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Multi angle  Waves and microstructures
Weinstein, Michael (Auteur de la Conférence) | CIRM (Editeur )

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The simulation of random heterogeneous materials is often very expensive. For instance, in a homogenization setting, the homogenized coefficient is defined from the so-called corrector function, that solves a partial differential equation set on the entire space. This is in contrast with the periodic case, where he corrector function solves an equation set on a single periodic cell. As a consequence, in the stochastic setting, the numerical approximation of the corrector function (and therefore of the homogenized coefficient) is a challenging computational task.
In practice, the corrector problem is solved on a truncated domain, and the exact homogenized coefficient is recovered only in the limit of infinitely large domains. As a consequence of this truncation, the approximated homogenized coefficient turns out to be stochastic, even though the exact homogenized coefficient is deterministic. One then has to resort to Monte-Carlo methods, in order to compute the expectation of the (approximated, apparent) homogenized coefficient within a good accuracy. Variance reduction questions thus naturally come into play, in order to increase the accuracy (e.g. reduce the size of the confidence interval) for a fixed computational cost. In this talk, we will present some variance reduction approaches to address this question.
The simulation of random heterogeneous materials is often very expensive. For instance, in a homogenization setting, the homogenized coefficient is defined from the so-called corrector function, that solves a partial differential equation set on the entire space. This is in contrast with the periodic case, where he corrector function solves an equation set on a single periodic cell. As a consequence, in the stochastic setting, the numerical ...

35B27 ; 60Hxx ; 35R60

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We discuss the rough path principle and some of its applications to problems of homogenization.

60H15 ; 35B27

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Traditionally homogenization asks whether average behavior can be discerned from Hamilton-Jacobi equations that are subject to high-frequency fluctuations in spatial variables. A similar question can be asked for the associated Hamiltonian ODEs. When the Hamiltonian function is convex in momentum variable, these two questions turn out to be equivalent. This equivalence breaks down for general Hamiltonian functions. In this talk I will give a dynamical system formulation for homogenization and address some result concerning weak and strong homogenization phenomena.
Traditionally homogenization asks whether average behavior can be discerned from Hamilton-Jacobi equations that are subject to high-frequency fluctuations in spatial variables. A similar question can be asked for the associated Hamiltonian ODEs. When the Hamiltonian function is convex in momentum variable, these two questions turn out to be equivalent. This equivalence breaks down for general Hamiltonian functions. In this talk I ...

35F21 ; 35B27 ; 60G10

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