We study the representative volume element (RVE) method, which is a method to approximately infer the effective behavior \(a_{\textrm{hom}}\) of a stationary random medium. The latter is described by a coefficient field a(x) generated from a given ensemble \(\langle \cdot \rangle \) and the corresponding linear elliptic operator \(-\nabla \cdot a\nabla \). In line with the theory of homogenization, the method proceeds by computing \(d=3\) correctors (d denoting the space dimension). To be numerically tractable, this computation has to be done on a finite domain: the so-called representative volume element, i.e., a large box with, say, periodic boundary conditions. The main message of this article is: Periodize the ensemble instead of its realizations. By this, we mean that it is better to sample from a suitably periodized ensemble than to periodically extend the restriction of a realization a(x) from the whole-space ensemble \(\langle \cdot \rangle \). We make this point by investigating the bias (or systematic error), i.e., the difference between \(a_{\textrm{hom}}\) and the expected value of the RVE method, in terms of its scaling w.r.t. the lateral size L of the box. In case of periodizing a(x), we heuristically argue that this error is generically \(O(L^{-1})\). In case of a suitable periodization of \(\langle \cdot \rangle \), we rigorously show that it is \(O(L^{-d})\). In fact, we give a characterization of the leading-order error term for both strategies and argue that even in the isotropic case it is generically non-degenerate. We carry out the rigorous analysis in the convenient setting of ensembles \(\langle \cdot \rangle \) of Gaussian type, which allow for a straightforward periodization, passing via the (integrable) covariance function. This setting has also the advantage of making the Price theorem and the Malliavin calculus available for optimal stochastic estimates of correctors. We actually need control of second-order correctors to capture the leading-order error term. This is due to inversion symmetry when applying the two-scale expansion to the Green function. As a bonus, we present a stream-lined strategy to estimate the error in a higher-order two-scale expansion of the Green function.

我们研究了代表性体积元 (RVE) 方法，这是一种近似推断平稳随机介质的有效行为\(a_{\textrm{hom}}\) 的方法。后者由从给定系综\(\langle \cdot \rangle \)和相应的线性椭圆算子\(-\nabla \cdot a\nabla \)生成的系数场a ( x )描述。根据均匀化理论，该方法通过计算\(d=3\)个校正器（d表示空间维度）。为了在数值上易于处理，这种计算必须在有限域上进行：所谓的代表性体积元素，即具有周期性边界条件的大盒子。这篇文章的主要信息是：周期化集成而不是它的实现。通过这种方式，我们的意思是从适当周期化的系综中采样比从全空间系综\(\langle \cdot \rangle \)中周期性地扩展实现a ( x )的限制要好。我们通过研究偏差（或系统误差）来说明这一点，即\(a_{\textrm{hom}}\)与 RVE 方法的预期值之间的差异，就横向尺寸L的缩放比例而言的盒子。在周期化a ( x )的情况下，我们启发式地认为这个错误通常是\(O(L^{-1})\)。在\(\langle \cdot \rangle \)的适当周期化的情况下，我们严格证明它是\(O(L^{-d})\)。事实上，我们给出了两种策略的前导误差项的特征，并认为即使在各向同性的情况下它通常也是非退化的。我们在合奏的方便设置中进行严格的分析\(\langle \cdot \rangle \)高斯类型，允许直接的周期化，通过（可积的）协方差函数传递。此设置还具有使 Price 定理和 Malliavin 微积分可用于校正器的最佳随机估计的优势。我们实际上需要控制二阶校正器来捕获前导误差项。这是由于将双尺度展开应用于格林函数时的反演对称性。作为奖励，我们提出了一种流线型策略来估计 Green 函数的高阶双尺度展开中的误差。