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We study the Representative Volume Element (RVE) method, which is a method toapproximately infer the effective behavior $a_{\text{hom}}$ of a stationaryrandom medium. The latter is described by a coefficient field $a(x)$ generatedfrom a given ensemble $\langle\cdot\rangle$ and the corresponding linearelliptic operator $-\nabla\cdot a\nabla$. In line with the theory ofhomogenization, the method proceeds by computing $d = 3$ correctors (d denotingthe space dimension).To be numerically tractable, this computation has to bedone on a finite domain: the so-called "representative" volume element, i. e. alarge box with, say, periodic boundary conditions. The main message of thisarticle is: Periodize the ensemble instead of its realizations. By this we meanthat it is better to sample from a suitably periodized ensemble than toperiodically extend the restriction of a realization $a(x)$ from thewhole-space ensemble $\langle\cdot\rangle$. We make this point by investigatingthe bias (or systematic error), i. e. the difference between $a_{\text{hom}}$and the expected value of the RVE method, in terms of its scaling w. r. t. thelateral size $L$ of the box. In case of periodizing $a(x)$, we heuristicallyargue that this error is generically $O(L^{-1})$. In case of a suitableperiodization of $\langle\cdot\rangle$, we rigorously show that it is$O(L^{-d})$. In fact, we give a characterization of the leading-order errorterm for both strategies, and argue that even in the isotropic case it isgenerically non-degenerate. We carry out the rigorous analysis in theconvenient setting of ensembles $\langle\cdot\rangle$ of Gaussian type withintegrable covariance, which allow for a straightforward periodization andwhich make the Price theorem and the Malliavin calculus available for optimalstochastic estimates of correctors.

Bias in the representative volume element method: periodize the ensemble instead of its realizations