We present an efficient method for the computation of homogenized coefficients of divergence-form operators with random coefficients. The approach is based on a multiscale representation of the homogenized coefficients. We then implement the method numerically using a finite-element method with hierarchical hybrid grids, which is a semi-implicit method allowing for significant gains in memory usage and execution time. Finally, we demonstrate the efficiency of our approach on two- and three-dimensional examples, for piecewise-constant coefficients with corner discontinuities. For moderate ellipticity contrast and for a precision of a few percentage points, our method allows to compute the homogenized coefficients on a laptop computer in a few seconds, in two dimensions, or in a few minutes, in three dimensions.
Several techniques have been explored to reduce the size of the fluctuations of estimators for the homogenized matrix. In particular, control variate techniques and the selection of special realizations of the coefficient field, called quasi-random structures'', have been explored, see @cite_34 @cite_9 for surveys. The latter approach, inspired by @cite_17 @cite_54 and, in the context of the homogenization of elliptic operators, advocated for in @cite_19 , has recently received a spectacular theoretical foundation in @cite_25 . We would find it very interesting to investigate how these techniques can be combined with those discussed in the present paper.
@cite_54 @cite_9 @cite_19 @cite_34 @cite_25 @cite_17
2010942428 2107101459 2085186189 2963487200
We introduce and analyze a numerical strategy to approximate effective coefficients in stochastic homogenization of discrete elliptic equations. In particular, we consider the simplest case possible: An elliptic equation on the @math -dimensional lattice @math with independent and identically distributed conductivities on the associated edges. Recent results by Otto and the author quantify the error made by approximating the homogenized coefficient by the averaged energy of a regularized corrector (with parameter @math ) on some box of finite size @math . In this article, we replace the regularized corrector (which is the solution of a problem posed on @math ) by some practically computable proxy on some box of size @math , and quantify the associated additional error. In order to improve the convergence, one may also consider @math independent realizations of the computable proxy, and take the arithmetic mean of the associated approximate homogenized coefficients. A natural optimization problem consists in properly choosing @math and @math in order to reduce the error at given computational complexity. Our analysis is sharp and allows us to give a clear answer to this question. In particular, we propose and analyze a numerical algorithm to approximate the homogenized coefficients, taking advantage of the (nearly) optimal scalings of the errors we derive. The efficiency of the approach is illustrated by a numerical study in dimension 2. We study quantitatively the effective large-scale behavior of discrete elliptic equations on the lattice ( Z^d ) with random coefficients. The theory of stochastic homogenization relates the random, stationary, and ergodic field of coefficients with a deterministic matrix of effective coefficients. This is done via the corrector problem, which can be viewed as a highly degenerate elliptic equation on the infinite-dimensional space of admissible coefficient fields. In this contribution we develop new quantitative methods for the corrector problem based on the assumption that ergodicity holds in the quantitative form of a Spectral Gap Estimate w.r.t. a Glauber dynamics on coefficient fields—as it is the case for independent and identically distributed coefficients. As a main result we prove an optimal decay in time of the semigroup associated with the corrector problem (i.e. of the generator of the process called “random environment as seen from the particle”). As a corollary we recover existence of stationary correctors (in dimensions (d>2 )) and prove new optimal estimates for regularized versions of the corrector (in dimensions (d 2 )). We also give a self-contained proof of a new estimate on the gradient of the parabolic, variable-coefficient Green’s function, which is a crucial analytic ingredient in our approach. As an application of these results, we prove the first (and optimal) estimates for the approximation of the homogenized coefficients by the popular periodization method in case of independent and identically distributed coefficients. We consider a random, uniformly elliptic coefficient field (a(x) ) on the (d )-dimensional integer lattice ( Z ^d ). We are interested in the spatial decay of the quenched elliptic Green function (G(a;x,y) ). Next to stationarity, we assume that the spatial correlation of the coefficient field decays sufficiently fast to the effect that a logarithmic Sobolev inequality holds for the ensemble ( ). We prove that all stochastic moments of the first and second mixed derivatives of the Green function, that is, ( | _x G(x,y)|^p ) and ( | _x _y G(x,y)|^p ), have the same decay rates in (|x-y| 1 ) as for the constant coefficient Green function, respectively. This result relies on and substantially extends the one by Delmotte and Deuschel (Probab Theory Relat Fields 133:358–390, 2005), which optimally controls second moments for the first derivatives and first moments of the second mixed derivatives of (G ), that is, ( | _x G(x,y)|^2 ) and ( | _x _y G(x,y)| ). As an application, we are able to obtain optimal estimates on the random part of the homogenization error even for large ellipticity contrast. We adapt and study a variance reduction approach for the homogenization of elliptic equations in divergence form. The approach, borrowed from atomistic simulations and solid-state science [23], [24], [25], consists in selecting random realizations that best satisfy some statistical properties (such as the volume fraction of each phase in a composite material) usually only obtained asymptotically. We study the approach theoretically in some simplified settings (one-dimensional setting, perturbative setting in higher dimensions), and numerically demonstrate its efficiency in more general cases.