Développement asymptotique | EPFL Graph Search

En mathématiques, un développement asymptotique d'une fonction f donnée dans un voisinage fixé est une somme finie de fonctions de référence qui donne une bonne approximation du comportement de la fonction f dans le voisinage considéré. Le concept de développement asymptotique a été introduit par Poincaré à propos de l'étude du problème à N corps de la mécanique céleste par la théorie des perturbations. La somme étant finie, la question de la convergence ne se pose pas. On parle parfois par abus de langage de « série asymptotique » pour une somme comprenant une infinité de termes. Cette somme infinie est le plus souvent formelle, car la série est en général divergente. Analyse asymptotique : comportement équivalent Comparaison asymptotique Introduction L'analyse asymptotique est une méthode d'analyse qui permet de classer les comportements de fonctions dans un voisinage donné en se concentrant sur certaines « tendances caractéristiques ». On l'exprime en généra

Effective models and numerical homogenization methods for long time wave propagation in heterogeneous media

Timothée Noé Pouchon

Modeling wave propagation in highly heterogeneous media is of prime importance in engineering applications of diverse nature such as seismic inversion, medical imaging or the design of composite materials. The numerical approximation of such multiscale physical models is a mathematical challenge. Indeed, to reach an acceptable accuracy, standard numerical methods require the discretization of the whole medium at the microscopic scale, which leads to a prohibitive computational cost. Homogenization theory ensures the existence of a homogenized wave equation, obtained from the original problem by a limiting process. As this equation does not depend on the microscopic scale, it is a good target for numerical methods. Unfortunately, for general media, the homogenized equation may not be unique and no formulas are available for its effective data. %Diverse numerical strategies have been developed to approximate a homogenized solution. Nevertheless, such formulas are known for media described by a locally periodic tensor. In that case, or more generally for problems with scale separation, methods such as the finite element heterogeneous multiscale method (FE-HMM) are proved to efficiently approximate the homogenized solution. For wave propagation in heterogeneous media, however, it is known that at large timescales the homogenized solution fails to describe the dispersive behavior of the original wave. Hence, a new equation that captures this dispersion is needed. In this thesis, we study such effective equations for long time wave propagation in heterogeneous media. The first result that we present holds in periodic media. Using the technique of asymptotic expansion, we obtain the characterization of a whole family of equations that describes the long time dispersive effects of the oscillating wave. The validity of our derivation is ensured by rigorous a priori error estimates. We also derive a numerical procedure for the computation of the tensors involved in the first order effective equations. This leads to a numerical homogenization method for long time wave propagation in periodic media. The second result that we present generalizes the procedure for deriving effective equations to arbitrary timescales. This generalization is also useful, for example, for the homogenization of the wave equation with high frequency initial data. We also provide a numerical procedure allowing to compute effective tensors of arbitrary order. The third result is the generalization of the family of first order effective equations from periodic to locally periodic media. A rigorous a priori error analysis is also derived in this situation. This constitutes the first analysis of effective models for the long time approximation of the wave equation in locally periodic media. In a second part of the thesis, we derive numerical homogenization methods for the long time approximation of the wave equation in locally periodic media. In one dimension, we analyze a modification of the FE-HMM called the FE-HMM-L. In higher dimensions, we design a spectral homogenization method. For both methods, we prove error estimates valid for large timescales and in arbitrarily large spatial domains. In particular, we show that these numerical homogenization methods converge to effective solutions that approximate the highly oscillatory wave equation over long time.

EPFL2017

A Covariance Formula For Topological Events Of Smooth Gaussian Fields

Alejandro Rivera

We derive a covariance formula for the class of 'topological events' of smooth Gaussian fields on manifolds; these are events that depend only on the topology of the level sets of the field, for example, (i) crossing events for level or excursion sets, (ii) events measurable with respect to the number of connected components of level or excursion sets of a given diffeomorphism class and (iii) persistence events. As an application of the covariance formula, we derive strong mixing bounds for topological events, as well as lower concentration inequalities for additive topological functionals (e.g., the number of connected components) of the level sets that satisfy a law of large numbers. The covariance formula also gives an alternate justification of the Harris criterion, which conjecturally describes the boundary of the percolation university class for level sets of stationary Gaussian fields. Our work is inspired by (Ann. Inst. Henri Poincare Probab. Stat. 55 (2019) 1679-1711), in which a correlation inequality was derived for certain topological events on the plane, as well as by (Asymptotic Methods in the Theory of Gaussian Processes and Fields (1996) Amer. Math. Soc.), in which a similar covariance formula was established for finite-dimensional Gaussian vectors.

INST MATHEMATICAL STATISTICS2020

Robust parameter estimation for the Ornstein-Uhlenbeck process

Sonja Rieder

In this paper, we derive elementary M- and optimally robust asymptotic linear (AL)-estimates for the parameters of an Ornstein-Uhlenbeck process. Simulation and estimation of the process are already well-studied, see Iacus (Simulation and inference for stochastic differential equations. Springer, New York, 2008). However, in order to protect against outliers and deviations from the ideal law the formulation of suitable neighborhood models and a corresponding robustification of the estimators are necessary. As a measure of robustness, we consider the maximum asymptotic mean square error (maxasyMSE), which is determined by the influence curve (IC) of AL estimates. The IC represents the standardized influence of an individual observation on the estimator given the past. In a first step, we extend the method of M-estimation from Huber (Robust statistics. Wiley, New York, 1981). In a second step, we apply the general theory based on local asymptotic normality, AL estimates, and shrinking neighborhoods due to Kohl et al. (Stat Methods Appl 19:333-354, 2010), Rieder (Robust asymptotic statistics. Springer, New York, 1994), Rieder (2003), and Staab (1984). This leads to optimally robust ICs whose graph exhibits surprising behavior. In the end, we discuss the estimator construction, i.e. the problem of constructing an estimator from the family of optimal ICs. Therefore we carry out in our context the One-Step construction dating back to LeCam (Asymptotic methods in statistical decision theory. Springer, New York, 1969) and compare it by means of simulations with MLE and M-estimator.

Springer-Verlag2012

Effective models and numerical homogenization methods for long time wave propagation in heterogeneous media

Timothée Noé Pouchon

EPFL2017