Yujiang Wu

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.

SSOR-like methods for saddle point problems

Saddle point problems arise in a wide variety of applications in computational and engineering. The aim of this paper is to present a SSOR-like iterative method for solving the saddle point problems. Here the convergence of this method is studied and specifically, the spectral radius and the optimal relaxation parameter of the iteration matrix are also investigated. Numerical experiments show that the SSOR-like method with a proper preconditioning matrix is better than SOR-like method presented by Golub et al. [G.H. Golub, X. Wu, and J.-Y. Yuan, SOR-like methods for augmented systems, BIT 41 (2001), pp. 71–85].

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.

Adaptive Synchronization of Fractional Neural Networks with Unknown Parameters and Time Delays

In this paper, the parameters identification and synchronization problem of fractional-order neural networks with time delays are investigated. Based on some analytical techniques and an adaptive control method, a simple adaptive synchronization controller and parameter update laws are designed to synchronize two uncertain complex networks with time delays. Besides, the system parameters in the uncertain network can be identified in the process of synchronization. To demonstrate the validity of the proposed method, several illustrative examples are presented.

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.

Numerical analysis and physical simulations for the time fractional radial diffusion equation

We do the numerical analysis and simulations for the time fractional radial diffusion equation used to describe the anomalous subdiffusive transport processes on the symmetric diffusive field. Based on rewriting the equation in a new form, we first present two kinds of implicit finite difference schemes for numerically solving the equation. Then we strictly establish the stability and convergence results. We prove that the two schemes are both unconditionally stable and second order convergent with respect to the maximum norm. Some numerical results are presented to confirm the rates of convergence and the robustness of the numerical schemes. Finally, we do the physical simulations. Some interesting physical phenomena are revealed; we verify that the long time asymptotic survival probability @Kt^-^@a, but independent of the dimension, where @a is the anomalous diffusion exponent.

Regularization methods for unknown source in space fractional diffusion equation

We discuss determining the unknown steady source in a space fractional diffusion equation and show that both the Fourier and wavelet dual least squares regularization methods work well for the ill-posed problem. The detailed error estimates are also strictly established for both of the methods. Moreover, the algorithm implementation and the corresponding numerical results are presented for the Fourier regularization method.