Constraint augmentation in pseudo-singularly perturbed linear programs

In this paper we study a linear programming problem with a linear perturbation introduced through a parameter ε > 0. We identify and analyze an unusual asymptotic phenomenon in such a linear program. Namely, discontinuous limiting behavior of the optimal objective function value of such a linear program may occur even when the rank of the coefficient matrix of the constraints is unchanged by the perturbation. We show that, under mild conditions, this phenomenon is a result of the classical Slater constraint qualification being violated at the limit and propose an iterative, constraint augmentation approach for resolving this problem.     

波系在液体中的弹性管梁上的反射和辐射

本文研究入射波系在液体中的半无限弹性管梁的开口端的反射和辐射问题,此波系由管梁上的挠曲波和管内、管外液体中相应的表面波(声波)所组成.利用Fourier变换,将这个半无限问题严格地归结为求解Wiener-Hopf型方程.然后将液体和管梁的密度比作为小参数,用摄动法求近似解.文章着重研究了反射系数的计算,还给出了远场的辐射型式曲线.     

On the vanishing viscosity limit for a 3-D system arising from the Keller-Segel model

In this paper, we consider the vanishing viscosity limit problem for a system arising from the Keller-Segel equations in three space dimensions. First, we construct an accurate approximate solution that incorporates the effects of boundary layers. Then, we prove the structural stability of the approximate solution as the chemical diffusion coefficient tends to zero. Our approach is based on the method of matched asymptotic expansions of singular perturbation theory and the classical energy estimates.     

Assessing Local Influence in Canonical Correlation Analysis

The first order local influence approach is adopted in this paper to assess the local influence of observations to canonical correlation coefficients, canonical vectors and several relevant test statistics in canonical correlation analysis. This approach can detect different aspects of influence due to different perturbation schemes. In this paper, we consider two different kinds, namely, the additive perturbation scheme and the case-weights perturbation scheme. It is found that, under the additive perturbation scheme, the influence analysis of any canonical correlation coefficient can be simplified to just observing two predicted residuals. To do the influence analysis for canonical vectors, a scale invariant norm is proposed. Furthermore, by choosing proper perturbation scales on different variables, we can compare the different influential effects of perturbations on different variables under the additive perturbation scheme. An example is presented to illustrate the effectiveness of the first order local influence approach.     

Analysis of the spectrum of a Cartesian Perfectly Matched Layer (PML) approximation to acoustic scattering problems

In this paper, we study the spectrum of the operator which results when the Perfectly Matched Layer (PML) is applied in Cartesian geometry to the Laplacian on an unbounded domain. This is often thought of as a complex change of variables or “complex stretching.” The reason that such an operator is of interest is that it can be used to provide a very effective domain truncation approach for approximating acoustic scattering problems posed on unbounded domains. Stretching associated with polar or spherical geometry lead to constant coefficient operators outside of a bounded transition layer and so even though they are on unbounded domains, they (and their numerical approximations) can be analyzed by more standard compact perturbation arguments. In contrast, operators associated with Cartesian stretching are non-constant in unbounded regions and hence cannot be analyzed via a compact perturbation approach. Alternatively, to show that the scattering problem PML operator associated with Cartesian geometry is stable for real nonzero wave numbers, we show that the essential spectrum of the higher order part only intersects the real axis at the origin. This enables us to conclude stability of the PML scattering problem from a uniqueness result given in a subsequent publication.     

Local Profiles for Elliptic Problems at Different Scales: Defects in,and Interfaces between Periodic Structures

Following-up on a previous work of ours, we present a general approach to approximate at the fine scale the solution to an elliptic equation with oscillatory coefficient when this coefficient consists of a “nice” (in the simplest possible case say periodic) function which is, in some sense to be made precise, perturbed. The approach is based on the determination of a local profile, solution to an equation similar to the corrector equation in classical homogenization. The well-posedness of that equation, in various functional settings depending upon the nature of the perturbation, is the purpose of this article. The case of a local perturbation is first addressed. The case of a more complex geometrical structure (such as the prototypical case of two different periodic structures separated by a common interface) is next discussed. Some related problems, and future directions of research are mentioned.     

On Stokes flow with variable and degenerate surface tension coefficient

Short-time existence, uniqueness, and regularity results are shown for the moving boundary problem of a free drop of liquid governed by the Stokes equations and driven by surface tension. The value of the surface tension coefficient is variable, not necessarily strictly positive, and transported with the flow on the moving surface.By a perturbation of identity approach, the problem is transformed into a nonlinear, nonlocal first order degenerate parabolic evolution equation on a fixed reference manifold. Its solvability is proved by deriving a priori estimates and using Galerkin approximations.     

Postoptimality Analyses of the Transportation Problem

We develop here a new approach to postoptimality analyses of the transportation problem (TP) through the use of perturbation analysis. In contrast to the usual sensitivity analysis the proposed methods allow for simultaneous, independent (or dependent) perturbation of the supplies, demands and transportation rates cost from their nominal values whilst preserving the current optimal basis. The essential parts of the calculation involve the same manipulations as the solution algorithm used in solving the parametric perturbed supply-demand version of the nominal TP. Furthermore, the coefficient of the perturbed parameters in the optimal parametric cost function is used to indicate the existence of a cheaper (or equivalent cost) solution than the current optimal one by shipping more total goods. The proposed approach permits computation of the maximal allowable additional units and distribution of these extra units in a systematic manner. The proposed unified procedures are demonstrated by numerical examples.     

Solving the flight perturbation problem with meta heuristics

When there is a perturbation in a carefully constructed aircraft schedule, e.g. an aircraft breakdown, it is important to minimize the negative consequences of this disturbance. Here, a tabu search and a simulated annealing approach to the flight perturbation problem are presented. The heuristics use a tree-search algorithm to find new schedules for the aircraft, and utilize a path relinking strategy to explore paths between structurally different solutions. The computational results indicate that the solution strategies, especially the tabu search, can be successfully used to solve the flight perturbation problem.     

Perturbation singulière d'un problème de frottement sec non monotone

In this Note we deal with a singularly perturbed system constituted by a differential inclusion which has a unique solution for each value of the perturbation parameter. The associated degenerated problem, that corresponds to a dynamic dry friction problem, has many solutions. We show that perturbed problem solutions converge to a particular solution of the degenerated problem when the perturbation parameter goes to zero. The singular perturbation approach allows an analysis of a criterion used to select a solution of the degenerated problem, and suggests a method to study more elaborated dry friction problems.     

Wave Propagation in Dissipative or Dispersive Non-linear Media

A singular perturbation method is developed to investigate onedimensional weak nonlinear waves in dissipative or dispersivemedia. Utilizing this method a boundary value problem for asystem of partial differential equations characterizing wavepropagation in homogeneous dissipative or dispersive media isstudied. In order to obtain a first-order uniformly valid solution,the problem is reduced to an initial value problem for scalarnon-linear partial differential equation. Some special casesarising from the structure of coefficient matrices are examinedand the method is extended to these cases. As an applicationof the perturbation method, various problems of wave propagationin a finite linear viscoelastic half-space are studied.     

Fuzzy perturbation analysis,part I: Directional perturbation

In this paper, a new idea, fuzzy perturbation, is advanced. The stability of the solutions of a fuzzy relation equation and the generalized solution of the unsolvable equation are defined by means of this idea. Thereupon, a part of the theory of fuzzy perturbation analysis, fuzzy directional perturbation, is formed. By use of the theory an approach to solving two open problems in the inverse problem of fuzzy multifactorial decision is made.     

Hyperbolic-parabolic singular perturbation for quasilinear equations of Kirchhoff type

We consider a hyperbolic-parabolic singular perturbation problem for a quasilinear equation of Kirchhoff type, and obtain parameter-dependent time decay estimates of the difference between the solutions of a quasilinear dissipative hyperbolic equation of Kirchhoff type and the corresponding quasilinear parabolic equation. For this purpose we show time decay estimates for hyperbolic-parabolic singular perturbation problem for linear equations with a time-dependent coefficient.     

二次规划逆问题的牛顿方法

针对二次规划逆问题,将其表达为带有互补约束的锥约束优化问题.借助于对偶理论,将问题转化为变量更少的线性互补约束非光滑优化问题.通过扰动的方法求解转化后的问题并证明了收敛性.采用非精确牛顿法求解扰动问题,给出了算法的全局收敛性与局部二阶收敛速度.最后通过数值实验验证了该算法的可行性.     

Some results on condition numbers of the scaled total least squares problem

Under the Golub-Van Loan condition for the existence and uniqueness of the scaled total least squares (STLS) solution, a first order perturbation estimate for the STLS solution and upper bounds for condition numbers of a STLS problem have been derived by Zhou et al. recently. In this paper, a different perturbation analysis approach for the STLS solution is presented. The analyticity of the solution to the perturbed STLS problem is explored and a new expression for the first order perturbation estimate is derived. Based on this perturbation estimate, for some STLS problems with linear structure we further study the structured condition numbers and derive estimates for them. Numerical experiments show that the structured condition numbers can be markedly less than their unstructured counterparts.     

Bifurcations from nondegenerate families of periodic solutions in Lipschitz systems

The paper addresses the problem of bifurcation of periodic solutions from a normally nondegenerate family of periodic solutions of ordinary differential equations under perturbations. The approach to solve this problem can be described as transforming (by a Lyapunov–Schmidt reduction) the initial system into one which is in the standard form of averaging, and subsequently applying the averaging principle. This approach encounters a fundamental problem when the perturbation is only Lipschitz (nonsmooth) as we do not longer have smooth Lyapunov–Schmidt projectors. The situation of Lipschitz perturbations has been addressed in the literature lately and the results obtained conclude the existence of the bifurcated branch of periodic solutions. Motivated by recent challenges in control theory, we are interested in the uniqueness problem. We achieve this in the case when the Lipschitz constant of the perturbation obeys a suitable estimate.     

Uniqueness in One Inverse Problem for the Elasticity System

We consider an inverse problem for the stationary elasticity system with constant Lame coefficients and a variable matrix coefficient depending on the spatial variables and frequency. The right-hand side contains a delta-function whose support (source) varies in some domain disjoint from the support of the variable coefficient. The inverse problem is to find the coefficient from the scattered wave measured at the same point at which the perturbation originates. A uniqueness theorem is proven. The proof bases on reduction of the inverse problem to a family of equations with the M. Riesz potential.     

Nonlinear Flow Between Permeable Disks Using Computer-Extended Series Method

The problem of two-dimensional, steady, nonlinear flow of an incompressible, viscous fluid between two parallel permeable disks is studied using the computer-extended series solution (CESS). The limitation of the classical regular perturbation technique (RPT) in solving this problem is highlighted and the CESS method in conjunction with Padé approximation is advocated to analyze the problem for much larger values of suction/injection Reynolds number R and to achieve higher accuracy. The skin-friction coefficient and coefficient of pressure distribution are evaluated for different values of R . The advantages of using CESS method over the RPT and numerical technique are discussed.     

Series solution for unsteady axisymmetric flow and heat transfer over a radially stretching sheet

This paper deals with the unsteady axisymmetric flow and heat transfer of a viscous fluid over a radially stretching sheet. The heat is prescribed at the surface. The modelled non-linear partial differential equations are solved using an analytic approach namely the homotopy analysis method. Unlike perturbation technique, this approach gives accurate analytic approximation uniformly valid for all dimensionless time. The explicit expressions for velocity, temperature and skin friction coefficient are developed. The influence of time on the velocity, temperature and skin friction coefficient is discussed.     

On perturbations of abstract fractional differential equations by nonlinear operators

We prove the unique solvability of a Cauchy-type problem for an abstract parabolic equation containing fractional derivatives and a nonlinear perturbation term. The result is applied to establish the solvability of the inverse coefficient problem for a fractional-order equation.