An improved proximal-based decomposition method for structured monotone variational inequalities

The proximal-based decomposition method was originally proposed by Chen and TebouUe(Math.Programming,1994,64:81-101 for solving convex minimization problems.This paper extends it to solving monotone variational inequalities associated with separable structures with the improvements that the restrictive assumptions on the involved parameters are much relaxed,and thus makes it practical to solve the subprob- lems easily.Without additional assumptions,global convergence of the new method is proved under the same mild assumptions on the problem's data as the original method.     

Three-step relaxed hybrid steepest-descent methods for variational inequalities

The classical variational inequality problem with a Lipschitzian and strongly monotone operator on a nonempty closed convex subset in a real Hilbert space is studied. A new three-step relaxed hybrid steepest-descent method for this class of variational inequalities is introduced.Strong convergence of this method is established under suitable assumptions imposed on the algorithm parameters.     

Projected subgradient method for non-Lipschitz set-valued mixed variational inequalities

A projected subgradient method for solving a class of set-valued mixed variational inequalities (SMVIs) is proposed when the mapping is not necessarily Lipschitz. Under some suitable conditions, it can be proven that the sequence generated by the method can strongly converge to the unique solution to the problem in the Hilbert spaces.     

关于变分不等式问题近似解的数值证明

基于文献[1]给出了一种数值证明变分不等式解的存在性方法。通过Hilbert空间中的Riesz表示定理,首先将变分不等式问题的迭代过程转化为一种不动点形式,再利用Schauder不动点定理构造了一个高效率的数值证明过程,即通过数值计算产生一个包含近似解的有界闭凸子集。非线性Helmholtz方程的算例说明这一方法的可行性和高效性。     

NEW SIMPLE SMOOTH MERIT FUNCTION FOR BOX CONSTRAINED VARIATIONAL INEQUALITIES AND DAMPED NEWTON TYPE METHOD

IntroductionLetX Rn,F:X→Rnbe given, the variational inequality, denoted byVI(X,F),is tofind a vectorx∈ Xsuch thatF(x)T(y-x)≥0, y∈ X. (1)DenoteN∶={1,2,…,n},whenX =[a,b]∶={x∈Rn|ai≤xi≤bi,i∈N},theVI(X,F)is called the box constrained variational ine     

Nonlinear implicit iterative method for solving nonlinear ill-posed problems

In the paper, we extend the implicit iterative method for linear ill-posed operator equations to solve nonlinear ill-posed problems. We show that under some conditions the error sequence of solutions of the nonlinear implicit iterative method is monotonically decreasing and, with this monotonicity, prove convergence of the new method for both the exact and perturbed equations.     

Approximate solution to a class of multivalued nonlinear mixed variational inclusions

IntroductionIn recent years,the theory of variational inclusion has appeared as an elegant andfascinating branch of pure and applied mathematics.This theory provides us with a convenientmathematical apparatus for uniformly studying a wide range of problem…     

A Newton type iterative method for heat-conduction inverse problems

An inverse problem for identification of the coefficient in heat-conduction equation is considered. After reducing the problem to a nonlinear ill-posed operator equation, Newton type iterative methods are considered. The implicit iterative method is applied to the linearized Newton equation, and the key step in the process is that a new reasonable a posteriori stopping rule for the inner iteration is presented. Numerical experiments for the new method as well as for Tikhonov method and Bakushikskii method are given, and these results show the obvious advantages of the new method over the other ones.     

A Newton type iterative inverse method for heat-conduction problems

An inverse problem for identification of the coefficient in heat-conduction equation is considered. After reducing the problem to a nonlinear ill-posed operator equation, Newton type iterative methods are considered. The implicit iterative method is applied to the linearized Newton equation, and the key step in the process is that a new reasonable a posteriori stopping rule for the inner iteration is presented. Numerical experiments for the new method as well as for Tikhonov method and Bakushikskii method are given, and these results show the obvious advantages of the new method over the other ones.     

基于Uzawa算法的弹塑性扭转问题的局部微分求积法

研究了由椭圆变分不等式描述的弹塑性扭转问题,构造了基于Uzawa算法的局部微分求积法,给出了数值算例,通过与有限元方法的比较,说明了方法的有效性。     

On the Mann and Ishikawa iterative approximation of solutions to variational inclusions with accretive type mappings in Banach spaces

The purpose of this paper is to introduce and study the existence of solutions and convergence of Mann and Ishikawa iterative processes for a class of variational inclusions with accretive type mappings in Banach spaces. The results presented in this paper extend and improve the corresponding results by Chang, Ding, Hassouni, Kazmi, Siddiqi, Zeng et al. Foundation item: the National Natural Science Foundation of China (19771058)     

The existence and uniqueness of solutions of generalized variational inequalities arising from elasticity with friction

ＩｎｔｒｏｄｕｃｔｉｏｎＡｔｐｒｅｓｅｎｔ,ｔｈｅｒｅｓｅａｒｃｈｏｎｖａｒｉａｔｉｏｎａｌｉｎｅｑｕａｌｉｔｉｅｓｃａｕｓｅｄｂｙｅｌａｓｔｉｃｉｔｙｗｉｔｈｆｒｉｃｔｉｏｎａｒｅｓｔｉｌｌｒｅｓｔｒｉｃｔｅｄｔｏｔｈｅｅｑｕｉｖａｌｅｎｔｍａｘｉｍｕｍｏｒｍｉｎｉｍｕｍｅｎｅｒｇｙｐｒｉｎｃｉｐｌｅｓ .Ｓｏｍｅｆａｍｏｕｓｅｘｐｅｒｔｓ,ＧＵＯＹｏｕ＿ｚｈｏｎｇ[1],ＺＨＯＵＳｈｕ＿ｚｈｉ[2 ],Ｇ .Ｆ .ＣａｒｅｙａｎｄＪ.Ｔ .Ｏｄｅｎ[3]ａｎｄＪ.Ｔ .ＯｄｅｎａｎｄＬ .Ｃａｍｐｏｓ[4 ],ａｌｌｈａｖ…     

An iterative parallel algorithm of finite element method

In this paper, a parallel algorithm with iterative form for solving finite element equation is presented. Based on the iterative solution of linear algebra equations, the parallel computational steps are introduced in this method. Also by using the weighted residual method and choosing the appropriate weighting functions, the finite element basic form of parallel algorithm is deduced. The program of this algorithm has been realized on the ELXSI-6400 parallel computer of Xi'an Jiaotong University. The computational results show the operational speed will be raised and the CPU time will be cut down effectively. So this method is one kind of effective parallel algorithm for solving the finite element equations of large-scale structures.     

On the existence and uniqueness of solutions for a class of variational inequalities with applications to the signorini problem in mechanics

In this paper,we introduce a new unified and general class of variational inequalities,and show some existence and uniqueness results of solutions for this kind of variationalinequalities.As an application,we utilize the results presented in this paper to study theSignorini problem in mechanics.     

PRECONDITIONED GAUSS-SEIDEL TYPE ITERATIVE METHOD FOR SOLVING LINEAR SYSTEMS

The preconditioned Gauss-Seidel type iterative method for solving linear systems, with the proper choice of the preconditioner, is presented. Convergence of the preconditioned method applied to Z-matrices is discussed. Also the optimal parameter is presented. Numerical results show that the proper choice of the preconditioner can lead to effective by the preconditioned Gauss-Seidel type iterative methods for solving linear systems.     

Adaptive variational multiscale method for the Stokes equations

An adaptive variational multiscale method for the Stokes equations is presented in this paper. We solve the coarse scale problem on the coarse mesh and approximate the fine scale solution by solving a series of local residual equations defined on some local fine grids, which can be implemented in parallel. In addition, we also propose a reliable local a posteriori error estimator and construct an adaptive algorithm based on the corresponding a posterior error estimate. Finally, numerical examples are presented to verify the algorithm.Copyright © 2012 John Wiley & Sons, Ltd.     

A natural neighbour method based on Fraeijs de Veubeke variational principle for materially non-linear problems

Abstract The natural neighbour method can be considered as one of many variants of the meshless methods. In the present paper, a new approach based on the Fraeijs de Veubeke （FdV） functional, which is initially developed for linear elasticity, is extended to the case of geometrically linear but materially non-linear solids. The new approach provides an original treatment to two classical problems： the numerical evaluation of the integrals over the domain A and the enforcement of boundary conditions of the type ui = hi on Su. In the absence of body forces （Fi = 0）, it will be shown that the calculation of integrals of the type fA .dA can be avoided and that boundary conditions of the type ui = hi on Su can be imposed in the average sense in general and exactly if hi is linear between two contour nodes, which is obviously the case for tTi = O.     

Modified iterative method for augmented system

The successive overrelaxation-like(SOR-like) method with the real parameters ω is considered for solving the augmented system. The new method is called the modified SOR-like(MSOR-like) method. The functional equation between the parameters and the eigenvalues of the iteration matrix of the MSOR-like method is given. Therefore, the necessary and sufficient condition for the convergence of the MSOR-like method is derived. The optimal iteration parameter ω of the MSOR-like method is derived. Finally, the proof of theorem and numerical computation based on a particular linear system are given, which clearly show that the MSOR-like method outperforms the SOR-like(Li, C. J., Li, B. J., and Evans, D. J. Optimum accelerated parameter for the GSOR method. Neural, Parallel Scientific Computations, 7(4), 453–462(1999)) and the modified symmetric SOR-like(MSSOR-like) methods(Wu, S. L., Huang, T. Z., and Zhao, X. L. A modified SSOR iterative method for augmented systems.     

Alternative formulations of isotropic hardening for Mises materials, and associated variational inequalities

This work provides insight into aspects of classical Mises–Hill plasticity, its extension to the Aifantis theory of gradient plasticity, and the formulations of both theories as variational inequalities. Firstly, it is shown that the classical isotropic hardening rule, which is dissipative in nature, may equally well be characterized via a defect energy—and, what is striking, this energetically based hardening rule mimics dissipative behavior by describing loading processes that are irreversible. A second aspect concerns the equivalence between the conventional form of the flow rule and its formulation in terms of dissipation. This equivalence has been previously established using the tools of convex analysis (cf., e.g., Han and Reddy, Plasticity: mathematical theory and numerical analysis, Springer, New York, 1999)—in the current work this equivalence is derived directly from the constitutive equations and the specific form of the dissipation, without recourse to such machinery. Variational inequalities corresponding to the dissipative and energetic forms of the flow rule are derived; these inequalities involve only the displacement and plastic strain and are well suited to computational studies. Finally, it is shown that the framework developed for the classical theory is easily extended to incorporate the gradient-plasticity theory of Aifantis (Trans ASME J Eng Mater Technol 106:326–330, 1984).