Spectral measure and approximation of homogenized coefficients

• 成果类型: Article
• 署名作者: Gloria, Antoine; Mourrat, Jean-Christophe
• 署名单位: Centre National de la Recherche Scientifique (CNRS); CNRS - National Institute for Mathematical Sciences (INSMI); Universite de Lille; Aix-Marseille Universite
• 刊物名称: PROBABILITY THEORY AND RELATED FIELDS
• ISSN/ISSBN: 0178-8051
• DOI: 10.1007/s00440-011-0370-7
• 发表日期: 2012
• 页码: 287-326
• 关键词: quenched invariance-principles; reversible markov-processes; random-walks; percolation; functionals; limit

摘要

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 a parts per thousand yen 2, there exists an explicit numerical strategy to approximate homogenized coefficients which converges at the rate of the central limit theorem.