Bias in the representative volume element method: periodize the ensemble instead of its realizations
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We study the Representative Volume Element (RVE) method, which is a method to approximately infer the effective behavior a_hom of a stationary random medium. The latter is described by a coefficient field a(x) generated from a given ensemble ⟨·⟩ and the corresponding linear elliptic operator -∇· a∇. 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 ⟨·⟩. We make this point by investigating the bias (or systematic error), i. e. the difference between a_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 ⟨·⟩, 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 ⟨·⟩ 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.
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