Bias in the Representative Volume Element method: periodize the ensemble instead of its realizations

Speaker: Felix Otto (Max Planck Institute, Leipzig)
Occasion: Mathematical and Computational Materials Science
Date: February 15, 2021

Abstract: We study the Representative Volume Element (RVE) method, which is a method to approximately infer the effective behavior $a_{\rm hom}$ of a stationary random medium, described by a coefficient field $a(x)$ and the corresponding linear elliptic operator $-\nabla\cdot a\nabla$. In line with the theory of homogenization, the method proceeds by computing $d=3$ ($d$ denoting the space dimension) correctors, however on a “representative” volume element, i.e. 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 $a_{\rm hom}$ 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.

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 Malliavin calculus available for optimal stochastic estimates of correctors. We actually need control of second-order correctors to capture the leading-order error term in the presence of cancellations due to point symmetry. This is joint work with Nicolas Clozeau, Marc Josien, and Qiang Xu.