this paper, we use a generalized version of the PIU method, named as generalized PIU (GPIU) method, to solve the singular saddle-point problems. The semi-convergence properties of the GPIU method are derived under suitable restrictions on the involved iteration parameters. Moreover, the quasi-optimal iteration parameters and the corresponding quasi-optimal semi-convergence factor are also determined. Numerical experiments are used to verify the feasibility and effectiveness of the GPIU method.

A note on semi-convergence of generalized parameterized inexact Uzawa method for singular saddle point problems

Recently, Zhang and Wang studied the generalized parameterized inexact Uzawa (GPIU) method for singular saddle point problems, see Zhang and Wang (Appl. Math. Comput. 219:4225---4231, 2013). In this note, we prove the semi-convergence of GPIU method by another method, which has weaken the conditions of GPIU method. Then, we analyze the spectral properties of the corresponding preconditioned matrix. Moreover, for solving singular saddle point problems, we give a preconditioned matrix for GPIU method, numerical experiments are given to illustrate the efficiency of GPIU method with the preconditioned matrix.

On generalized parameterized inexact Uzawa method for a block two-by-two linear system

Recently, Chen and Jiang [F. Chen, Y.-L. Jiang, A generalization of the inexact parameterized Uzawa methods for saddle point problems, Appl. Math. Comput. 206 (2008) 765–771] presented a parameterized inexact Uzawa (PIU) algorithm for solving symmetric saddle point problems, where the (1, 2)- and the (2, 1)-blocks are the transpose of each other. In this paper, we extend the PIU method to the block two-by-two linear system by allowing the (1, 2)-block to be not equal to the transpose of the (2, 1)-block and the (2, 2)-block may not be zero. We prove that the iteration method is convergent under certain conditions. With different choices of the parameter matrices, we obtain several new algorithms for solving the block two-by-two linear system. Numerical experiments confirm our theoretical results and show that our method is feasible and effective.

A note on block-diagonally preconditioned PIU methods for singular saddle point problems

Recently, Bai and Wang [On parameterized inexact Uzawa methods for generalized saddle point problems, Linear Algebra Appl. 428 (2008), pp. 2900–2932] and Gao and Kong [Block diagonally preconditioned PIU methods of saddle point problems, Appl. Math. Comput. 216 (2010), pp. 1880–1887] discussed the parameterized inexact Uzawa (PIU) method and the preconditioned parameterized inexact Uzawa (PPIU) method. In this paper, we further study the block-diagonally preconditioned PIU methods for solving singular saddle point problems, and give the corresponding convergence analysis.

On parameterized inexact Uzawa methods for generalized saddle point problems

For the large sparse saddle point problems, Bai et al. recently studied a class of parameterized inexact Uzawa (PIU) methods [Z.-Z. Bai, B.N. Parlett, Z.-Q. Wang. On generalized successive overrelaxation methods for augmented linear systems, Numer. Math. 102 (2005) 1–38]. For a special case that the (1,1)-block is solved exactly, they determined the convergence domain and computed the optimal iteration parameters and the corresponding optimal convergence factor for the induced method. In this paper, we develop these methods to the large sparse generalized saddle point problems. For the obtained parameterized inexact Uzawa method, we prove its convergence under suitable restrictions on the iteration parameters. In particular, we determine its quasi-optimal iteration parameters and the corresponding quasi-optimal convergence factor for the saddle point problems. Furthermore, This PIU method is generalized to obtain a framework of accelerated variants of the parameterized inexact Uzawa methods for solving both symmetric and nonsymmetric generalized saddle point problems by using the techniques of vector extrapolation, matrix relaxation and inexact iteration.