广义Lipschitz伪压缩映射黏滞迭代逼近方法的强收敛

K是Banach空间E的一个非空闭凸子集,T:K→K是一个广义Lipschitz伪压缩映射.对Lipschitz强伪压缩映射f:K→K和x_1∈K,序列{x_n}由下式定义:x_n+1=(1-α_n-β_n)x_n+α_nf(x_n)+β_nTx_n.在{α_n}与{β_n}满足合适条件的情况下,每当{z∈K;μ_n‖x_n-z‖~2=inf_(y∈K)μ_n‖x_n-y‖~2}∩F(T)≠φ时,{x_n}强收敛到T的某个不动点x~*.     

Convergence Results of the ERM Method for Nonlinear Stochastic Variational Inequality Problems

This paper considers the expected residual minimization (ERM) method proposed by Luo and Lin (J. Optim. Theory Appl. 140:103–116, 2009) for a class of stochastic variational inequality problems. Different from the work mentioned above, the function involved is assumed to be nonlinear in this paper. We first consider a quasi-Monte Carlo method for the case where the underlying sample space is compact and show that the ERM method is convergent under very mild conditions. Then, we suggest a compact approximation approach for the case where the sample space is noncompact. This work was supported in part by Project 10771025 supported by NSFC and SRFDP 20070141063 of China.     

Exploiting equalities in polynomial programming

We propose a novel approach for solving polynomial programs over compact domains with equality constraints. By means of a generic transformation, we show that existing solution schemes for the, typically simpler, problem without equalities can be used to address the problem with equalities.     

线性过程的强逼近和重对数律

本文讨论由独立同分布随机变量列产生的线性过程的泛函型重对数律和强逼近, 同时又给出由NA随机变量列产生的线性过程的重对数律.     

Alternative subcell discretisations for viscoelastic flow: Velocity-gradient approximation

Under subcell discretisation for viscoelastic flow, we have given further consideration to the compatibility of function spaces for stress/velocity-gradient approximation [see F. Belblidia, H. Matallah, B. Puangkird, M.F. Webster, Alternative subcell discretisations for viscoelastic flow: stress interpolation, J. Non-Newtonian Fluid Mech. 146 (2007) 59–78]. This has been conducted through the three scheme discretisations (quad-fe(par), fe(sc) and fe/fv(sc)). In this companion study, we have extended the application of an original implementation for velocity-gradient approximation, being of localised superconvergent recovered form, continuous and quadratic on the parent fe-triangular element. This has led to the consideration of both localised (pointwise) and global (Galerkin weighted-residual) approximations for velocity-gradient, highlighting some of their advantages and disadvantages. The global form is equivalent to the discontinuous elastico–viscous stress splitting (DEVSS-type) technique of Fortin and co-workers. Each representation, local or global, is based on linear/quadratic order upon parent or subcell element stencils. We consider Oldroyd modelling and the contraction flow benchmark, covering abrupt and rounded-corner planar geometries. The localised superconvergent quadratic velocity-gradient treatment affords strong stability and accuracy properties for the three scheme discretisations considered. Through associated analysis and iterative solution processes, we have successfully linked global approximations to their localised counterparts, depicting the inadequacy of inaccurate but stable versions through their corresponding solution features. These issues pervade all formulations, coupled or pressure-correction, and in focusing on velocity-gradient approximation, also apply universally to all discrete representations of stress. The inaccuracy of the global treatment can be somewhat repaired through an increase in (mass) iteration number. The efficiency of localised schemes (and associated properties) is particularly attractive over their global alternatives, being less restrictive to choice of spatial-order (higher-order). Conversely, global implementations are more restrictive in satisfaction of the space inclusion principle. Localised schemes come into their own when chosen to represent strongly localised solution features, such as arise in non-smooth flows. Analysis has also proved helpful in clarifying that space inclusion (extended LBB-condition) is a non-necessary convergence condition in the viscoelastic context.Overall, the localised-quadratic velocity-gradient treatment for both linear (subcell) and quadratic (parent) stress interpolation has achieved both stability and accuracy. Under DEVSS-type approximations (global), once function spaces for stress and velocity-gradients have been selected, this choice dictates the state of system consistency. Additionally, stability gains are recognised through the further application of strain-rate-stabilisation procedures.     

Global Convergence of a Nonlinear Programming Method Using Convex Approximations

The method of moving asymptotes (MMA) and its globally convergent extension SCP (sequential convex programming) are known to work well for certain problems arising in structural optimization. In this paper, the methods are extended for a general mathematical programming framework and a new scheme to update certain penalty parameters is defined, which leads to a considerable improvement in the performance. Properties of the approximation functions are outlined in detail. All convergence results of the traditional methods are preserved.     

Scheduling networks of queues: Heavy traffic analysis of a simple open network

We consider a queueing network with two single-server stations and two types of customers. Customers of type A require service only at station 1 and customers of type B require service first at station 1 and then at station 2. Each server has a different general service time distribution, and each customer type has a different general interarrival time distribution. The problem is to find a dynamic sequencing policy at station 1 that minimizes the long-run average expected number of customers in the system.The scheduling problem is approximated by a dynamic control problem involving Brownian motion. A reformulation of this control problem is solved, and the solution is interpreted in terms of the queueing system in order to obtain an effective sequencing policy. Also, a pathwise lower bound (for any sequencing policy) is obtained for the total number of customers in the network. We show via simulation that the relative difference between the performance of the proposed policy and the pathwise lower bound becomes small as the load on the network is increased toward the heavy traffic limit.     

Resolving Controversies in the Application of the Method of Multiple Scales and the Generalized Method of Averaging

I compare application of the method of multiple scales with reconstitution and the generalized method of averaging for determining higher-order approximations of three single-degree-of-freedom systems and a two-degree-of-freedom system. Three implementations of the method of multiple scales are considered, namely, application of the method to the system equations expressed as second-order equations, as first-order equations, and in complex-variable form. I show that all of these methods produce the same modulation equations.I address the problem of determining higher-order approximate solutions of the Duffing equation in the case of primary resonance. I show that the conclusions of Rahman and Burton that the method of multiple scales, the generalized method of averaging, and Lie series and transforms might lead to incorrect results, in that spurious solutions occur and the obtained frequency–response curves bear little resemblance to the actual response, is the result of their using parameter values for which the neglected terms are the same order as the retained terms. I show also that spurious solutions cannot be avoided, in general, in any consistent expansion and their presence does not constitute a limitation of the methods. In particular, I show that, for the Duffing equation, the second-order frequency–response equation does not possess spurious solutions for the case of hardening nonlinearity, but possesses spurious solutions for the case of softening nonlinearity. For sufficiently small nonlinearity, the spurious solutions are far removed from the actual response. But as the strength of the nonlinearity increases, these solutions move closer to the backbone and eventually distort it. This is not a drawback of the perturbation methods but an indication of an application of the analysis for parameter values outside the range of validity of the expansion.Also, I address the problem of obtaining non-Hamiltonian modulation equations in the application of the method of multiple scales to multi-degree-of-freedom Hamiltonian systems written as second-order equations in time and how this problem can be overcome by attacking the state-space form of the governing equations. Moreover, I show that application of a variation of the method of Rahman and Burton to multi-degree-of-freedom systems leads to results that do not agree with those obtained with the generalized method of averaging.Contributed by Prof. R.A. Ibrahim.     

Higher-order analysis of near-tip fields around an interfacial crack between two dissimilar power law hardening materials

By means of an asymptotic expansion method of a regular series, an exact higher-order analysis has been carried out for the near-tip fields of an interfacial crack between two different elastic-plastic materials. The condition of plane strain is invoked. Two group of solutions have been obtained for the crack surface conditions: (1) traction free and (2) frictionless contact, respectively. It is found that along the interface ahead of crack tip the stress fields are co-order continuous while the displacement fields are cross-order continuous. The zone of dominance of the asymptotic solutions has been estimated. The project supported by the National Natural Science Foundation of China