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Showing papers on "Asymptotology published in 2012"


Book
14 Dec 2012
TL;DR: In this article, the authors provide an account of the theory which combines classical sensitivity analysis in shape optimization with asymptotic analysis by means of compound expansion for elliptic boundary value problems.
Abstract: The topological derivative is defined as the first term (correction) of the asymptotic expansion of a given shape functional with respect to a small parameter that measures the size of singular domain perturbations, such as holes, inclusions, defects, source-terms and cracks. Over the last decade, topological asymptotic analysis has become a broad, rich and fascinating research area from both theoretical and numerical standpoints. It has applications in many different fields such as shape and topology optimization, inverse problems, imaging processing and mechanical modeling including synthesis and/or optimal design of microstructures, fracture mechanics sensitivity analysis and damage evolution modeling. Since there is no monograph on the subject at present, the authors provide here the first account of the theory which combines classical sensitivity analysis in shape optimization with asymptotic analysis by means of compound asymptotic expansions for elliptic boundary value problems. This book is intended for researchers and graduate students in applied mathematics and computational mechanics interested in any aspect of topological asymptotic analysis. In particular, it can be adopted as a textbook in advanced courses on the subject and shall be useful for readers interested on the mathematical aspects of topological asymptotic analysis as well as on applications of topological derivatives in computation mechanics.

253 citations


Book
10 Mar 2012
TL;DR: In this article, the analysis of the solutions of a system of Ordinary Differential Equations with turning points is studied. But the authors focus on the complex case and do not consider the real case.
Abstract: 1. The Analytic Theory of Differential Equations.- 1. Analyticity of the Solutions of a System of Ordinary Differential Equations.- 2. Regular Singular Points.- 3. Irregular Singular Points.- 2. Second-Order Equations on the Real Line.- 1. Transformations of Second-Order Equations.- 2. WKB-Bounds.- 3. Asymptotic Behaviour of Solutions of a Second-Order Equation for Large Values of the Parameter.- 4. Systems of Two Equations Containing a Large Parameter.- 5. Systems of Equations Close to Diagonal Form.- 6. Asymptotic Behaviour of the Solutions for Large Values of the Argument.- 7. Dual Asymptotic Behaviour.- 8. Counterexamples.- 9. Roots of Constant Multiplicity.- 10. Problems on Eigenvalues.- 11. A Problem on Scattering.- 3. Second-Order Equations in the Complex Plane.- 1. Stokes Lines and the Domains Bounded by them.- 2. WKB-Bounds in the Complex Plane.- 3. Equations with Polynomial Coefficients. Asymptotic Behaviour of a Solution in the Large.- 4. Equations with Entire or Meromorphic Coefficients.- 5. Asymptotic Behaviour of the Eigenvalues of the Operator -d2 / dx2 + ?2q(x). Self-Adjoint Problems.- 6. Asymptotic Behaviour of the Discrete Spectrum of the Operator -y? + ?2q(x)y. Non-Self-Adjoint Problems.- 7. The Eigenvalue Problem with Regular Singular Points.- 8. Quasiclassical Approximation in Scattering Problems.- 9. Sturm-Liouville Equations with Periodic Potential.- 4. Second-Order Equations with Turning Points.- 1. Simple Turning Points. The Real Case.- 2. A Simple Turning Point. The Complex Case.- 3. Some Standard Equations.- 4. Multiple and Fractional Turning Points.- 5. The Fusion of a Turning Point and Regular Singular Point.- 6. Multiple Turning Points. The Complex Case.- 7. Two Close Turning Points.- 8. Fusion of Several Turning Points.- 5. nth-Order Equations and Systems.- 1. Equations and Systems on a Finite Interval.- 2. Systems of Equations on a Finite Interval.- 3. Equations on an Infinite Interval.- 4. Systems of Equations on an Infinite Interval.- 5. Equations and Systems in the Complex Plane.- 6. Turning Points.- 7. A Problem on Scattering, Adiabatic Invariants and a Problem on Eigenvalues.- 8. Examples.- References.

173 citations


Posted Content
25 Jun 2012
TL;DR: This work gives a new representation of Pickands' constants, which arise in the study of extremes for a variety of Gaussian processes, and resolves the long-standing problem of devising a reliable algorithm for estimating these constants.
Abstract: This work gives a new representation of Pickands' constants, which arise in the study of extremes for a variety of Gaussian processes. Using this representation, we resolve the long-standing problem of devising a reliable algorithm for estimating these constants. A detailed error analysis illustrates the strength of our approach. (Joint with Ton Dieker, Georgia Institute of Technology)

57 citations


Journal ArticleDOI
TL;DR: It is shown that using values of the integrand at certain complex points close to the critical points can actually yield a higher asymptotic order approximation to the integral.
Abstract: Asymptotic expansions for oscillatory integrals typically depend on the values and derivatives of the integrand at a small number of critical points. We show that using values of the integrand at certain complex points close to the critical points can actually yield a higher asymptotic order approximation to the integral. This superinterpolation property has interesting ramifications for numerical methods based on exploiting asymptotic behaviour. The asymptotic convergence rates of Filon-type methods can be doubled at no additional cost. Numerical steepest descent methods already exhibit this high asymptotic order, but their analyticity requirements can be significantly relaxed. The method can be applied to general oscillators with stationary points as well, through a simple change of variables.

39 citations


Journal ArticleDOI
TL;DR: In this paper, an elliptic self-adjoint first order pseudodifferential operator acting on columns of complex-valued half-densities over a connected compact manifold without boundary is considered.
Abstract: We consider an elliptic self-adjoint first order pseudodifferential operator acting on columns of complex-valued half-densities over a connected compact manifold without boundary. The eigenvalues of the principal symbol are assumed to be simple but no assumptions are made on their sign, so the operator is not necessarily semi-bounded. We study the following objects: the propagator (time-dependent operator which solves the Cauchy problem for the dynamic equation), the spectral function (sum of squares of Euclidean norms of eigenfunctions evaluated at a given point of the manifold, with summation carried out over all eigenvalues between zero and a positive lambda) and the counting function (number of eigenvalues between zero and a positive lambda). We derive explicit two-term asymptotic formulae for all three. For the propagator "asymptotic" is understood as asymptotic in terms of smoothness, whereas for the spectral and counting functions "asymptotic" is understood as asymptotic with respect to lambda tending to plus infinity.

21 citations


Journal ArticleDOI
TL;DR: A Chapman-Enskog-type asymptotic expansion is introduced and an effective system of equations describing the late-time/stiff relaxation singular limit is derived, and a new finite volume discretization is proposed which allows for a discrete version of the same effective asymPTotic system.
Abstract: We investigate the late-time asymptotic behavior of solutions to nonlinear hyperbolic systems of conservation laws containing stiff relaxation terms. First, we introduce a Chapman-Enskog-type asymptotic expansion and derive an effective system of equations describing the late-time/stiff relaxation singular limit. The structure of this new system is discussed and the role of a mathematical entropy is emphasized. Second, we propose a new finite volume discretization which, in late-time asymptotics, allows us to recover a discrete version of the same effective asymptotic system. This is achieved provided we suitably discretize the relaxation term in a way that depends on a matrix-valued free-parameter, chosen so that the desired asymptotic behavior is obtained. Our results are illustrated with several models of interest in continuum physics, and numerical experiments demonstrate the relevance of the proposed theory and numerical strategy.

21 citations



Journal ArticleDOI
TL;DR: In this article, the coefficients of the asymptotic expansions of the kernel of the Berezin-Toeplitz quantization were computed using the stationary phase formula of Melin-Sjostrand.
Abstract: We give new methods for computing the coefficients of the asymptotic expansions of the kernel of Berezin–Toeplitz quantization obtained recently by Ma–Marinescu, and of the composition of two Berezin–Toeplitz quantizations. Our main tool is the stationary phase formula of Melin–Sjostrand.

18 citations


Journal ArticleDOI
TL;DR: In this paper, the authors investigated boundary value and spectral problems for an elliptic differential equation with rapidly oscillating coefficients in a thin perforated region with rapidly changing thickness and described algorithms for solutions of such problems in thin regions with different limiting dimensions.
Abstract: Boundary value and spectral problems for an elliptic differential equation with rapidly oscillating coefficients in a thin perforated region with rapidly changing thickness are investigated. Descriptions of asymptotic algorithms for solutions of such problems in thin regions with different limiting dimensions are combined. For a mixed inhomogeneous boundary value problem a corrector is constructed and an asymptotic estimate in the corresponding Sobolev space is established. Asymptotic bounds for eigenvalues and eigenfunctions of the Neumann spectral problems are also found. Full asymptotic expansions for the eigenvalues and eigenfunctions are constructed under certain symmetry assumptions about the structure of the thin perforated region and the coefficients of the equations. Bibliography: 21 titles.

14 citations


Journal ArticleDOI
TL;DR: In this article, an asymptotic analysis of the Boltzmann equations (Riccati differential equations) that describe the physics of thermal dark-matter-relic abundances is presented.
Abstract: This paper presents an asymptotic analysis of the Boltzmann equations (Riccati differential equations) that describe the physics of thermal dark-matter-relic abundances. Two different asymptotic techniques are used, boundary-layer theory, which makes use of asymptotic matching, and the delta expansion, which is a powerful technique for solving nonlinear differential equations. Two different Boltzmann equations are considered. The first is derived from general relativistic considerations and the second arises in dilatonic string cosmology. The global asymptotic analysis presented here is used to find the long-time behavior of the solutions to these equations. In the first case the nature of the so-called freeze-out region and the post-freeze-out behavior is explored. In the second case the effect of the dilaton on cold dark-matter abundances is calculated and it is shown that there is a large-time power-law fall off of the dark-matter abundance. Corrections to the power-law behavior are also calculated.

14 citations


Journal ArticleDOI
TL;DR: In this paper, the authors proposed a new model to obtain asymptotic distributions near zero and compute the limiting distribution for the Ornstein-Uhlenbeck process driven by a fractional Brownian motion.
Abstract: Consider an Ornstein-Uhlenbeck process driven by a fractional Brownian motion. It is an interesting problem to find criteria for whether the process is stable or has a unit root, given a finite sample of observations. Recently, various asymptotic distributions for estimators of the drift parameter have been developed. We illustrate through computer simulations and through a Stein's bound that these asymptotic distributions are inadequate approximations of the finite-sample distribution for moderate values of the drift and the sample size. We propose a new model to obtain asymptotic distributions near zero and compute the limiting distribution. We show applications to regression analysis and obtain hypothesis tests and their asymptotic power.

Journal ArticleDOI
TL;DR: In this article, the authors present the asymptotic properties of maximum quasi-likelihood estimators (MQLEs) in generalized linear models with adaptive designs under some mild regular conditions.
Abstract: In this paper, we present the asymptotic properties of maximum quasi-likelihood estimators (MQLEs) in generalized linear models with adaptive designs under some mild regular conditions. The existence of MQLEs in quasi-likelihood equation is discussed. The rate of convergence and asymptotic normality of MQLEs are also established. The results are illustrated by Monte-Carlo simulations.

Journal ArticleDOI
TL;DR: In this paper, the authors consider linear Volterra-renewal integral equations (VIEs) whose solutions depend on a space variable, via a map transformation, and investigate the asymptotic properties of the solutions.

Journal ArticleDOI
TL;DR: A framework that extends the look-ahead estimator to a broader range of applications is studied and a general asymptotic theory for the estimator is provided, where both L1 consistency and L2 asymPTotic normality are established.
Abstract: The look-ahead estimator is used to compute densities associated with Markov processes via simulation We study a framework that extends the look-ahead estimator to a broader range of applications We provide a general asymptotic theory for the estimator, where both L1 consistency and L2 asymptotic normality are established The L2 asymptotic normality implies √n convergence rates for L2 deviation

Journal ArticleDOI
TL;DR: In this paper, the authors studied the nonsteady Stokes flow in a thin tube structure composed by two thin rectangles with lateral elastic boundaries which are connected by a domain with rigid boundaries.
Abstract: We study the nonsteady Stokes flow in a thin tube structure composed by two thin rectangles with lateral elastic boundaries which are connected by a domain with rigid boundaries. After a variational approach of the problem which gives us existence, uniqueness, regularity results, and some a priori estimates, we construct an asymptotic solution. The existence of a junction region between the two rectangles imposes to consider, as part of the asymptotic solution, some boundary layer correctors that correspond to this region. We present and solve the problems for all the terms of the asymptotic expansion. For two different cases, we describe the order of steps of the algorithm of solving the problem and we construct the main term of the asymptotic expansion. By means of the a priori estimates, we justify our asymptotic construction, by obtaining a small error between the exact and the asymptotic solutions.

Journal ArticleDOI
TL;DR: In this article, a class of second order nonlinear functional differential equations is considered and the generalized Riccati transformation and integral averaging technique are used to obtain new oscillation criteria and asymptotic behavior.
Abstract: This paper is concerned with oscillatory and asymptotic behavior of solutions of a class of second order nonlinear functional differential equations. By using the generalized Riccati transformation and the integral averaging technique, new oscillation criteria and asymptotic behavior are obtained for all solutions of the equation. Our results generalize and improve some known theorems.

Book
26 Jan 2012
TL;DR: In this article, the authors discuss the role of Computation and its application in various areas of computer science, including numerical methods, asymptotics and perturbation theory, and system decomposition.
Abstract: Role of Computation.- Outline of Numerical.- Methods.- Asymptotics and Perturbation Theory.- Asymptotology and System Decomposition.- Multiple Scales in Computation.- Examples and Applications.


Journal ArticleDOI
TL;DR: Asymptotic behavior for solutions of neutral difference equations dealing with mechanical vibration problems and other related topics, and some theorems have been obtained, are studied in this article, where the authors consider the problem of finding the optimal solution of a neutral difference equation.
Abstract: Asymptotic behavior is studied for solutions of neutral difference equations dealing with mechanical vibration problems and other related topics, and some theorems have been obtained.

Book ChapterDOI
01 Jan 2012
TL;DR: In this article, several asymptotic methods have been developed for the derivation of the evolution equations which describe how some dynamical variables evolve in time and space, which are not integrable by analytic methods.
Abstract: Many physical systems involving nonlinear wave propagation include the effects of dispersion, dissipation, and/or the inhomogeneous property of the medium. The governing equations are usually derived from conservation laws. In simple cases, these equations are hyperbolic. However, in general, the physical processes involved are so complex that the governing equations are very complicated, and hence, are not integrable by analytic methods. So, special attention is given to seeking mathematical methods which lead to a less complicated problem, yet retain all of the important physical features. In recent years, several asymptotic methods have been developed for the derivation of the evolution equations which describe how some dynamical variables evolve in time and space.


Dissertation
14 Mar 2012
TL;DR: In this paper, the application of perturbation expansion techniques for the solution perturbed problems, precisely differential equations, is addressed and the results are compared with the exact solution of the original problem.
Abstract: The main purpose of this thesis is to address the application of perturbation expansion techniques for the solution perturbed problems, precisely differential equations. When a large or small parameter ‘"’ known as the perturbation parameter occurs in a mathematical model, then the model problem is known as a perturbed problem. Asymptotic expansion technique is a method to get the approximate solution using asymptotic series for model perturbed problems. The asymptotic series may not and often do not converge but in a truncated form of only two or three terms, provide a useful approximation to the original problem. Though the perturbed differential equations can be solved numerically by using various numerical schemes, but the asymptotic techniques provide an awareness of the solution before one compute the numerical solution. Here, perturbation expansion for some model algebraic and differential equations are considered and the results are compared with the exact solution.

Dissertation
01 Jan 2012
Abstract: An Asymptotic Minimax Analysis of Nonlocal Means on Edges


Journal ArticleDOI
TL;DR: In this article, the authors dealt with the existence, uniqueness and asymptotic behaviours of the solutions of equations governing a two-dimensional axi-symmetric flow past a right-angled wedge.
Abstract: This paper deals with the existence, uniqueness and asymptotic behaviours, as the similarity variable tends to infinity, of the solutions of equations governing a two-dimensional axi-symmetric flow past a right-angled wedge. The results pertaining to the existence and uniqueness of the solutions are based on topological arguments, whereas the results on the asymptotic behaviour are based on the asymptotic integration of second order linear differential equations.