Convergence of a Generalized USOR Iterative Method for Augmented Systems

Zhang and Shang (2010) have presented the Uzawa-SOR (USOR) algorithm to solve augmented systems. In this paper, we establish a generalizedUzawa-SOR (GUSOR)method for solving augmented systems, which is the extension of theUSORmethod.We prove the convergence of the proposed method under suitable restrictions on the iteration parameters. Lastly, numerical experiments are carried out and experimental results show that our proposed method with appropriate parameters has faster convergence rate than the USOR method.


Introduction
We consider the solution of systems of linear equations of the 2-by-2 block structure as follows: where  ∈  × is a symmetric and positive definite matrix and  ∈  × is a matrix of full column rank, ,  ∈   , ,  ∈   , and  ≥ .  denotes the transpose of the matrix .We assume that they are of the appropriate dimensions whenever the zero matrix  and the identity matrix  are used in this paper.The linear systems (1) appears in many different applications of scientific computing, such as the mixed finite element for incompressible flow problems when some form of pressure stabilization is included in the discretization, constrained optimization [1], computational fluid dynamics, and Stokes problems, of constrained least squares problems, and generalized least squares problems [2][3][4][5]; see [6,7] and references therein.A large variety of methods for solving linear systems of the form (1) can be found in the literature.Yuan and Iusem [3,5] presented variants of the SOR method and preconditioned conjugate gradient methods.Golub et al. [8] proposed SORlike algorithms for solving the augmented systems, which was further accelerated and generalized by GSOR method in [9].Darvishi and Hessari [10] studied the SSOR method.Zhang and Lu [11] studied a GSSOR (generalized SSOR) method.Recently, Zhang and Shang [12] proposed the Uzawa-SOR method and studied its convergence.Bai and Wang [13] established and studied the parameterized inexact Uzawa (PIU) method for solving the corresponding saddle point problems, which was also discussed convergence conditions for matrix splitting iteration methods in [14].
The remainder of the paper is organized as follows.In Section 2, we establish a generalized Uzawa-SOR (GUSOR) method for solving augmented systems and analyze convergence of the corresponding method in Section 3. Numerical results are presented in Section 4. At last, we give some remarks in Section 5.

Generalized Uzawa-SOR (GUSOR) Method
For the sake of simplicity, we rewrite system (1) as where  ∈  × is a symmetric and positive definite matrix and  ∈  × is a matrix of full column rank.Let  be decomposed as  =  −  −  in which  is the diagonal of ,  is the strict lower part of , and  is the strict upper part of  with  and  being nonzero reals.
To construct the generalized USOR method, we consider the following splitting: where ∈ R × is a prescribed a symmetric positive definite matrix and  +  = 1.Let  and  be two nonzero reals, let   ∈  × and   ∈  × be the m-by-m and the n-by-n identity matrices, respectively, and let Ω be given parameter matrices of the form Then we consider the following generalized SOR iteration scheme for solving the augmented linear system (2): or equivalently, where More precisely, we have the following algorithmic description of this GUSOR method.

Convergence of the GUSOR Method
In this section, we will analyze a sufficient condition for parameters , , and  in the generalized Uzawa-SOR (GUSOR) method to solve augmented systems (2).We will use the following notations and definitions.For a vector ,  * denotes the complex conjugate transpose of the vector . min () and  max () denote the minimum and maximum eigenvalues of the Hermitian matrix , respectively, and () denotes the spectral radius of .We also assumed that the parameters of , , and  used in this paper are positive real numbers.
Note that the iteration matrix of the proposed methods is (, , ); therefore, the GUSOR method is convergent if and only if the spectral radius of the matrix (, , ), defined in ( 8) is less than one; that is, ((, , )) < 1.
Let  be an eigenvalue of (, , ) and let (   ) be the corresponding eigenvector.Then we have or equivalently, Lemma 2. Let  be a symmetric positive definite matrix and  a matrix of full column rank.If  is an eigenvalue of the iteration matrix (, , ), then  ̸ = 1.
Proof.If  = 1, since  and  are positive real numbers, then from ( 12), we have  = − −1  and    = 0.It follows that    −1  = 0 from the above relations, which leads to  = 0 and  = 0.This is a contradiction to the assumption that (   ) is an eigenvector of the iteration matrix (, , ).Lemma 3. Let  be a symmetric positive definite matrix, and  a matrix of full column rank.If  is an eigenvalue of the iteration matrix (, , ) and (   ) is an eigenvector of the iteration matrix (, , ) corresponding to the eigenvalue , then  ̸ = 0.Moreover, if  = 0, then || < 1.
Proof.The method of proof is exactly the same as in [12], here we omit the proof of Lemma 3.
Now we are in the position to establish the convergence of the proposed methods.The following theorem presents a sufficient condition for guaranteeing the convergence of the GUSOR method.

Numerical Experiments
In this section, we provide numerical experiments to examine the feasibility and effectiveness of GUSOR method for solving the saddle point problem (1) and compare the results between the GUSOR method and the USOR method provided in [12].We report the number of iterations (IT), norm of absolution residual vectors (RES), the elapsed CPU time (CPU), and the spectral radius of corresponding iterative matrix denoted by .Here, RES is defined as with (   ,    )  being the final approximate solution, where ‖ ⋅ ‖ refers to  2 -norm.We choose the right-hand vector (  ,   )  ∈  + such that the exact solution of the augmented linear system (1) is (  * ,   * )  = (1, 1, . . ., 1)  ∈  + .All numerical experiments are carried out on a PC equipped with Intel Core i3 2.3 GHz CPU and 2.00 GB RAM memory Using MATLAB R2010a.Example 6 (see [9]).Let the augmented system (1) in which with ⊗ is the Kronecker product symbol and ℎ = 1/( + 1) and  = tridiag(, , ) is a tridiagonal matrix with  , = ,  −1, = ,  ,+1 =  for appropriate .
For this example,  = 2 2 and  =  2 .Hence, the total number of variables is  +  = 3 2 .We choose the matrix  as an approximation to the matrix    −1 , according to three cases listed in Table 1.
In our experiments, all runs with respect to both USOR method and GUSOR method are started from the initial vector which is set to the zero vector and terminate if the current iteration satisfies ERR < 10 In Tables 2, 3, and 4, we list the values of (, ) which are same as in [12], IT, RES, CPU, and the spectral radii of corresponding iterative matrices for various problem sizes (m, n), respectively.They clearly show that the GUSOR method is more effective than the USOR method on convergence rate, computing speed, and the spectral radii of corresponding iterative matrices.IT and CPU of our proposed method are nearly half of the USOR if  is smaller.However, the relaxed parameters  of GUSOR method are not optimal values and only lie in the convergence region of the method.The determination of optimum values of the parameters needs further study.
Remark 7. When  = 0, in this case, the proposed method is the one in [12].Through experiment results, we find the optimal relaxation of  seems to be about 0.5.

Conclusions
In this paper, we propose the GUSOR method for the solution of the saddle point problems and analyze the convergence of GUSOR method.When chosen the relaxed parameters , the spectral radii of the iteration matrices, IT and CPU with the proposed method are smaller than those in [12], which is shown through numerical experiments.Particularly, one may discuss how to select the set of optimal parameters for accelerating the convergence of the considered method effectively.The optimal choice of this set of parameters is valuably studied which is our future work.

Table 1 :
Choices of the matrix .

Table 2 :
Numerical results for Example 6 in Case I with  = 0.4.

Table 3 :
Numerical results for Example 6 in Case II with  = 0.6.

Table 4 :
Numerical results for Example 4.1 in Case III with  = 0.4.