The necessity of Shishkin decompositions

In the present paper, we study model singularly perturbed convection-diffusion problems with exponential boundary layers. It has been believed for some time that only a complete splitting of the exact solution into regular and layer parts provides the information necessary for the study of the uniform convergence properties of numerical methods for these problems on layer-adapted grids (such as Shishkin meshes). In the present paper, we give new proofs of uniform interpolation error estimates for linear and bilinear interpolation; these proofs are based on the older a priori bounds derived by Kellogg and Tsan [1].     

修正乘子交替方向法求解三个可分离算子的凸优化

指出直接推广的经典乘子交替方向法对三个算子的问题不能保证收敛的原因, 并且给出将其改造成收敛算法的相应策略. 同时, 在一个统一框架下, 证明了修正的乘子交替方向法的收敛性和遍历意义下具有O(1/t)~收敛速率.     

Helmholtz方程周期Green函数及其偏导数截断误差收敛阶的分析

在应用边界元方法求解Helmholtz方程周期边值问题时,需要构造以周期Green函数或其偏导数为核函数的积分算子形式的解.由于Helmholtz方程的周期Green函数G~P是一个函数项级数,该级数的通项是Hankel函数,在数值求解中,需要对其进行截断,从而很有必要研究其截断误差.本文根据Hankel函数在变量趋于无穷大时的渐近展开式,并结合Abel不等式,证明了G~P及其一阶偏导和二阶混合偏导一致收敛,且其截断误差收敛阶均为O(1/p~(1/2)).最后,通过数值实验验证了理论证明的正确性.本文的证明方法也可被用于证明其它一些方程周期Green函数的收敛性问题.     

Geometric Properties of Symmetric Spaces with Applications to Orlicz–Lorentz Spaces

We deal with the basic convexity properties –rotundity, and uniform, local uniform and full rotundity –- for symmetric spaces. A characterization of Orlicz–Lorentz spaces with the Kadec–Klee property for pointwise convergence is given. These results are applied to obtain criteria of convexity properties for Orlicz–Lorentz sequence spaces, and some new proofs of the sufficiency part of criteria for rotundity and uniform rotundity for Orlicz–Lorentz function spaces.     

Diffusion approximation of a multitype re-entrant line under smaller-buffer-first-served policy

This paper studies a multitype re-entrant line under smaller-buffer-first-served policy, which is an extension of first-buffer-first-served re-entrant line. We prove a heavy traffic limit theorem. The key to the proof is to prove the uniform convergence of the corresponding critical fluid model.     

Newton's method for nonlinear ordinary and partial differential equations

A unified approach is presented for proving the local, uniform and quadratic convergence of the approximate solutions and a-posteriori error bounds obtained by Newton's method for systems of nonlinear ordinary or partial differential equations satisfying an inverse-positive property. An important step is to show that, at each iteration, the linearized problem is inverse-positive. Many classes of problems are shown to satisfy this property. The convergence proofs depend crucially on an error bound derived previously by Rosen and the author for quasilinear elliptic, parabolic and hyperbolic problems.     

Convergence analysis of gradient descent stochastic algorithms

This paper proves convergence of a sample-path based stochastic gradient-descent algorithm for optimizing expected-value performance measures in discrete event systems. The algorithm uses increasing precision at successive iterations, and it moves against the direction of a generalized gradient of the computed sample performance function. Two convergence results are established: one, for the case where the expected-value function is continuously differentiable; and the other, when that function is nondifferentiable but the sample performance functions are convex. The proofs are based on a version of the uniform law of large numbers which is provable for many discrete event systems where infinitesimal perturbation analysis is known to be strongly consistent.     

Well posedness,large deviations and ergodicity of the stochastic 2D Oldroyd model of order one

In this work, we establish the unique global solvability of the stochastic two dimensional viscoelastic fluid flow equations, arising from the Oldroyd model for the non-Newtonian fluid flows perturbed by multiplicative Gaussian noise. A local monotonicity property of the linear and nonlinear operators and a stochastic generalization of the Minty–Browder technique are exploited in the proofs. The Laplace principle for the strong solution of the stochastic system is established in a suitable Polish space using a weak convergence approach. The Wentzell–Freidlin large deviation principle is proved using the well known results of Varadhan and Bryc. The large deviations for shot time are also considered. We also establish the existence of a unique ergodic and strongly mixing invariant measure for the stochastic system with additive Gaussian noise, using the exponential stability of strong solutions.     

样条变差缩减算子迭代极限的一个简单证明

一、引言在[1]和[2]中,胡莹生、徐叔贤利用 Markov 链终极条件概率的有关结果,确定了一类变差缩减算子的迭代极限.本文采用作者在[3]中使用的技巧,利用多项式样条的若干基本知识,不但简单地得出了[1,2]的结果,并且给出了迭代收敛速度的估计.对于等距分划的三次样条,所给的误差估计在某种意义上是最好的.     

Effective metastability of Halpern iterates in CAT(0) spaces

This paper provides an effective uniform rate of metastability (in the sense of Tao) on the strong convergence of Halpern iterations of nonexpansive mappings in CAT(0) spaces. The extraction of this rate from an ineffective proof due to Saejung is an instance of the general proof mining program which uses tools from mathematical logic to uncover hidden computational content from proofs. This methodology is applied here for the first time to a proof that uses Banach limits and hence makes a substantial reference to the axiom of choice.     

Local limit theorems for multiplicative free convolutions

This paper describes the quality of convergence to an infinitely divisible law relative to free multiplicative convolution. We show that convergence in distribution for products of identically distributed and infinitesimal free random variables implies superconvergence of their probability densities to the density of the limit law. Superconvergence to the marginal law of free multiplicative Brownian motion at a specified time is also studied. In the unitary case, the superconvergence to free Brownian motion and that to the Haar measure are shown to be uniform over the entire unit circle, implying further a free entropic limit theorem and a universality result for unitary free Lévy processes. Finally, the method of proofs on the positive half-line gives rise to a new multiplicative Boolean to free Bercovici–Pata bijection.     

Unfolding homogenization in doubly periodic media and applications

We define a reiterated unfolding operator for a doubly periodic domain presenting two periodicity scales. Then we show how to apply it to the homogenization of both linear and nonlinear problems. The main novelty is that this method allows the use of test functions with one scale of periodicity only and it considerably simplifies the proofs of the convergence results. We illustrate this new approach on a Poisson problem with Dirichlet boundary conditions and on the flow of a power law fluid in a doubly periodic porous medium.     

A quasistatic contact problem with slip‐dependent coefficient of friction

We consider a mathematical model which describes the bilateral quasistatic contact of a viscoelastic body with a rigid obstacle. The contact is modelled with a modified version of Coulomb's law of dry friction and, moreover, the coefficient of friction is assumed to depend either on the total slip or on the current slip. In the first case, the problem depends upon contact history. We present the classical formulations of the problems, the variational formulations and establish the existence and uniqueness of a weak solution to each of them, when the coefficient of friction is sufficiently small. The proofs are based on classical results for elliptic variational inequalities and fixed point arguments. We also study the dependence of the solutions on the perturbations of the friction coefficient and obtain a uniform convergence result. Copyright © 1999 John Wiley & Sons, Ltd.     

关于Dini-Lipschitz型逼近定理的统一处理方法方法

This survey paper studies the approximation of (polynomial) processes for which the operator norms do not form a bounded sequence. In view of familiar direct estimates and quantitative uniform boundedness principles, a unified approach is given to results concerning the equivalence of Dini-Lipschitz-type conditions with (strong) convergence on (smoothness) classes. Emphasis is laid upon the necessity of these conditions, essential ingredients of the proofs are suitable modifications of the familiar gliding hump method. Apart from the classical results concerned with Fourier partial sums, explicit applications are treated for (trigonometric as well as algebralc) Lagrange interpolation, interpolatory quadrature rules based upon Jacobl knots, multipliers or strong convergence, and for Bochner-Riesz means of multivariate Fourier series for parameter values below the critical index.     

Numerical approximation of entropy solutions for hyperbolic integro-differential equations

Summary. Scalar hyperbolic integro-differential equations arise as models for e.g. radiating or self-gravitating fluid flow. We present finite volume schemes on unstructured grids applied to the Cauchy problem for such equations. For a rather general class of integral operators we show convergence of the approximate solutions to a possibly discontinuous entropy solution of the problem. For a specific model problem in radiative hydrodynamics we introduce a convergent fully discrete finite volume scheme. Under the assumption of sufficiently fast spatial decay of the entropy solution we can even establish the convergence rate h^1/4|ln(h)| where h denotes the grid parameter. The convergence proofs rely on appropriate variants of the classical Kruzhkov method for local balance laws together with a truncation technique to cope with the nonlocal character of the integral operator.Mathematics Subject Classification (2000): 35L65, 35Q35, 65M15     

A family of iterative methods for computing the approximate inverse of a square matrix and inner inverse of a non-square matrix

Based on a quadratical convergence method, a family of iterative methods to compute the approximate inverse of square matrix are presented. The theoretical proofs and numerical experiments show that these iterative methods are very effective. And, more importantly, these methods can be used to compute the inner inverse and their convergence proofs are given by fundamental matrix tools.     

On quantitative versions of theorems due to F.E. Browder and R. Wittmann

This paper is another case study in the program of logically analyzing proofs to extract new (typically effective) information (‘proof mining’). We extract explicit uniform rates of metastability (in the sense of T. Tao) from two ineffective proofs of a classical theorem of F.E. Browder on the convergence of approximants to fixed points of nonexpansive mappings as well as from a proof of a theorem of R. Wittmann which can be viewed as a nonlinear extension of the mean ergodic theorem. The first rate is extracted from Browder's original proof that is based on an application of weak sequential compactness (in addition to a projection argument). Wittmann's proof follows a similar line of reasoning and we adapt our analysis of Browder's proof to get a quantitative version of Wittmann's theorem as well. In both cases one also obtains totally elementary proofs (even for the strengthened quantitative forms) of these theorems that neither use weak compactness nor the existence of projections anymore. In this way, the present article also discusses general features of extracting effective information from proofs based on weak compactness. We then extract another rate of metastability (of similar nature) from an alternative proof of Browder's theorem essentially due to Halpern that already avoids any use of weak compactness. The paper is concluded by general remarks concerning the logical analysis of proofs based on weak compactness as well as a quantitative form of the so-called demiclosedness principle. In a subsequent paper these results will be utilized in a quantitative analysis of Baillon's nonlinear ergodic theorem.     

A Quasi-Neutral Limit in a Hydrodynamic Model for Charged Fluids

 The combined quasineutral and relaxation time limit for a bipolar hydrodynamic model is considered. The resulting limit problem is a nonlinear diffusion equation describing a neutral fluid. We make use of various entropy functions and the related entropy productions in order to obtain strong enough uniform bounds. The necessary strong convergence of the densities is obtained by using a generalized version of the “div-curl” Lemma and monotonicity methods.     

Convergence of approximate solutions for waves in a stratified fluid with approximated density distribution

Equations defining in linear approximation the wave motions in an arbitrarily stratified fluid are derived. Investigation of convergence of wave solutions with approximations ρ_n (z) of the mean density profile ρ_o (z) shows that uniform convergence of ρ_n (z) to ρ_o (z) is the sufficient condition of convergence of wave equation solutions. The convergence of solutions is uniform on sets of upper bound wave numbers and lower bound phase velocities of waves. Examples that show that when the continuous function ρ_o (z) is approximated by step-wide functions ρ_n (z) the convergence of solutions for internal waves is not uniform over the whole set of admissible wave numbers and phase velocities of waves.     

Incompressible reacting flows

We establish steady-state convergence results for a system of
reaction-convection-diffusion equations that model in particular combustion phenomena in the presence of nontrivial incompressible fluid motion. Despite the presence of the convection terms, we find that the asymptotic behavior of the system is identical to the case we have previously considered in which the velocity field was set equal to zero. In particular we are again able to establish the convergence of solutions to steady-states and to explicitly calculate the steady-states from the initial and boundary data. Key to our analysis is the establishment of high-order uniform bounds on the temperature and mass fraction components, a process significantly complicated by the presence of the convection terms.