Numerical approximation of effective coefficients in stochastic homogenization of discrete elliptic equations

Antoine Gloria

ESAIM: Mathematical Modelling and Numerical Analysis (2011)

  • Volume: 46, Issue: 1, page 1-38
  • ISSN: 0764-583X

Abstract

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We introduce and analyze a numerical strategy to approximate effective coefficients in stochastic homogenization of discrete elliptic equations. In particular, we consider the simplest case possible: An elliptic equation on the d-dimensional lattice ℤ d with independent and identically distributed conductivities on the associated edges. Recent results by Otto and the author quantify the error made by approximating the homogenized coefficient by the averaged energy of a regularized corrector (with parameter T) on some box of finite size L. In this article, we replace the regularized corrector (which is the solution of a problem posed on ℤ d ) by some practically computable proxy on some box of size R≥L, and quantify the associated additional error. In order to improve the convergence, one may also consider N independent realizations of the computable proxy, and take the empirical average of the associated approximate homogenized coefficients. A natural optimization problem consists in properly choosing T, R, L and N in order to reduce the error at given computational complexity. Our analysis is sharp and sheds some light on this question. In particular, we propose and analyze a numerical algorithm to approximate the homogenized coefficients, taking advantage of the (nearly) optimal scalings of the errors we derive. The efficiency of the approach is illustrated by a numerical study in dimension 2.

How to cite

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Gloria, Antoine. "Numerical approximation of effective coefficients in stochastic homogenization of discrete elliptic equations." ESAIM: Mathematical Modelling and Numerical Analysis 46.1 (2011): 1-38. <http://eudml.org/doc/222176>.

@article{Gloria2011,
abstract = { We introduce and analyze a numerical strategy to approximate effective coefficients in stochastic homogenization of discrete elliptic equations. In particular, we consider the simplest case possible: An elliptic equation on the d-dimensional lattice $\mathbb\{Z\}^d$ with independent and identically distributed conductivities on the associated edges. Recent results by Otto and the author quantify the error made by approximating the homogenized coefficient by the averaged energy of a regularized corrector (with parameter T) on some box of finite size L. In this article, we replace the regularized corrector (which is the solution of a problem posed on $\mathbb\{Z\}^d$) by some practically computable proxy on some box of size R≥L, and quantify the associated additional error. In order to improve the convergence, one may also consider N independent realizations of the computable proxy, and take the empirical average of the associated approximate homogenized coefficients. A natural optimization problem consists in properly choosing T, R, L and N in order to reduce the error at given computational complexity. Our analysis is sharp and sheds some light on this question. In particular, we propose and analyze a numerical algorithm to approximate the homogenized coefficients, taking advantage of the (nearly) optimal scalings of the errors we derive. The efficiency of the approach is illustrated by a numerical study in dimension 2. },
author = {Gloria, Antoine},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis},
keywords = {Stochastic homogenization; effective coefficients; difference operator; numerical method; error estimate},
language = {eng},
month = {7},
number = {1},
pages = {1-38},
publisher = {EDP Sciences},
title = {Numerical approximation of effective coefficients in stochastic homogenization of discrete elliptic equations},
url = {http://eudml.org/doc/222176},
volume = {46},
year = {2011},
}

TY - JOUR
AU - Gloria, Antoine
TI - Numerical approximation of effective coefficients in stochastic homogenization of discrete elliptic equations
JO - ESAIM: Mathematical Modelling and Numerical Analysis
DA - 2011/7//
PB - EDP Sciences
VL - 46
IS - 1
SP - 1
EP - 38
AB - We introduce and analyze a numerical strategy to approximate effective coefficients in stochastic homogenization of discrete elliptic equations. In particular, we consider the simplest case possible: An elliptic equation on the d-dimensional lattice $\mathbb{Z}^d$ with independent and identically distributed conductivities on the associated edges. Recent results by Otto and the author quantify the error made by approximating the homogenized coefficient by the averaged energy of a regularized corrector (with parameter T) on some box of finite size L. In this article, we replace the regularized corrector (which is the solution of a problem posed on $\mathbb{Z}^d$) by some practically computable proxy on some box of size R≥L, and quantify the associated additional error. In order to improve the convergence, one may also consider N independent realizations of the computable proxy, and take the empirical average of the associated approximate homogenized coefficients. A natural optimization problem consists in properly choosing T, R, L and N in order to reduce the error at given computational complexity. Our analysis is sharp and sheds some light on this question. In particular, we propose and analyze a numerical algorithm to approximate the homogenized coefficients, taking advantage of the (nearly) optimal scalings of the errors we derive. The efficiency of the approach is illustrated by a numerical study in dimension 2.
LA - eng
KW - Stochastic homogenization; effective coefficients; difference operator; numerical method; error estimate
UR - http://eudml.org/doc/222176
ER -

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