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Asymptotology

About: Asymptotology is a research topic. Over the lifetime, 1319 publications have been published within this topic receiving 35831 citations.


Papers
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Journal ArticleDOI
TL;DR: Maslov asymptotic ray theory is used to describe body waves in inhomogeneous media, but caustics, shadows, critical points, etc. have to be treated as special cases as discussed by the authors.
Abstract: Asymptotic ray theory is widely used to describe body waves in inhomogeneous media, but caustics, shadows, critical points, etc. have to be treated as special cases. Unfortunately, these singularities are often the points of greatest interest as they are caused by inhomogeneities in the model. Transform methods, e.g., the reflectivity method and WKBJ seismograms, are used to investigate waves at these singular points but are restricted to laterally homogeneous media. Maslov asymptotic theory uses the ideas of asymptotic ray theory and transform methods, combining the advantages—simplicity and generality—of both techniques. In this paper, Maslov asymptotic theory is developed for the computation of body-wave seismograms. The eikonal equation of asymptotic ray theory is equivalent to Hamilton9s canonical equations, and the ray trajectories can be considered in the phase space of position and slowness. Normal asymptotic ray theory gives the wave solution in the spatial domain. However, the asymptotic solution for other generalized coordinates in phase space can also be found. For instance, normal transform methods find the solution in the mixed domain where the horizontal slowness replaces the coordinate. Maslov asymptotic theory extends this idea to inhomogeneous media, and the asymptotic solution in a mixed domain (position and slowness) is obtained by a canonical transformation from the spatial domain. The method is useful as the singularities in the mixed and spatial domains are at different locations, and Maslov theory provides a uniform result, combining the solutions in the different domains. These transforms between the mixed-frequency and spatial-time domains are evaluated exactly using the WKBJ seismogram algorithm. This avoids the oscillatory integrals of asymptotic theory and stabilizes the numerical solution by providing the smoothed, discrete seismograms directly. The result is a rigorous but simple method for computing body-wave seismograms in inhomogeneous media. The theory is developed in outline, and numerical examples are included.

260 citations

Journal ArticleDOI
TL;DR: The approach is based on an extended delta method and appears to be particularly suitable for deriving asymptotics of the optimal value of stochastic programs.
Abstract: In this paper we discuss a general approach to studying asymptotic properties of statistical estimators in stochastic programming. The approach is based on an extended delta method and appears to be particularly suitable for deriving asymptotics of the optimal value of stochastic programs. Asymptotic analysis of the optimal value will be presented in detail. Asymptotic properties of the corresponding optimal solutions are briefly discussed.

260 citations

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

Journal ArticleDOI
TL;DR: In this article, the authors give a simple characterization of the uniform asymptotic stability of equations in terms of Lyapunov functions and a new sufficient condition is given for uniform stability.
Abstract: In this paper we give a simple characterization of the uniform asymptotic stability of equations $\dot x = - P(t)x$ where $P(t)$ is a bounded piecewise continuous symmetric positive semi-definite matrix. In the course of developing this characterization, a new and general sufficient condition is given for uniform asymptotic stability in terms of Lyapunov functions. The stability of this type of equation has come up in various control theory contexts (identification, optimization and filtering).

233 citations


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Performance
Metrics
No. of papers in the topic in previous years
YearPapers
20231
20222
20181
201725
201626
201526