The semi-convergence of GSI method for singular saddle point problems

Transcription

1 Bull. Math. Soc. Sci. Math. Roumanie Tome 57(05 No., 04, The semi-convergence of GSI method for singular saddle point problems by Shu-Xin Miao Abstract Recently, Miao Wang considered the GSI method for solving nonsingular saddle point problems studied the convergence of the GSI method. In this paper, we prove the semi-convergence of the GSI method when it is applied to solve the singular saddle point problems. Key Words: GSI method, Singular saddle point problem, Semi-convergence, Iterative method. 00 Mathematics Subject Classification: Primary 65F0, Secondary 65F50. Introduction Consider the saddle point problems of the form ( ( A B x B T 0 y ( b = q, (. where A R m m is symmetric positive definite, B R m n is a matrix of rank r, b R m q R n are given vectors, with m n. When r = n, note that the coefficient matrix is nonsingular the saddle point problem (. has a unique solution. When r < n, the coefficient matrix is singular, in such case, we assume that the saddle point problem (. is consistent []. The saddle point problem (. appears in many engineering scientific computing applications such as constrained optimization, the finite element method to Stokes equations, fluid dynamics weighted linear squares problem [6, 3, 7, 9]. (. is also termed as a Karsh- Kuhn-Tucker (KKT system, or an augmented system, or an equilibrium system [0,, ]. For its property of large sparsity, (. is suitable for being solved by the iterative methods. For nonsingular saddle point problem (., there are many efficient iterative methods have been studied in the literature [3, 7, 4, 9, 7,, 3,, 8], see [4] for a comprehensive survey. Recently, Miao Wang studied the generalized stationary iterative (GSI method

2 94 S.-X. Miao [0] for nonsingular saddle point problem (., see [8]. Note that the GSI method, studied in [8], includes SOR-like method [3] GAOR method [4] as special cases. In most cases, the matrix B is full column rank in scientific computing engineering applications, but not always. If r < n, (. is a singular saddle point problem. Zheng, Bai Yang [] show that the GSOR method [3] can be used to solve the singular saddle point problem (., it is semi-convergent. Li et al. [5, 6] give the semi-convergent analyses of the GSSOR method [] the inexact Uzawa methods [9, 8] for the singular saddle point problem (.. In this paper, the GSI method for solving singular saddle point problem (. is further investigated the semi-convergence conditions are proposed, which generalize the result of Miao Wang [8] for the nonsingular saddle point problems to the singular saddle point problems. Throughout this paper, for A R m m, A T, σ(a ρ(a denote the transpose, the spectral set the spectral radius of the matrix A, respectively. I n is the identity matrix with order n. The semi-convergence of the GSI method Firstly, some basic concepts lemmas are given for latter use. For a matrix C R p p, we call C = M N a splitting if M is nonsingular. Let T = M N, then solving linear systems Cz = c is equivalent to considering the following iterative scheme x k+ = T x k + c, k = 0,,,. (. When C is nonsingular, for any initial vector x 0 the iteration scheme (. converges to the exact solution of the system of linear equations Cz = c if only if ρ(t <. But for the singular matrix C, we have σ(t ρ(t, so that one can require only the semiconvergence of the iterative scheme (.. By [5], the iterative scheme (. is semi-convergent if only if the following three conditions are satisfied: ( The spectral radius of the iterative matrix T is equal to one, i.e., ρ(t = ; ( The elementary divisors of the iterative matrix T associated with λ = σ(t are linear, i.e., rank(i p T = rank(i p T, here rank( denotes the rank of the corresponding matrix; (3 If λ σ(t with λ =, then λ =, i.e., ϑ(t <, where ϑ(t = max{ λ, λ σ(t, λ } is called the semi-convergence factor of the iterative scheme (.. We call a matrix T is semi-convergent provided it satisfies the above three conditions, iterative method (. is semi-convergent if T is a semi-convergent matrix. When C is singular, the semi-convergence property about the iteration scheme (. are described in the following two lemmas.

3 The semi-convergence of GSI method 95 Lemma. [5] Let C = M N with M nonsingular, T = M N. Then for any initial vector x 0 the iterative scheme (. is semi-convergent to a solution x of the system of linear equations Cz = c if only if the matrix T is semi-convergent. Lemma. [] Let H R l l with positive integers l. Then the partitioned matrix ( H 0 T = L is semi-convergent if only if either of the following conditions holds true: ( L = 0 H is semi-convergent; ( ρ(h <. Secondly, we review the GSI method presented in [8]. Following [3], we rewrite system (. as ( ( ( A B x b B T = (. 0 y q for the sake of simplicity. For the coefficient matrix of the linear system (., we make the following splitting: ( ( ( A B αa 0 (α A B B T 0 βb T αq ( βb T, (.3 αq where α β are real parameters with α 0, Q R n n is a nonsingular matrix. With the splitting (.3, the GSI iterative scheme [0] for augmented system (. is defined as ( x (k+ y (k+ = T α,β ( x (k y (k I t + (αd βl ( b q, k = 0,,, (.4 where ( ( T α,β = α Im α A B α β α Q B T I n α Q B T A B is the GSI iteration matrix. (.5 Algorithm. GSI Method:. Given the initial vectors x (0, y (0, the relaxation parameters α, β the nonsingular matrix Q.. For i = 0,,, until convergence, compute { x (k+ = ( α x(k + α A (b By (k, y (k+ = y (k + α Q { B T [x (k+ + ( βx (k ] q }. Remark. It is worth mentioning that the GSI method (.4 becomes the SOR-like method [3] when α = ω β =, the GAOR method [4] when α = ω β = γ ω.

4 96 S.-X. Miao Finally, we discuss the semi-convergence of the GSI method. When the coefficient matrix of (. is nonsingular, the convergence of GSI method is studied in [8]. When r < n, the coefficient matrix of (. is singular, the following theorem describes the semi-convergence property when the GSI method is applied to solve the singular saddle point poblem (.. Theorem. Let A R m m, Q R n n be symmetric positive definite matrices, B R m n be a matrix of rank r with r < n. Denote the largest eigenvalues of the matrix Q B T A B by µ max. Then the GSI method is semi-convergent to a solution of the singular saddle point problem (. if α β satisfies α > max {, µmax } α < β < α(α +. (.6 µ max µ max Proof: By Lemma, we only need to demonstrate the semi-convergence of the iteration matrix T α,β, defined by equation (.5, of the GSI method. Let B = U(B r, 0V be the singular value decomposition of B, where B r = (Σ r, 0 T R m n with Σ r = diag(σ, σ,, σ r, U, V are unitary matrices. Then ( U 0 P = 0 V is an (m + n-by-(m + n unitary matrix. Let T α,β = P T α,β P, then the matrix T α,β has the same eigenvalues with matrix T α,β. Here, we have used ( to denote the conjugate transpose of the corresponding complex matrix. Hence, we only need to demonstrate the semi-convergence of the matrix T α,β. Define matrices  = U AU, B = U BV Q = V QV. Then it holds that B = (B r, 0 ( Q V = Q V V Q V V Q V V Q, V here we have partitioned the unitary matrix V into the block form V = (V, V confirmly to the partition of the matrix B. Through direct operations, we have U A BV = (U A U(U BV =  (B r, 0 = ( B r, 0, V Q B T U = (V Q V (V B T U ( V = Q V Br T V Q V Br T

5 The semi-convergence of GSI method 97 Hence, V Q B T A BV = (V Q V (V B T U(U A U(U BV = Q (B r, 0 T  (B r, 0 = Q ( B T r  B r n r. (.7 T α,β = P T α,β P ( ( = α Im α U A BV α β α V Q B T U I n α V Q B T A BV ( Ĥα,β 0 =, L α,β I n r where Ĥ α,β = ( ( α β α α α Im αâ B r (V Q V B T r I r α (V Q V B T r  B r ( α β L α,β = (V Q V Br T, α (V Q V Br T  B r. As L α,β 0, from Lemma we know that the matrix T α,β is semi-convergent if only if ρ(ĥα,β <. Hence, in what follows, we will give out the restrictions for the parameters α β such that ρ(ĥα,β <. Consider the following nonsingular saddle point problem (  Br Br T 0 ( x ŷ = ( b q. (.8 If the coefficient matrix of the nonsingular saddle point problem (.8 be splitted as ( ( (  Br αâ Br T = 0 (α  0 βbr T α Q B r ( βbr T α Q with the preconditioning matrix Q = (V Q V, then the GSI method for solving (.8 can be well defined, the iteration matrix is Ĥα,β. From Theorem.4 in [8], we know that ρ(ĥα,β < if α β satisfies α > max {, µmax } α < β < α(α +, µ max µ max

6 98 S.-X. Miao here µ max is the largest eigenvalue of the matrix Q BT r  B r. By (.7, we can see that µ max is also the largest eigenvalue of the matrix Q B T A B. The proof is completed. The GSI method has the SOR-like method [3] the GAOR method [4] as its special cases, thus from Theorem, we can derive the following corollaries directly. Corollary. Let A Q be symmetric positive definite, B is a matrix of rank r with r < n. Denote the largest eigenvalues of the matrix Q B T A B by µ max. Then the SOR-like method is semi-convergent to a solution of the singular saddle point problems (. if 0 < ω < µ max. Corollary. Let A Q be symmetric positive definite, B is a matrix of rank r with r < n. Denote the largest eigenvalues of the matrix Q B T A B by µ max. Then the GAOR method is semi-convergent to a solution of the singular saddle point problems (. if ω γ satisfies 0 < ω < min{, µmax } ω µ max < γ < ω + ω ωµ max. Note that min{, µmax }, from Theorem.3 in [5], we known that the condition in Corollary is the sufficient but necessary condition. 3 Conclusion In this paper, we consider the GSI method for solving singular saddle point problems. The semi-convergence conditions are given, which generalize the result of Miao Wang [8] for nonsingular saddle point problems to singular saddle point problems. Meanwhile, from the proof of Theorem, we note that the semi-convergence factor of the GSI method for the singular saddle point problem (. is the convergence factor of the GSI method for the nonsingular saddle point problem (.8. References [] Z.-Z. Bai, G.H. Golub, Accelerated Hermitian skew-hermitian splitting iteration methods for saddle-point problems, IMA J. Numer. Anal., 7 (007, 3. [] Z.-Z. Bai, G.H. Golub, J.-Y. Pan, Preconditioned Hermitian skew-hermitian splitting methods for non-hermitian positive semidefinite linear systems, Numer. Math., 98 (004, 3.

7 The semi-convergence of GSI method 99 [3] Z.Z. Bai, B.N. Parlett, Z.Q. Wang, On generalized successive overrelaxation methods for augmented systems, Numer. Math., 0 (005, 38. [4] M. Benzi, G. H. Golub, J. Liesen, Numerical solution of saddle point problems, Acta. Numerica., 4 (005, 37. [5] A. Berman, R. J. Plemmons, Nonnegative Matrices in the Mathematical Sciences, SIAM, Philadelphia, PA, 994. [6] J.T. Betts, Practical Methods for Optimal Control using Nonlinear Programming, SIAM, Philadelphia, PA, 00. [7] Y.H. Cao, Y.Q. Lin, Y.M. Wei, Nolinear Uzawa methods for sloving nonsymmetric saddle point problems, J. Appl. Math. Comput., (006,. [8] F. Chen, Y.-L. Jiang, A generalization of the inexact parameterized Uzawa methods for saddle point problems, Appl. Math. Comput., 06 (008, [9] M.-R. Cui, A sufficient condition for the inexact Uzawa algorithm for saddle point problems. J. Comput. Appl. Math., 39 (00, [0] N. Dyn, W.E. Ferguson, The numerical solution of equality constrained quadratic programming problems, Math. Comput., 4 (983, [] M. Fortin, R. Glowinski, Augmented Lagrangian Methods: Applications to the Numerical Solution of Boundary Value Problems, North-Holl, Amsterdam, 983. [] P.E. Gill, W. Murray, M.H. Wright, Practical Optimization, Academic Press, New York, 98. [3] G.H. Golub, X. Wu, J.Y. Yuan, SOR-like methods for augmented systems, BIT, 4 (00, [4] C.J. Li, Z. Li, Y.Y. Nie, D.J. Evans, Generalized AOR method for the augmented system, International Journal of Computer Mathematics, 8 ( [5] J.-L. Li, T.-Z Huang, Semi-convegence analysis of the inexact Uzawa method for singular saddle point problems, Rev. Un. Mat. Argentina, 53 (0, [6] J.-L. Li, T.-Z Huang, D. Luo, The semi-convergence of generalized SSOR method for singular augmented systems, International of Numerical Analysis Modelling, 9 (0, [7] Z. Li, C.J. Li, D.J. Evans, T. Zheng, Two-parameter GSOR method for augmented system, International Journal of Computer Mathematics, 8 ( [8] S.-X. Miao, K. Wang, On generalized stationary iterative method for solving the saddle point problems, J. Appl. Math. Comput., 35 (0,

8 00 S.-X. Miao [9] Y.M. Wei, X.Y. Yu, R.Y. Zhang, Preconditioned conjugate gradient method generalized successive over relaxation method for weighted least squares problems, International Journal of Computer Mathematics, 8 (004, [0] J.H. Yun, S.W. Kim, Generalized stationary iterative method for solving linear systems, Korean J. Comput. Appl. Math., 5 (998, [] G.-F. Zhang, Q.-H. Lu, On generalized symmetric SOR method for augmented system, J. Comput. Appl. Math., 9 (008, [] B. Zheng, Z.-Z. Bai, X. Yang, On semi-convergence of parameterized Uzawa methods for singular saddle point problems, Linear Algebra Appl., 43 (009, Received: 4..0, Accepted: College of Mathematics Statistics, Northwest Normal University, Lanzhou, , P.R. China shuxinmiao@gmail.com