Ai-Li Yang

Lopsided PMHSS iteration method for a class of complex symmetric linear systems

Based on the preconditioned modified Hermitian and skew-Hermitian splitting (PMHSS) iteration method, we introduce a lopsided PMHSS (LPMHSS) iteration method for solving a broad class of complex symmetric linear systems. The convergence properties of the LPMHSS method are analyzed, which show that, under a loose restriction on parameter α, the iterative sequence produced by LPMHSS method is convergent to the unique solution of the linear system for any initial guess. Furthermore, we derive an upper bound for the spectral radius of the LPMHSS iteration matrix, and the quasi-optimal parameter α⋆ which minimizes the above upper bound is also obtained. Both theoretical and numerical results indicate that the LPMHSS method outperforms the PMHSS method when the real part of the coefficient matrix is dominant.

Pinning adaptive anti-synchronization between two general complex dynamical networks with non-delayed and delayed coupling

Abstract This paper investigates the anti-synchronization (AS) problem of two general complex dynamical networks with non-delayed and delayed coupling using pinning adaptive control method. Based on Lyapunov stability theory and Barbalat lemma, a sufficient condition is derived to guarantee the AS between two networks with non-delayed and delayed coupling. Numerical simulations are also presented to show the effectiveness of the proposed AS criterion.

A generalized preconditioned HSS method for non-Hermitian positive definite linear systems

Based on the HSS (Hermitian and skew-Hermitian splitting) and preconditioned HSS methods, we will present a generalized preconditioned HSS method for the large sparse non-Hermitian positive definite linear system. Our method is essentially a two-parameter iteration which can extend the possibility to optimize the iterative process. The iterative sequence produced by our generalized preconditioned HSS method can be proven to be convergent to the unique solution of the linear system. An exact parameter region of convergence for the method is strictly proved. A minimum value for the upper bound of the iterative spectrum is derived, which is relevant to the eigensystem of the products formed by inverse preconditioner and splitting. An efficient preconditioner based on incremental unknowns is presented for the actual implementation of the new method. The optimality and efficiency are effectively testified by some comparisons with numerical results.

On semi-convergence of the Uzawa-HSS method for singular saddle-point problems

Based on the Hermitian and skew-Hermitian splitting (HSS) iteration scheme, an efficient Uzawa-HSS iteration method has been proposed to solve the nonsingular saddle-point problems. In this paper, we discuss the feasibility of the Uzawa-HSS method used for solving singular saddle-point problems. The semi-convergence properties of the Uzawa-HSS iteration method are carefully analyzed, which show that the iterative sequence generated by the Uzawa-HSS method converges to a solution of the singular saddle-point problem if the iteration parameters satisfy suitable restrictions. Numerical results verify the robustness and efficiency of the Uzawa-HSS method.

Parameterized preconditioned Hermitian and skew-Hermitian splitting iteration method for saddle-point problems

By utilizing the preconditioned Hermitian and skew-Hermitian splitting (PHSS) iteration technique, we establish a parameterized PHSS (PPHSS) iteration method for non-Hermitian positive semidefinite linear saddle-point systems. The PPHSS method is essentially a two-parameter iteration which covers standard PHSS iteration and can extend the possibility to optimize the iterative process. The iterative sequence produced by the PPHSS method is proved to be convergent to the unique solution of the saddle-point problem when the iteration parameters satisfy a proper condition. In addition, for a special case of the PPHSS iteration method, we derive the optimal iteration parameter and the corresponding optimal convergence factor. Numerical experiments demonstrate the effectiveness and robustness of the PPHSS method both used as a solver and as a preconditioner for Krylov subspace methods.

Modified accelerated parameterized inexact Uzawa method for singular and nonsingular saddle point problems

Abstract Recently, Bai and Wang (2008) [9] presented an efficient accelerated parameterized inexact Uzawa (APIU) method for solving both symmetric and nonsymmetric nonsingular saddle point problems. In this work, we propose a modified APIU (MAPIU) method for solving saddle point problems, which is an extension of the APIU iteration method and covers a series of existing iterative methods. The MAPIU method can be applied not only to the nonsingular case but also to the singular case. The convergence properties for the nonsingular saddle point problems and the semi-convergence properties for the singular one are carefully discussed under suitable restrictions. We prove that the MAPIU method has a wider convergence (or semi-convergence) region than the APIU 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.

The semi-convergence properties of MHSS method for a class of complex nonsymmetric singular linear systems

We use the modified Hermitian and skew-Hermitian splitting (MHSS) iteration method to solve a class of complex nonsymmetric singular linear systems. The semi-convergence properties of the MHSS method are studied by analyzing the spectrum of the iteration matrix. Moreover, after investigating the semi-convergence factor and estimating its upper bound for the MHSS iteration method, an optimal iteration parameter that minimizes the upper bound of the semi-convergence factor is obtained. Numerical experiments are used to illustrate the theoretical results and examine the effectiveness of the MHSS method served both as a preconditioner for GMRES method and as a solver.

The modified shift-splitting preconditioners for nonsymmetric saddle-point problems

Abstract For a nonsymmetric saddle-point problem, a modified shift-splitting (MSS) preconditioner is proposed based on a splitting of the nonsymmetric saddle-point matrix. By removing the shift term of the ( 1 , 1 ) -block of the MSS preconditioner, a local MSS (LMSS) preconditioner is also presented. Both of the two preconditioners are easy to be implemented since they have simple block structures. The convergence properties of the two iteration methods induced respectively by the MSS and the LMSS preconditioners are carefully analyzed. Numerical experiments are illustrated to show the robustness and efficiency of the MSS and the LMSS preconditioners used for accelerating the convergence of the generalized minimum residual method.

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 A‡b