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Foundations of Computational Mathematics

, Volume 19, Issue 2, pp 435–483 | Cite as

Efficient Methods for the Estimation of Homogenized Coefficients

  • J.-C. MourratEmail author
Article
  • 71 Downloads

Abstract

The main goal of this paper is to define and study new methods for the computation of effective coefficients in the homogenization of divergence-form operators with random coefficients. The methods introduced here are proved to have optimal computational complexity and are shown numerically to display small constant prefactors. In the spirit of multiscale methods, the main idea is to rely on a progressive coarsening of the problem, which we implement via a generalization of the Green–Kubo formula. The technique can be applied more generally to compute the effective diffusivity of any additive functional of a Markov process. In this broader context, we also discuss the alternative possibility of using Monte Carlo sampling and show how a simple one-step extrapolation can considerably improve the performance of this alternative method.

Keywords

Homogenization Multiscale methods 

Mathematics Subject Classification

82B80 35B27 82B28 

Notes

Acknowledgements

I would like to thank Josselin Garnier for an inspiring talk which motivated me to revisit this problem, Tony Lelièvre for his helpful feedback and Harmen Stoppels for his precious help with the Julia language. This work has been partially supported by the ANR Grant LSD (ANR-15-CE40-0020-03).

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Copyright information

© SFoCM 2018

Authors and Affiliations

  1. 1.DMA, Ecole normale supérieure, CNRSPSL Research UniversityParisFrance

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