Pham Ky Anh

Parallel hybrid extragradient methods for pseudomonotone equilibrium problems and nonexpansive mappings

In this paper we propose and analyze three parallel hybrid extragradient methods for finding a common element of the set of solutions of equilibrium problems involving pseudomonotone bifunctions and the set of fixed points of nonexpansive mappings in a real Hilbert space. Based on parallel computation we can reduce the overall computational effort under widely used conditions on the bifunctions and the nonexpansive mappings. A simple numerical example is given to illustrate the proposed parallel algorithms.

Modified hybrid projection methods for finding common solutions to variational inequality problems

In this paper we propose several modified hybrid projection methods for solving common solutions to variational inequality problems involving monotone and Lipschitz continuous operators. Based on differently constructed half-spaces, the proposed methods reduce the number of projections onto feasible sets as well as the number of values of operators needed to be computed. Strong convergence theorems are established under standard assumptions imposed on the operators. An extension of the proposed algorithm to a system of generalized equilibrium problems is considered and numerical experiments are also presented.

Parallel methods for regularizing systems of equations involving accretive operators

In this paper, two parallel methods for solving systems of accretive operator equations in Banach spaces are studied. The convergence analysis of the methods in both free-noise and noisy data cases is provided.

Parallel Hybrid Methods for a Finite Family of Relatively Nonexpansive Mappings

In this article, we propose two parallel hybrid methods for finding a common fixed point of a finite family of relatively nonexpansive mappings. The strong convergence of the methods is established and their effectiveness are examined by numerical experiments. Thanks to the parallel computation, we can reduce the overall computational effort under widely used assumptions on mappings and spaces.

On the solvability of initial-value problems for nonlinear implicit difference equations

Our aim is twofold. First, we propose a natural definition of index for linear nonautonomous implicit difference equations, which is similar to that of linear differential-algebraic equations. Then we extend this index notion to a class of nonlinear implicit difference equations and prove some existence theorems for their initial-value problems.

Parallel regularized Newton method for nonlinear ill-posed equations

We introduce a regularized Newton method coupled with the parallel splitting-up technique for solving nonlinear ill-posed equations with smooth monotone operators. We analyze the convergence of the proposed method and carry out numerical experiments for nonlinear integral equations.

On discrete analogues of nonlinear implicit differential equations

This paper deals with some classes of nonlinear implicit difference equations obtained via discretization of nonlinear differential-algebraic or partial differential-algebraic equations. The unique solvability of discretized problems is proved and the compatibility between index notions for nonlinear differential-algebraic equations and nonlinear implicit difference equations is studied.

Novel Hybrid Methods for Pseudomonotone Equilibrium Problems and Common Fixed Point Problems

ABSTRACT The purpose of this article is to introduce some hybrid algorithms for finding a common element of the solution sets of pseudomonotone equilibrium problems and the fixed point sets of nonexpansive mappings in real Hilbert spaces. Our algorithms combine Mann’s iterative methods and Armijo line-search with parallel splitting-up and hybrid techniques. The strong convergence of the proposed algorithms are established without the assumption on the Lipschitz-type condition for the bifunctions involved.

PARALLEL ITERATIVE REGULARIZATION ALGORITHMS FOR LARGE OVERDETERMINED LINEAR SYSTEMS

In this paper, we study the performance of some parallel iterative regularization methods for solving large overdetermined systems of linear equations.

Stability of periodically switched discrete-time linear singular systems

Abstract In this paper we present some necessary and sufficient conditions for the stability of periodically switched discrete-time linear index-1 singular system, (PSSS). In particular, it is proved that, if at least one subsystem of a PSSS is asymptotically stable, then there is a switching rule, so that the whole system is also uniformly exponentially stable. Furthermore, for a periodically switched control system with no stable subsystems, there exist a switching rule and feedback matrices, such that the obtained PSSS is uniformly exponentially stable.