Showing papers on "Asymptotology published in 2005"
TL;DR: In this paper, the authors explore quadrature methods for highly oscillatory integrals, which approximate the integral as a linear combination of function values and derivatives, with coefficients that may depend on frequency.
Abstract: In this paper, we explore quadrature methods for highly oscillatory integrals. Generalizing the method of stationary phase, we expand such integrals into asymptotic series in inverse powers of the frequency. The outcome is two families of methods, one based on a truncation of the asymptotic series and the other extending an approach implicit in the work of [Filon (Filon 1928 Proc. R. Soc. Edinb. 49 , 38–47)][1]. Both kinds of methods approximate the integral as a linear combination of function values and derivatives, with coefficients that may depend on frequency. We determine asymptotic properties of these methods, proving, perhaps counterintuitively, that their performance drastically improves as frequency grows. The paper is accompanied by numerical results that demonstrate the potential of this set of ideas.
[1]: #ref-4
335 citations
TL;DR: In this article, the case of the steady-state Navier-Stokes equations for an incompressible fluid and a no-slip condition prescribed on the boundary of an arbitrary shaped obstacle is addressed.
Abstract: The aim of the topological asymptotic analysis is to provide an asymptotic expansion of a shape functional with respect to the size of a small inclusion inserted inside the domain. The main field of application is shape optimization. This paper addresses the case of the steady-state Navier-Stokes equations for an incompressible fluid and a no-slip condition prescribed on the boundary of an arbitrary shaped obstacle. The two and three dimensional cases are treated for several examples of cost functional and a numerical application is presented.
78 citations
Book•
18 Jul 2005
TL;DR: Asymptotic methods provide considerable physical insight and understanding of diffraction mechanisms and are very useful in the design of electromagnetic devices such as radar targets and antennas.
Abstract: Asymptotic methods provide considerable physical insight and understanding of diffraction mechanisms and are very useful in the design of electromagnetic devices such as radar targets and antennas. However, difficulties can arise when trying to solve problems using multipole and asymoptotic methods together, such as in radar cross section objects.
53 citations
Dissertation•
01 Jan 2005
TL;DR: In this article, a software package called RobASt is developed by means of the statistics software R. This software package is based on the asymptotic theory of robustness.
Abstract: In the framework of this dissertation a software package – the R bundle RobASt – by means of the statistics software R has been developed. It includes all robust procedures introduced throughout the thesis. The dissertation itself consists of five parts and starts with a brief motivation, which makes precise why robust statistics is necessary. After that a detailed summary in German and English is given. Part I provides a description of the asymptotic theory of robustness (Chapter 1) which forms the basis of this thesis. It is based on Chapters 4 and 5 of Rieder (1994). Chapter 2 provides supplements to the asymptotic theory of robustness which have proved necessary for this thesis. More precisely, it contains results about: properties of the optimally robust influence curves (ICs), how one should proceed in an optimal way if the neighborhood radius is unknown – as mostly in practice, and the construction of estimates by means of the one-step method. At the end of Chapter 2 convergence of robust models is introduced which is related to the concept of convergence of experiments of Le Cam. Part II deals with optimally robust estimators for some non-standard models in robust statistics. These models are covered by the R package ROptEst which makes use of S4 classes and methods and is part of the R bundle RobASt. More precisely, the binomial (Chapter 3) and Poisson (Chapter 4) model, the exponential scale and Gumbel location model (Chapter 5) as well as the Gamma model (Chapter 6) are investigated. In particular, the binomial and Poisson model are used to study convergence of robust models. Using exponential scale and Gumbel location one can show that there is a connection between certain scale and location models via a log-transformation which also holds for the corresponding optimally robust ICs. Finally, the Gamma model is used to demonstrate how differentiable parameter transformations can be estimated in an optimally robust way. In Part III robust regression with random regressor and unknown error scale (Chapter 7) is treated where it is distinguished between simultaneous and separate estimation. In both cases the optimally robust estimators as well as robust estimators for several narrower classes of M estimators are considered. All these estimators are implemented in the R packages ROptRegTS and RobRex which are part of the R bundle RobASt. Numerical comparisons for several regressor distributions show that the various suboptimal M estimators may have very small but also huge efficiency losses. A further comparison of these and several other well-known robust estimators in case of normal location and scale is made in Chapter 8. These location and scale estimators are implemented in the R package RobLox which is part of the R bundle RobASt. In Part IV (Chapter 9) robust adaptivity in terms of two asymptotic MSE problems is defined. Hence, adaptivity is no longer only a dichotomous criterion but can be evaluated quantitatively in terms of efficiency loss. The various regression and time series models considered…
34 citations
TL;DR: Based on the asymptotic distribution of quadratic form in Gaussian random variables, two formulations for the case when the number of random variables n→∞ is provided are proposed, which result in simple closed-form expressions for the probability of failure of an engineering structure.
Abstract: In the reliability analysis of safety critical complex engineering structures, a very large number of the system parameters can be considered as random variables. The difficulty in computing the failure probability using the classical first- and second-order reliability methods (FORM and SORM) increases rapidly with the number of variables or ‘dimension’. There are mainly two reasons behind this. The first is the increase in computational time with the increase in the number of random variables. In principle, this problem can be handled with superior computational tools. The second reason, which is perhaps more fundamental, is that there are some conceptual difficulties typically associated with high dimensions. This means that even when one manages to carry out the necessary computations, the application of existing FORM and SORM may still lead to incorrect results in high dimensions. This paper is aimed at addressing this issue. Based on the asymptotic distribution of quadratic form in Gaussian random variables, two formulations for the case when the number of random variables n/N is provided. The first is called ‘strict asymptotic formulation’ and the second is called ‘weak asymptotic formulation’. Both approximations result in simple closed-form expressions for the probability of failure of an engineering structure. The proposed asymptotic approximations are compared with existing approximations and Monte Carlo simulations using numerical examples.
21 citations
19 citations
01 Jan 2005
TL;DR: In this paper, the global almost sure asymptotic stability of the trivial solution of nonlinear stochastic difference equations with in-the-arithmetic-meansense monotone drift part and diffusive part driven by independent (but not necessarily identically distributed) random variables is proven under appropriate conditions in IR1.
Abstract: Global almost sure asymptotic stability of the trivial solution of some nonlinear stochastic difference equations with in-the-arithmetic-meansense monotone drift part and diffusive part driven by independent (but not necessarily identically distributed) random variables is proven under appropriate conditions in IR1. This result can be used to verify asymptotic stability of stochastic-numerical methods such as partially drift-implicit trapezoidal methods for nonlinear stochastic differential equations with variable step sizes.
13 citations
TL;DR: In this paper, the authors studied the asymptotic approximation of some functions defined by the q-Jackson integrals, for a fix q ∈]0, 1[.
Abstract: This paper aims to study the asymptotic approximation of some functions defined by the q-Jackson integrals, for a fix q ∈]0, 1[. For this purpose, we shall attempt to extend the classical methods b...
13 citations
TL;DR: The asymPTotic robustness of the normal theory asymptotic biases of the least-squares estimators of the parameters in covariance structures against the violation of normality is shown.
11 citations
TL;DR: In this paper, the authors studied the asymptotic expansion for solution of singularly perturbed equation for functional Markovian evolution in road traffic and found the view of regular and singular parts of solution.
Abstract: We study the asymptotic expansion for solution of singularly perturbed equation for functional of Markovian evolution in Rd. The view of regular and singular parts of solution is found.
TL;DR: The Hartman-Wintner theorem, the Harris-Lutz theorem, and the Eastham theorem were shown to be asymptotic results for perturbed linear difference systems in this article.
TL;DR: In this paper, the authors investigated the equivalence of the concepts of strong stability and t∞-similarity for nonlinear differential systems and showed that they are equivalent under strong stability.
TL;DR: In this article, the authors present a review of some results concerning delay estimation by continuous time observations of solutions of stochastic differential equations in two asymptotics: small noise limit and large samples limit.
Abstract: We present a review of some results concerning delay estimation by continuous time observations of solutions of stochastic differential equations in two asymptotics. The first one corresponds to small noise limit and the second to large samples limit. In both cases we describe the properties of the maximum likelihood estimator and Bayesian estimators with especial attention to asymptotic efficiency of the estimators. We show that the first asymptotic corresponds to regular problems of mathematical statistics and the second is close to non regular problems. In small noise asymptotics we give the next after the Gaussian term of asymptotic expansion of the maximum likelihood estimator.
Dissertation•
01 Jan 2005
TL;DR: In this article, the effect of a uniformly pulsating free-s tream with the leading edge of a body is considered, and the effect on transition is investigated by using the Parabolized Stability Equation (PSE).
Abstract: We consider the interaction of a uniformly pulsating free-s tream with the leading edge of a body, and consider its effect on transition. The free-st r am is assumed to be incompressible, high Reynolds number flow parallel to the chord of t he body, with a small, unsteady, perturbation of a single harmonic frequency. We p resent a method which calculates Tollmien-Schlichting (T-S) wave amplitudes downs tream of the leading edge, by a combination of an asymptotic receptivity approach in the l eading edge region and a numerical method which marches through the Orr-Sommerfeld egion. The asymptotic receptivity analysis produces a three deck eigenmode which , in its far downstream limiting form, produces an upstream initial condition for our num erical Parabolized Stability Equation (PSE). Downstream T-S wave amplitudes are calculated for the flat pl ate, and good comparisons are found with the Orr-Sommerfeld asymptotics ava ilable in this region. The importance of theO(Re− 1 2 ) term of the asymptotics is discussed, and, due to the complexity in calculating this term, we show the importance of n umerical methods in the Orr-Sommerfeld region to give accurate results. We also discuss the initial transients present for certain p rameter ranges, and show that their presence appears to be due to the existence of high er T-S modes in the initial upstream boundary condition. Extensions of the receptivity/PSE method to the parabola an d the Rankine body are considered, and a drop in T-S wave amplitudes at lower branch is observed for both bodies, as the nose radius increases. The only exception to this trend occurs for the Rankine body at very large Reynolds numbers, which are not accessible in experiments, where a double maximum of the T-S wave amplitude at lower branch is ob served. The extension of the receptivity/PSE method to experimenta lly realistic bodies is also considered, by using slender body theory to model the invisc id flow around a modified super ellipse to compare with numerical studies.
Journal Article•
Journal Article•
TL;DR: Asymptotic expansion of integrals has been studied extensively in the literature as discussed by the authors, where the authors focus on special functions defined through integrals and consider their approximation by means of asymptotics.
Abstract: Abstract. In the first part we discuss the concept of asymptotic expansion and its importance in applications. We focus our attention on special functions defined through integrals and consider their approximation by means of asymptotic expansions. We explain the general ideas of the theory of asymptotic expansions of integrals and describe two classical methods (Watson’s lemma and the saddle point method) and modern methods (distributional methods). In the second part we apply these ideas to approximate (in an asymptotic sense) polynomials of the Askey table in terms of simpler polynomials of the Askey table. We consider two different types of asymptotic expansions that have been recently developed: i) some parameter of the polynomial is large or ii) the degree (and perhaps the variable too) of the polynomial is large. For each situation we employ a different asymptotic method. In the first case we use the technique of “matching of the generating functions at the origin”. In the second one we employ a modified version of the saddle point method together with the theory of two-point Taylor expansions. In the first case the asymptotic expansion results in a finite sum of polynomials. In the second one the asymptotic expansion is a convergent infinite series of polynomials. We conclude the paper with information on other recent developments in the research on asymptotic expansions of integrals.
TL;DR: The concordance method of asymptotic expansions applied for constructing uniform expansion of singularly-perturbed partial differential equations and systems is presented in this paper, where the concordances are applied to construct uniform expansion.
Abstract: The concordance method of asymptotic expansions applied for constructing uniform asymptotic expansions of singularly-perturbed partial differential equations and systems is presented.
TL;DR: The general fourth Painleve transcendent is studied to find a group of its asymptotics and the corresponding monodromic data, and its existence and ”uniqueness“ are proved.
Abstract: We use the uniform asymptotics method proposed by A. P. Bassom et
al. (1998) to study the general fourth Painleve transcendent, find a
group of its asymptotics and the corresponding monodromic data,
and prove its existence and ”uniqueness.“
TL;DR: In this paper, an asymptotic analysis of elastic stress in an infinite solid whose boundary is subject to a rapid thermal load is presented, and the problem under consideration couples the wave equation and the heat equation.
Abstract: In this article, we present an asymptotic analysis of waves of elastic stress in an infinite solid whose boundary is subject to a rapid thermal load. The problem under consideration couples the wave equation and the heat equation, and the asymptotic approximation of the solution requires three-scaled variables. The asymptotic approximation is supplied with a rigorous remainder estimate and is illustrated numerically.
TL;DR: In this paper, a simple analytical approach for defining the asymptotic behavior of shell elements is proposed to define and analyse the deformation modes of selected benchmark problems suitable for the evaluation of the shell elements.
TL;DR: In this article, the existence and asymptotic behavior of positive solutions to a quasilinear elliptic problem with Neumann condition was investigated and shown to be true.
Abstract: In this paper we investigate the existence and asymptotic behavior of positive solutions to a quasilinear elliptic problem with Neumann condition.
TL;DR: The asymptotic theory of initial value problems for a class of nonlinear perturbed Klein-Gordon equations in two space dimensions is considered in this article, where the well-posedness of solutions of the initial value problem in twice continuous differentiable space was obtained according to the equivalent integral equation of the KG equation.
Abstract: The asymptotic theory of initial value problems for a class of nonlinear perturbed Klein-Gordon equations in two space dimensions is considered. Firstly, using the contraction mapping principle, combining some priori estimates and the convergence of Bessel function, the well-posedness of solutions of the initial value problem in twice continuous differentiable space was obtained according to the equivalent integral equation of initial value problem for the Klein-Gordon equations. Next, formal approximations of initial value problem was constructed by perturbation method and the asymptotic validity of the formal approximation is got. Finally, an application of the asymptotic theory was given, the asymptotic approximation degree of solutions for the initial value problem of a specific nonlinear Klein-Gordon equation was analyzed by using the asymptotic approximation theorem.
01 Jan 2005
TL;DR: In this paper, an interesting application of analytic iteration theory and classical complex analysis to determine some new (and old) results in asymptotic enumeration is presented. But the method treats functional equations of a particular form, which have a natural interpretation in terms of combinatorial generating functions.
Abstract: This work oers an interesting application of analytic iteration theory and classical complex analysis to determine some new (and old) results in asymptotic enumeration. The method treats functional equations of a particular form, which have a natural interpretation in terms of combinatorial generating functions. Partition lattice chains and Takeuchi numbers are among the applications of this method presented here.
Journal Article•
TL;DR: In this article, a class of semi-linear Robin problems is considered and the existence and asymptotic behavior of its solution are studied more carefully using stretched variables, the formal asymptonotic expansion of solution for the problem is constructed and the uniform validity of the solution is obtained by using the method of upper and lower solution.
Abstract: A class of semi-linear Robin problem is considered. Under appropriate assumptions, the existence and asymptotic behavior of its solution are studied more carefully. Using stretched variables, the formal asymptotic expansion of solution for the problem is constructed and the uniform validity of the solution is obtained by using the method of upper and lower solution.
TL;DR: In this paper, asymptotic equivalence between nonlinear difference systems and their variational systems was studied, and the equivalence was shown to hold for the variational variational system as well.
Abstract: We study asymptotic equivalence between nonlinear difference system and its variational system .