Efficient Methods for the Estimation of Homogenized Coefficients

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.

Computing homogenized coefficientsviamultiscale representation and hierarchical hybrid grids

“…A result comparable to Theorem 1.1 was proved in [52] in the context of discrete finitedifference equations. Besides the adaptation to the continuous setting, there are two main differences between the present result and the one obtained in [52].…”

“…A result comparable to Theorem 1.1 was proved in [52] in the context of discrete finitedifference equations. Besides the adaptation to the continuous setting, there are two main differences between the present result and the one obtained in [52]. The first one is that the quantities on the right side of (1.6) are averages against a smooth mask, while only box averages could be handled in [52].…”

“…By the scaling of the central limit theorem, in order to measure this quantity within a precision δ > 0, we need to average over at least Cδ −2 unit cells. Similarly, as was shown in [52, Proposition 1.1], it is impossible to compute an approximation of the homogenized matrix at precision δ if one observes only o(δ −2 ) unit cells (the statement in [52] is written for finite-difference equations, but the proof applies essentially verbatim to the continuous setting).…”

“…Compared with the discrete setting of finite-difference operators investigated in [52], the case of continuous differential operators considered here poses crucial new challenges from a computational perspective. With applications such as those in materials science in mind, it is natural to consider piecewise constant coefficient fields.…”

“…The method we propose here, by combining this idea with a multiscale decomposition, enables to take fuller advantage of this idea. We refer to [52] for a detailed comparison between the single-scale and the multiscale approaches. As is shown in [2], the benefits of the multiscale approach can be seen even in the setting of periodic coefficient fields, if we operate under the constraint that the lattice of periods is unknown.…”