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

Foundations of Computational Mathematics, May 2023

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.

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

Foundations of Computational Mathematics https://doi.org/10.1007/s10208-023-09613-y Bias in the Representative Volume Element method: Periodize the Ensemble Instead of Its Realizations Nicolas Clozeau1 · Marc Josien2 · Felix Otto3 · Qiang Xu4 Received: 1 June 2022 / Revised: 22 February 2023 / Accepted: 27 February 2023 © The Author(s) 2023 Abstract We study the representative volume element (RVE) method, which is a method to approximately infer the effective behavior ahom 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 Communicated by Endre Suli. NC has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (Grant Agreement No. 948819). B Nicolas Clozeau Marc Josien Felix Otto Qiang Xu 1 Institute of Science and Technology Austria (ISTA), Am Campus 1, 3400 Klosterneuburg, Austria 2 CEA, DES, IRESNE, DEC, Cadarache, 13108 Saint-Paul-Lez-Durance, France 3 Max Planck Institute for Mathematics in the Sciences, Inselstrasse 22, 04103 Leipzig, Germany 4 School of Mathematics and Statistics, Lanzhou University, 222 Tianshui S Rd, Chengguan District, Lanzhou 730000, China 123 Foundations of Computational Mathematics (or systematic error), i.e., the difference between ahom 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, 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. Keywords Stochastic homogenization · Random media · Representative volume element method · Gaussian calculus Mathematics Subject Classification 35B27 · 65N99 Contents 1 Introduction and Statement of Rigorous Result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Uniformly Elliptic Coefficient Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 The RVE Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Two Strategies of Periodizing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4 Fluctuations and Bias . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5 Assumptions and Formulation of Rigorous Result . . . . . . . . . . . . . . . . . . . . . . . . . 2 Theorem 1: Refinement and Main Ideas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Representation Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Approximation by Second-Order Homogenization . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Refinement of Rigorous Result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Small Contrast Regime and Non-degeneracy . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5 Isotropic Ensembles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Structure of the Proof of Theorem 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Massive Approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Re-summation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Stochastic Corrector Estimates Up to Second Order . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Estimate of Homogenization Error to Second Order, Application to the Green Function . . . . . 4 Heuristic Result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Proofs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Proof of Theorem 2: Asymptotic of the Bias . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Proof of Proposition 2: Limit T ↑ ∞ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 Proof of Lemma 1: Fluctuation Estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4 Proof of Corollary 1: Limit L ↑ ∞ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5 Proof of Lemma 4: Improved Caccioppoli Inequality . . . . . . . . . . . . . . . . . . . . . . . 5.6 Proof of Lemma 3: Annealed Estimate on the Two-Scale Expansion Error . . . . . . . . . . . . 5.7 Proof of Proposition 4: Annealed Error Estimate on the Expansion of the Green Function . . . . Appendix A. 2nd -Order Two-Scale Expansion of the Mixed Derivative of the Massive Green Function . Appendix B. Intermediate Results for the Heuristic Result of Sect. 4 . . . . . . . . . . . . . . . . . . . sym B.1. Argument for (101): Symmetry of c L . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123 Foundations of Computational Mathematics B.2. Computation of the Integral (118) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B.3. Argument for (104): Representation Formula for the L-Derivative . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 Introduction and Statement of Rigorous Result 1.1 Uniformly Elliptic Coefficient Fields The basic objects of this paper are λ-uniformly elliptic tensor fields a = a(x) (that are not necessarily symmetric) in d-dimensional space, by which we mean that for all points x ξ · a(x)ξ ≥ λ|ξ |2 and ξ · a(x)ξ ≥ |a(x)ξ |2 for (...truncated)


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Clozeau, Nicolas, Josien, Marc, Otto, Felix, Xu, Qiang. Bias in the Representative Volume Element method: Periodize the Ensemble Instead of Its Realizations, Foundations of Computational Mathematics, 2023, pp. 1-83, DOI: 10.1007/s10208-023-09613-y