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Author: Antoine Gloria
Publisher:
ISBN:
Category :
Languages : en
Pages : 148
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Book Description
Alors que la plupart des questions sont résolues en homogénéisation périodique des équations elliptiques, ceci n'est pas du tout le cas de l'homogénéisation stochastique, même dans le cadre élémentaire des équations linéaires. En effet, il n'est pas connu si l'équation dite du correcteur admet des solutions stationnaires. Dans la première partie de ce document nous abordons cette question et analysons des méthodes numériques de calcul des coefficients homogénéisés. En particulier nous montrons qu'il existe des correcteurs stationnaires à partir de la dimension trois d'espace pour des coefficients à longueur de corrélation finie. Cette théorie nous a permis d'analyser et donner des estimations de convergence optimales pour des algorithmes standard de calcul des coefficients homogénéisés. Dans la seconde partie de ce manuscrit, nous nous concentrons sur des questions plus qualitatives dans le cas d'équations non linéaires et en particulier d'élasticité non linéaire. Partant d'un modèle discret de chaînes de polymères en interaction, nous obtenons un modèle continu d'élasticité non linéaire par homogénéisation quand la taille typique des chaînes tend vers zéro. La densité d'énergie du matériau obtenu satisfait les propriétés qualitatives importantes des matériaux polymériques : indifférence matérielle, isotropie, ellipticité stricte. Nous insistons également sur les aspects numériques pour le calcul de cette densité d'énergie et son utilisation pratique.
Author: Antoine Gloria
Publisher:
ISBN:
Category :
Languages : en
Pages : 148
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Book Description
Alors que la plupart des questions sont résolues en homogénéisation périodique des équations elliptiques, ceci n'est pas du tout le cas de l'homogénéisation stochastique, même dans le cadre élémentaire des équations linéaires. En effet, il n'est pas connu si l'équation dite du correcteur admet des solutions stationnaires. Dans la première partie de ce document nous abordons cette question et analysons des méthodes numériques de calcul des coefficients homogénéisés. En particulier nous montrons qu'il existe des correcteurs stationnaires à partir de la dimension trois d'espace pour des coefficients à longueur de corrélation finie. Cette théorie nous a permis d'analyser et donner des estimations de convergence optimales pour des algorithmes standard de calcul des coefficients homogénéisés. Dans la seconde partie de ce manuscrit, nous nous concentrons sur des questions plus qualitatives dans le cas d'équations non linéaires et en particulier d'élasticité non linéaire. Partant d'un modèle discret de chaînes de polymères en interaction, nous obtenons un modèle continu d'élasticité non linéaire par homogénéisation quand la taille typique des chaînes tend vers zéro. La densité d'énergie du matériau obtenu satisfait les propriétés qualitatives importantes des matériaux polymériques : indifférence matérielle, isotropie, ellipticité stricte. Nous insistons également sur les aspects numériques pour le calcul de cette densité d'énergie et son utilisation pratique.
Author: Scott Armstrong
Publisher: Springer
ISBN: 3030155455
Category : Mathematics
Languages : en
Pages : 548
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Book Description
The focus of this book is the large-scale statistical behavior of solutions of divergence-form elliptic equations with random coefficients, which is closely related to the long-time asymptotics of reversible diffusions in random media and other basic models of statistical physics. Of particular interest is the quantification of the rate at which solutions converge to those of the limiting, homogenized equation in the regime of large scale separation, and the description of their fluctuations around this limit. This self-contained presentation gives a complete account of the essential ideas and fundamental results of this new theory of quantitative stochastic homogenization, including the latest research on the topic, and is supplemented with many new results. The book serves as an introduction to the subject for advanced graduate students and researchers working in partial differential equations, statistical physics, probability and related fields, as well as a comprehensive reference for experts in homogenization. Being the first text concerned primarily with stochastic (as opposed to periodic) homogenization and which focuses on quantitative results, its perspective and approach are entirely different from other books in the literature.
Author: Nicolas Clozeau
Publisher:
ISBN:
Category :
Languages : en
Pages : 0
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Book Description
This PhD thesis aims at a better understanding of the quantitative theory of the stochastic homogenization of elliptic equations and systems. In Chapter 2, we investigate the case of linear elliptic systems with random coefficients and long-range correlation. We adopt a parabolic approach and, by combining tools from probability (in the form of logarithmic Sobolev inequalities) and regularity theory, we optimally quantify the time decay of the parabolic semigroup with an explicit dependence on the correlation length. In Chapter 3, we turn to the analysis of nonlinear elliptic equations and systems with strongly monotone coefficients. Under a short-range correlation assumption, we prove optimal estimates on the correctors and the two-scale expansion, by developing new perturbative large-scale estimates for the linearized operator. In Chapter 4 and 5 we prove estimates on the bias in the Representative Volume Element method applied to linear elliptic equations. Using a periodization in law of the coefficients instead of considering a more classical method based on “snapshot” of the media, we establish the optimal rate of convergence of the method with respect to the size of the box by performing the first order expansion of the error. This result is obtained by combining a general formula from Gaussian calculus in the form of Price's formula that we generalise in the infinite-dimensional setting (in Chapter 4) and a two-scale expansion result of the Green's function of the random linear elliptic operator together with stochastic estimates on the correctors (in Chapter 5).
Author: Mark J. Bowick
Publisher: American Mathematical Soc.
ISBN: 1470429195
Category : Mathematics
Languages : en
Pages : 342
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Book Description
A co-publication of the AMS, IAS/Park City Mathematics Institute, and Society for Industrial and Applied Mathematics Articles in this volume are based on lectures presented at the Park City summer school on “Mathematics and Materials” in July 2014. The central theme is a description of material behavior that is rooted in statistical mechanics. While many presentations of mathematical problems in materials science begin with continuum mechanics, this volume takes an alternate approach. All the lectures present unique pedagogical introductions to the rich variety of material behavior that emerges from the interplay of geometry and statistical mechanics. The topics include the order-disorder transition in many geometric models of materials including nonlinear elasticity, sphere packings, granular materials, liquid crystals, and the emerging field of synthetic self-assembly. Several lectures touch on discrete geometry (especially packing) and statistical mechanics. The problems discussed in this book have an immediate mathematical appeal and are of increasing importance in applications, but are not as widely known as they should be to mathematicians interested in materials science. The volume will be of interest to graduate students and researchers in analysis and partial differential equations, continuum mechanics, condensed matter physics, discrete geometry, and mathematical physics. Titles in this series are co-published with the Institute for Advanced Study/Park City Mathematics Institute. Members of the Mathematical Association of America (MAA) and the National Council of Teachers of Mathematics (NCTM) receive a 20% discount from list price. NOTE: This discount does not apply to volumes in this series co-published with the Society for Industrial and Applied Mathematics (SIAM).
Author: Martin Simon
Publisher: Springer
ISBN: 3658109939
Category : Mathematics
Languages : en
Pages : 153
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Book Description
This monograph is concerned with the analysis and numerical solution of a stochastic inverse anomaly detection problem in electrical impedance tomography (EIT). Martin Simon studies the problem of detecting a parameterized anomaly in an isotropic, stationary and ergodic conductivity random field whose realizations are rapidly oscillating. For this purpose, he derives Feynman-Kac formulae to rigorously justify stochastic homogenization in the case of the underlying stochastic boundary value problem. The author combines techniques from the theory of partial differential equations and functional analysis with probabilistic ideas, paving the way to new mathematical theorems which may be fruitfully used in the treatment of the problem at hand. Moreover, the author proposes an efficient numerical method in the framework of Bayesian inversion for the practical solution of the stochastic inverse anomaly detection problem.
Author: Ecole normale supérieure (France)
Publisher:
ISBN:
Category : Science
Languages : en
Pages : 268
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Book Description
Author: Carlos E. Kenig
Publisher: American Mathematical Soc.
ISBN: 1470461277
Category : Education
Languages : en
Pages : 345
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Book Description
The origins of the harmonic analysis go back to an ingenious idea of Fourier that any reasonable function can be represented as an infinite linear combination of sines and cosines. Today's harmonic analysis incorporates the elements of geometric measure theory, number theory, probability, and has countless applications from data analysis to image recognition and from the study of sound and vibrations to the cutting edge of contemporary physics. The present volume is based on lectures presented at the summer school on Harmonic Analysis. These notes give fresh, concise, and high-level introductions to recent developments in the field, often with new arguments not found elsewhere. The volume will be of use both to graduate students seeking to enter the field and to senior researchers wishing to keep up with current developments.
Author: Martin Heida
Publisher:
ISBN:
Category :
Languages : en
Pages :
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Book Description
In this work we deal with the stochastic homogenization of the initial boundary value problems of monotone type. The models of monotone type under consideration describe the deformation behaviour of inelastic materials with a microstructure which can be characterised by random measures. Based on the Fitzpatrick function concept we reduce the study of the asymptotic behaviour of monotone operators associated with our models to the problem of the stochastic homogenization of convex functionals within an ergodic and stationary setting. The concept of Fitzpatrick's function helps us to introduce and show the existence of the weak solutions for rate-dependent systems. The derivations of the homogenization results presented in this work are based on the stochastic two-scale convergence in Sobolev spaces. For completeness, we also present some two-scale homogenization results for convex functionals, which are related to the classical Gamma-convergence theory.
Author: Zhongwei Shen
Publisher: Springer
ISBN: 3319912143
Category : Mathematics
Languages : en
Pages : 295
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Book Description
This monograph surveys the theory of quantitative homogenization for second-order linear elliptic systems in divergence form with rapidly oscillating periodic coefficients in a bounded domain. It begins with a review of the classical qualitative homogenization theory, and addresses the problem of convergence rates of solutions. The main body of the monograph investigates various interior and boundary regularity estimates that are uniform in the small parameter e>0. Additional topics include convergence rates for Dirichlet eigenvalues and asymptotic expansions of fundamental solutions, Green functions, and Neumann functions. The monograph is intended for advanced graduate students and researchers in the general areas of analysis and partial differential equations. It provides the reader with a clear and concise exposition of an important and currently active area of quantitative homogenization.
Author: Grigorios A. Pavliotis
Publisher: Springer
ISBN: 1493913239
Category : Mathematics
Languages : en
Pages : 345
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Book Description
This book presents various results and techniques from the theory of stochastic processes that are useful in the study of stochastic problems in the natural sciences. The main focus is analytical methods, although numerical methods and statistical inference methodologies for studying diffusion processes are also presented. The goal is the development of techniques that are applicable to a wide variety of stochastic models that appear in physics, chemistry and other natural sciences. Applications such as stochastic resonance, Brownian motion in periodic potentials and Brownian motors are studied and the connection between diffusion processes and time-dependent statistical mechanics is elucidated. The book contains a large number of illustrations, examples, and exercises. It will be useful for graduate-level courses on stochastic processes for students in applied mathematics, physics and engineering. Many of the topics covered in this book (reversible diffusions, convergence to equilibrium for diffusion processes, inference methods for stochastic differential equations, derivation of the generalized Langevin equation, exit time problems) cannot be easily found in textbook form and will be useful to both researchers and students interested in the applications of stochastic processes.
