Some variance reduction methods for numerical stochastic homogenization

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 stoc...

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Bibliographic Details
Published in:Philosophical transactions of the Royal Society of London. A 2016-04, Vol.374 (2066), p.20150168
Main Authors: Blanc, X., Le Bris, C., Legoll, F.
Format: Article
Language:English
Subjects:
Elliptic Partial Differential Equations
Mathematical Modelling In Materials Science
Mathematics
Numerical Analysis
Stochastic Homogenization
Variance Reduction
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Summary: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.
ISSN:1364-503X
0264-3820
1471-2962
DOI:10.1098/rsta.2015.0168