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Spectral measure and approximation of homogenized coefficients
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  • Published: 02 June 2011

Spectral measure and approximation of homogenized coefficients

  • Antoine Gloria1 &
  • Jean-Christophe Mourrat2 

Probability Theory and Related Fields volume 154, pages 287–326 (2012)Cite this article

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Abstract

This article deals with the numerical approximation of effective coefficients in stochastic homogenization of discrete linear elliptic equations. The originality of this work is the use of a well-known abstract spectral representation formula to design and analyze effective and computable approximations of the homogenized coefficients. In particular, we show that information on the edge of the spectrum of the generator of the environment viewed by the particle projected on the local drift yields bounds on the approximation error, and conversely. Combined with results by Otto and the first author in low dimension, and results by the second author in high dimension, this allows us to prove that for any dimension d ≥ 2, there exists an explicit numerical strategy to approximate homogenized coefficients which converges at the rate of the central limit theorem.

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Author information

Authors and Affiliations

  1. Project-team SIMPAF & Laboratoire Paul Painlevé, UMR 8524, INRIA Lille - Nord Europe & Université Lille 1, Villeneuve-d’Ascq, France

    Antoine Gloria

  2. Centre de Mathématiques et Informatique (CMI), Université de Provence, Marseille Cedex, France

    Jean-Christophe Mourrat

Authors
  1. Antoine Gloria
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  2. Jean-Christophe Mourrat
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Corresponding author

Correspondence to Antoine Gloria.

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Cite this article

Gloria, A., Mourrat, JC. Spectral measure and approximation of homogenized coefficients. Probab. Theory Relat. Fields 154, 287–326 (2012). https://doi.org/10.1007/s00440-011-0370-7

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  • Received: 19 August 2010

  • Revised: 13 May 2011

  • Published: 02 June 2011

  • Issue Date: October 2012

  • DOI: https://doi.org/10.1007/s00440-011-0370-7

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Keywords

  • Stochastic homogenization
  • Spectral theory
  • Ergodic theory
  • Numerical method

Mathematics Subject Classification (2000)

  • 35B27
  • 37A30
  • 65C50
  • 65N99
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