Bias in the Representative Volume Element method: Periodize the Ensemble Instead of Its Realizations

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