On parameterized inexact Uzawa methods for generalized saddle point problems

Abstract 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.

[1]  Bai Zhongzhi.  On the convergence of the generalized matrix multisplitting relaxed methods , 1995 .

[2]  Yongzhong Song,et al.  On the convergence of the MAOR method , 1997 .

[3]  O. Axelsson.  Iterative solution methods , 1995 .

[4]  Gene H. Golub,et al.  SOR-like Methods for Augmented Systems , 2001 .

[5]  Jun Zou,et al.  An Iterative Method with Variable Relaxation Parameters for Saddle-Point Problems , 2001, SIAM J. Matrix Anal. Appl..

[6]  Generalized extrapolation principle and convergence of some generalized iterative methods , 1993 .

[7]  John J. H. Miller.  On the Location of Zeros of Certain Classes of Polynomials with Applications to Numerical Analysis , 1971 .

[8]  A. Wathen,et al.  Minimum residual methods for augmented systems , 1998 .

[9]  Zhong-Zhi Bai.  Modified Block SSOR Preconditioners for Symmetric Positive Definite Linear Systems , 2001, Ann. Oper. Res..

[10]  Gene H. Golub,et al.  Matrix computations , 1983 .

[11]  Convergence for a general form of the GAOR method and its application to the MSOR method , 1994 .

[12]  Beresford N. Parlett,et al.  On generalized successive overrelaxation methods for augmented linear systems , 2005, Numerische Mathematik.

[13]  On the convergence of the generalized AOR method , 1997 .

[14]  王德人,et al.  GENERALIZED MATRIX MULTISPLITTING RELAXATION METHODS AND THEIR CONVERGENCE , 1993 .

[15]  Zhi-Hao Cao.  Fast Uzawa algorithms for solving non-symmetric stabilized saddle point problems , 2004, Numer. Linear Algebra Appl..

[16]  Apostol T. Vassilev,et al.  Analysis of the Inexact Uzawa Algorithm for Saddle Point Problems , 1997 .

[17]  Raymond H. Chan,et al.  Conjugate Gradient Methods for Toeplitz Systems , 1996, SIAM Rev..

[18]  Michel Fortin,et al.  Mixed and Hybrid Finite Element Methods , 2011, Springer Series in Computational Mathematics.

[19]  G. Golub,et al.  Inexact and preconditioned Uzawa algorithms for saddle point problems , 1994 .

[20]  Zhi-Hao Cao.  Fast uzawa algorithm for generalized saddle point problems , 2003 .

[21]  Changjun Li,et al.  A generalized successive overrelaxation method for least squares problems , 1998 .

[22]  M. Fortin,et al.  Augmented Lagrangian methods : applications to the numerical solution of boundary-value problems , 1983 .

[23]  M. Ng.  Preconditioning of elliptic problems by approximation in the transform domain , 1997 .

[24]  J. Gillis,et al.  Matrix Iterative Analysis , 1961 .

[25]  Zhong-Zhi Bai.  A class of modified block SSOR preconditioners for symmetric positive definite systems of linear equations , 1999, Adv. Comput. Math..

[26]  Zhong-Zhi Bai,et al.  Structured preconditioners for nonsingular matrices of block two-by-two structures , 2005, Math. Comput..