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

ESAIM: Mathematical Modelling and Numerical Analysis (2011)

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

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topGloria, 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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