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

2011 ◽  
Vol 46 (1) ◽  
pp. 1-38 ◽  
Author(s):  
Antoine Gloria
Keyword(s):  
Elliptic Equations ◽  
Numerical Approximation ◽  
Stochastic Homogenization ◽  
Effective Coefficients

scholarly journals Some variance reduction methods for numerical stochastic homogenization

2016 ◽  
Vol 374 (2066) ◽  
pp. 20150168 ◽  
Author(s):  
X. Blanc ◽  
C. Le Bris ◽  
F. Legoll
Keyword(s):  
Monte Carlo ◽  
Variance Reduction ◽  
Stochastic Homogenization ◽  
Numerical Homogenization ◽  
Variance Reduction Techniques ◽  
Reduction Techniques ◽  
Effective Coefficients ◽  
Engineering Sciences ◽  
Reduction Methods ◽  
The Cost

We give an overview of a series of recent studies devoted to variance reduction techniques for numerical stochastic homogenization. Numerical homogenization requires that a set of problems is solved at the microscale, the so-called corrector problems. In a random environment, these problems are stochastic and therefore need to be repeatedly solved, for several configurations of the medium considered. An empirical average over all configurations is then performed using the Monte Carlo approach, so as to approximate the effective coefficients necessary to determine the macroscopic behaviour. Variance severely affects the accuracy and the cost of such computations. Variance reduction approaches, borrowed from other contexts in the engineering sciences, can be useful. Some of these variance reduction techniques are presented, studied and tested here.

scholarly journals REDUCTION OF THE RESONANCE ERROR — PART 1: APPROXIMATION OF HOMOGENIZED COEFFICIENTS

2011 ◽  
Vol 21 (08) ◽  
pp. 1601-1630 ◽  
Author(s):  
ANTOINE GLORIA
Keyword(s):  
Elliptic Equations ◽  
Length Scale ◽  
Numerical Homogenization ◽  
Cell Problem ◽  
Typical Size ◽  
Effective Coefficients ◽  
Homogenization Methods

This paper is concerned with the approximation of effective coefficients in homogenization of linear elliptic equations. One common drawback among numerical homogenization methods is the presence of the so-called resonance error, which roughly speaking is a function of the ratio ε/η, where η is a typical macroscopic length scale and ε is the typical size of the heterogeneities. In the present work, we propose an alternative for the computation of homogenized coefficients (or more generally a modified cell-problem), which is a first brick in the design of effective numerical homogenization methods. We show that this approach drastically reduces the resonance error in some standard cases.

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