Random walk in random environment, corrector equation and homogenized coefficients: From theory to numerics, back and forth

A. C. Egloffe, A. Gloria, J. C. Mourrat, T. N. Nguyen

Research output: Contribution to journalArticle

Abstract

This article is concerned with numerical methods to approximate effective coefficients in stochastic homogenization of discrete linear elliptic equations, and their numerical analysis - which has been made possible by recent contributions on quantitative stochastic homogenization theory by two of the authors, and by Neukamm and Otto. This article makes the connection between our theoretical results and computations. We give a complete picture of the numerical methods found in the literature, compare them in terms of known (or expected) convergence rates and empirically study them. Two types of methods are presented: methods based on the corrector equation and methods based on random walks in random environments. The numerical study confirms the sharpness of the analysis (which it completes by making precise the prefactors, next to the convergence rates), supports some of our conjectures and calls for new theoretical developments.

Original languageEnglish (US)
Pages (from-to)499-545
Number of pages47
JournalIMA Journal of Numerical Analysis
Volume35
Issue number2
DOIs
StatePublished - Mar 15 2015

Keywords

  • Monte-Carlo method
  • discrete elliptic equations
  • effective coefficients
  • quantitative estimates
  • random environment
  • random walk
  • stochastic homogenization

ASJC Scopus subject areas

  • Mathematics(all)
  • Computational Mathematics
  • Applied Mathematics

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