论文标题
代表卷元素方法中的偏差：定期合奏，而不是实现
Bias in the representative volume element method: periodize the ensemble instead of its realizations
论文作者
论文摘要
我们研究了代表性音量元素（RVE）方法，该方法是一种近似固定随机介质的有效行为$ a _ {\ text {hom}} $的方法。后者由从给定的合奏$ \ langle \ cdot \ rangle $和相应的线性椭圆操作员$ - \ nabla \ cdot a \ nabla $生成的系数字段$ a（x）$。与均质化理论一致，该方法是通过计算$ d = 3 $校正器（表示空间维度）来进行的。要在数值上进行操作，该计算必须在有限的域上进行：所谓的“代表性”音量元素，即。 e。一个大盒子，例如周期性的边界条件。本文的主要信息是：定期合奏，而不是实现。这是指从合适的周期合奏中采样比定期扩展实现$ a（x）$从整个空间集合$ \ langle \ cdot \ cdot \ rangle $中的限制。我们通过研究偏差（或系统错误）来提出这一点。 e。 $ a _ {\ text {hom}} $与RVE方法的预期值之间的差异，就其缩放w而言。 r。 t。盒子的横向尺寸$ l $。如果定期$ a（x）$，我们启发性地认为，此错误通常为$ o（l^{ - 1}）$。如果适合$ \ langle \ cdot \ rangle $，我们严格地表明它是$ o（l^{ - d}）$。实际上，我们给出了两种策略的前阶误差项的特征，并认为即使在各向同性的情况下，它也是通常的非分类。我们在合奏的方便环境中进行了严格的分析，$ \ langle \ cdot \ rangle $ y t of Gaussian类型具有集成协方差，这允许直接分期化，并使价格定理和Malliavin Colculus可用于最佳的正确随机性估计。
We study the Representative Volume Element (RVE) method, which is a method to approximately infer the effective behavior $a_{\text{hom}}$ 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 $a_{\text{hom}}$ 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 with integrable covariance, which allow for a straightforward periodization and which make the Price theorem and the Malliavin calculus available for optimal stochastic estimates of correctors.
