Title :

Quantitative version of the Kipnis-Varadhan theorem and Monte-Carlo approximation of homogenized coefficients

Author(s) :

Gloria, Antoine [Auteur]
SImulations and Modeling for PArticles and Fluids [SIMPAF]
Mourrat, Jean-Christophe [Auteur]
Département de Mathématiques - EPFL

Journal title :

The Annals of Applied Probability

Pages :

1544-1583

Publisher :

Institute of Mathematical Statistics (IMS)

Publication date :

2013

ISSN :

1050-5164

English keyword(s) :

Effective coefficients
Monte Carlo method
Quantitative estimates
Random environment
Random walk
Stochastic homogenization

HAL domain(s) :

Mathématiques [math]/Analyse numérique [math.NA]

English abstract : [en]

This article is devoted to the analysis of a Monte-Carlo method to approximate effective coefficients in stochastic homogenization of discrete elliptic equations. We consider the case of independent and identically distributed ...
Show more >This article is devoted to the analysis of a Monte-Carlo method to approximate effective coefficients in stochastic homogenization of discrete elliptic equations. We consider the case of independent and identically distributed coefficients, and adopt the point of view of the random walk in a random environment. Given some final time $t>0$, a natural approximation of the homogenized coefficients is given by the empirical average of the final squared positions rescaled by $t$ of $n$ independent random walks in $n$ independent environments. Relying on a quantitative version of the Kipnis-Varadhan theorem combined with estimates of spectral exponents obtained by an original combination of pde arguments and spectral theory, we first give a sharp estimate of the error between the homogenized coefficients and the expectation of the rescaled final position of the random walk in terms of $t$. We then complete the error analysis by quantifying the fluctuations of the empirical average in terms of $n$ and $t$, and prove a large-deviation estimate, as well as a central limit theorem. Our estimates are optimal, up to a logarithmic correction in dimension $2$.Show less >

Language :

Anglais

Popular science :

Non

Collections :

• Laboratoire Paul Painlevé - UMR 8524

Source :

Harvested from HAL