Convergence of Polynomial Level Sets

In this paper we give a characterization of pointwise and uniform convergence of sequences of homogeneous polynomials on a Banach space by means of the convergence of their level sets. Results are obtained both in the real and the complex cases, as well as some generalizations to the nonhomogeneous case and to holomorphic functions in the complex case. Kuratowski convergence of closed sets is used in order to characterize pointwise convergence. We require uniform convergence of the distance function to get uniform convergence of the sequence of polynomials.

     

Further arguments for slice convergence in nonreflexive spaces

It is shown that no notion of set convergence at least as strong as Wijsman convergence but not as strong as slice convergence can be preserved in superspaces. We also show that such intermediate notions of convergence do not always admit representations analogous to those given by Attouch and Beer for slice convergence, and provide a valid reformulation. Some connections between bornologies and the relationships between certain gap convergences for nonconvex sets are also observed.Research supported in part by an NSERC research grant and by the Shrum endowment.NSERC postdoctoral fellow.     

f-Konvergenz undf-Stetigkeit von Operatorenfolgen auf metrischen Räumen

Summary Equicontinuity of the operators of a given sequence of nonlinear operators is one of the necessary and sufficient conditions for continuous convergence of this sequence. This lemma, which is due to Rinow, gives a generalization of the Banach-Steinhaus-theorem. Hence it led to some generalizations of the Lax-Richt-myer-theory of difference approximations for initial value problems. The equicontinuity in this case correspondends to numerical stability. But often continuous convergence is a too strong demand in the theory of nonlinear numerical problems (for instance in the case of difference schemes for quasilinear partial differential equations), whereas a restriction to only pointwise convergence possibly leads to numerical instability. Therefore in this paper a set of definitions of convergence is considered lying between pointwise and continuous convergence. Sorts of continuity are described which are as characteristic for these kinds of convergence as equicontinuity for continuous convergence. As an numerical application we study the connection between the solution-depending stability and the sensitiveness to perturbations of difference schemes for quasilinear initial value problems.     

Affine invariant local convergence theorems for inexact Newton-like methods

Affine invariant sufficient conditions are given for two local convergence theorems involving inexact Newton-like methods. The first uses conditions on the first Fréchet-derivative whereas the second theorem employs hypotheses on the second. Radius of convergence as well as rate of convergence results are derived. Results involving superlinear convergence and known to be true for inexact Newton methods are extended here. Moreover, we show that under hypotheses on the second Fréchet-derivative our radius of convergence is larger than the corresponding one in [10]. This allows a wider choice for the initial guess. A numerical example is also provided to show that our radius of convergence is larger than the one in [10].     

L-模糊化拓扑空间中的Moore-Smith收敛理论

This paper presents a definition of L-fuzzifying nets and the related L-fuzzifying generalized convergence spaces. The Moore-Smith convergence is established in L-fuzzifying topology. It is shown that the category of L-fuzzifying generalized convergence spaces is a cartesianclosed topological category which embeds the category of L-fuzzifying topological spaces as a reflective subcategory.     

HOMOCENTRIC CONVERGENCE BALL OF THE SECANT METHOD

A local convergence theorem and five semi-local convergence theorems of the secant method are listed in this paper.For every convergence theorem,a convergence ball is respectively introduced,where the hypothesis conditions of the corresponding theorem can be satisfied.Since all of these convergence balls have the same center x~*,they can be viewed as a homocentric ball. Convergence theorems are sorted by the different sizes of various radii of this homocentric ball, and the sorted sequence represents the degree of weakness on the conditions of convergence theorems.     

Approximate Predetermined Convergence Properties of the Gibbs Sampler

This article aims to provide a method for approximately predetermining convergence properties of the Gibbs sampler. This is to be done by first finding an approximate rate of convergence for a normal approximation of the target distribution. The rates of convergence for different implementation strategies of the Gibbs sampler are compared to find the best one. In general, the limiting convergence properties of the Gibbs sampler on a sequence of target distributions (approaching a limit) are not the same as the convergence properties of the Gibbs sampler on the limiting target distribution. Theoretical results are given in this article to justify that under conditions, the convergence properties of the Gibbs sampler can be approximated as well. A number of practical examples are given for illustration.     

Convergence analysis of finite volume scheme for nonlinear aggregation population balance equation

In this work, we introduce the convergence analysis of the recently developed finite volume scheme to solve a pure aggregation population balance equation that is of substantial interest in many areas such as chemical engineering, aerosol physics, astrophysics, polymer science, pharmaceutical sciences, and mathematical biology. The notion of the finite volume scheme is to conserve total mass of the particles in the system by introducing weight in the formulation. The consistency of the finite volume scheme is also analyzed thoroughly as it is an influential factor. The convergence study of the numerical scheme shows second order convergence on uniform, nonuniform smooth (geometric) as well as on locally uniform meshes independent of the aggregation kernel. Moreover, the first‐order convergence is shown when the finite volume scheme is implemented on oscillatory and random meshes. In order to check the accuracy, the numerical experimental order of convergence is also computed for the physically relevant as well as analytically tractable kernels and validated against its analytical results.     

Weak convergence of inner superposition operators

The equivalence of the weak (pointwise) and strong convergence of a sequence of inner superposition operators is proved as well as the criteria for such convergence are provided. Besides, the problems of continuous weak convergence of such operators and of representation of a limit operator are studied.

     

不动点迭代法的一点注记

对于迭代函数不满足收敛定理假定条件的情况 ,提出了一种简单方法 .此方法对于迭代函数满足收敛定理假定条件的情况 ,可以加速序列收敛 .最后给出了实例和程序 .     

Almost sure convergence of branching processes

A general theorem concerning the almost sure convergence of some nonhomogeneous Markov chains, whose conditional distributions satisfy a certain convergence condition, is given. This result applied to branching processes with infinite mean yields almost sure convergence for a large class of processes converging in distribution, as well as a characterization of the limiting distribution function.     

Stepsize analysis for descent methods

The convergence rates of descent methods with different stepsize rules are compared. Among the stepsize rules considered are: constant stepsize, exact minimization along a line, Goldstein-Armijo rules, and stepsize equal to that which yields the minimum of certain interpolatory polynomials. One of the major results shown is that the rate of convergence of descent methods with the Goldstein-Armijo stepsize rules can be made as close as desired to the rate of convergence of methods that require exact minimization along a line. Also, a descent algorithm that combines a Goldstein-Armijo stepsize rule with a secant-type step is presented. It is shown that this algorithm has a convergence rate equal to the convergence of descent methods that require exact minimization along a line and that, eventually (i.e., near the minimum), it does not require a search to determine an acceptable stepsize.     

A convergence analysis of the iterative aggregation method with one parameter

A method for accelerating linear iterations in a Banach space is studied as a linear iterative method in an augmented space, and sufficient conditions for convergence are derived in the general case and in ordered Banach spaces. An acceleration of convergence takes place if an auxiliary functional is chosen sufficiently close to a dual eigenvector associated with a dominant simple eigenvalue of the iteration operator; in this case, the influence of this eigenvalue on the asymptotic rate of convergence is eliminated. Quantitative estimates and bounds on convergence are given.     

多场址问题的信赖域算法

多场址问题是一类重要的不可微凸规划问题，国内外已有许多学者对其进行研究，并提出了一 算法。但如文「２」中所述，大多数算法或无收敛收保证，或在较强的条件下才保证收敛，本文提出一类解多场址问题的信赖域算法，并在极弱的条件下证明该类算法的全局收敛性。     

A geometric theory for preconditioned inverse iteration IV: On the fastest convergence cases

In order to compute the smallest eigenvalue together with an eigenfunction of a self-adjoint elliptic partial differential operator one can use the preconditioned inverse iteration scheme, also called the preconditioned gradient iteration. For this iterative eigensolver estimates on the poorest convergence have been published by several authors. In this paper estimates on the fastest possible convergence are derived. To this end the convergence problem is reformulated as a two-level constrained optimization problem for the Rayleigh quotient. The new convergence estimates reveal a wide range between the fastest possible and the slowest convergence.     

m步非线性多重分裂Newton法的收敛性

本文讨论了多重分裂算法在求解一类非线性方程组的全局收敛性和单侧收敛性．当用研步Newton法来代替求得每个非线性多重分裂子问题的近似解时，同样给出相应收敛性结论．数值算例证实了算法的有效性．     

Weak Convergence of n-Particle Systems Using Bilinear Forms

The paper is concerned with the weak convergence of n-particle processes to deterministic stationary paths as ${n \rightarrow \infty}$ . A Mosco type convergence of a class of bilinear forms is introduced. The Mosco type convergence of bilinear forms results in a certain convergence of the resolvents of the n-particle systems. Based on this convergence a criterion in order to verify weak convergence of invariant measures is established. Under additional conditions weak convergence of stationary n-particle processes to stationary deterministic paths is proved. The method is applied to the particle approximation of a Ginzburg-Landau type diffusion. The present paper is in close relation to the paper [9]. Different definitions of bilinear forms and versions of Mosco type convergence are introduced. Both papers demonstrate that the choice of the form and the type of convergence relates to the particular particle system.     

一类并行数值Schwarz格式的收敛性及比较定理

本文对［１］给出的一类求解椭圆型偏微分方程的并行数值Ｓｃｈｗａｒｚ格式作了进一步的分析，给出了一个新的收敛定理，根据该收敛定理，可以把文［１］中的主要收敛快慢进行了比较，得出了一些比较结果。     

OPTAC: a portable software package for analyzing and comparing optimization methods by visualization

A software package for analyzing and comparing optimization methods is presented. This package displays, using different colors, the regions of convergence to the minima of a given function for various optimization methods. It displays also the rate of their convergence as well as the regions of divergence of these methods. Moreover, this package gives quantitative information regarding the total convergence area in a specific domain for various minima.     

Computing the region of convergence for power series in many real variables: A ratio-like test

We give an elementary proof that the region of convergence for a power series in many real variables is a star-convex domain but not, in general, a convex domain. In doing so, we deduce a natural higher-dimensional analog of the so-called ratio test from univariate power series. From the constructive proof of this result, we arrive at a method to approximate the region of convergence up to a desired accuracy. While most results in the literature are for rather specialized classes of multivariate power series, the method devised here is general. As far as applications are concerned, note that while theorems such as the Cauchy-Kowalevski theorem (and its generalizations to many variables) grant the existence of a region of convergence for a multivariate Taylor series to certain PDEs under appropriate restrictions, they do not give the actual region of convergence. The determination of the maximal region of convergence for such a series solution to a PDE is one application of our result.