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Special quasirandom structures: A selection approach for stochastic homogenization

  • Claude Le Bris EMAIL logo , Frédéric Legoll and William Minvielle


We adapt and study a variance reduction approach for the homogenization of elliptic equations in divergence form. The approach, borrowed from atomistic simulations and solid-state science [23], [24], [25], consists in selecting random realizations that best satisfy some statistical properties (such as the volume fraction of each phase in a composite material) usually only obtained asymptotically. We study the approach theoretically in some simplified settings (one-dimensional setting, perturbative setting in higher dimensions), and numerically demonstrate its efficiency in more general cases.

Funding source: ONR

Award Identifier / Grant number: N00014-12-1-0383

Funding source: EOARD

Award Identifier / Grant number: FA8655-13-1-3061

Funding source: ANR

Award Identifier / Grant number: Investments for the Future ANR-11-LABX-022-01

The first two authors would like to thank E. Cancès for introducing them to the SQS approach in the context of solid state physics, pointing out to them [23], [24], [25], as well as for several stimulating discussions in the early stages of this work. The authors also thank J.-D. Deuschel for his helpful comments and for pointing out [10], and X. Blanc for his remarks on a draft version of this article.


1 A. Anantharaman, R. Costaouec, C. Le Bris, F. Legoll and F. Thomines, Introduction to numerical stochastic homogenization and the related computational challenges: Some recent developments, Multiscale Modeling and Analysis for Materials Simulation, Lect. Notes Series 22, World Scientific, Hackensack (2011), 197–272. 10.1142/9789814360906_0004Search in Google Scholar

2 A. Bensoussan, J.-L. Lions and G. Papanicolaou, Asymptotic Methods in Periodic Structures, Stud. Math. Appl. 5, North-Holland, Amsterdam, 1978. Search in Google Scholar

3 C. Bernardin and S. Olla, Thermodynamics and non-equilibrium macroscopic dynamics of chains of anharmonic oscillators, preprint 2014, Search in Google Scholar

4 X. Blanc, R. Costaouec, C. Le Bris and F. Legoll, Variance reduction in stochastic homogenization: The technique of antithetic variables, Numerical Analysis of Multiscale Computations, Lecture Notes Comput. Sci. Eng. 82, Springer, Berlin (2012), 47–70. 10.1007/978-3-642-21943-6_3Search in Google Scholar

5 X. Blanc, R. Costaouec, C. Le Bris and F. Legoll, Variance reduction in stochastic homogenization using antithetic variables, Markov Process. Related Fields 18 (2012), 1, 31–66; preliminary version available at Search in Google Scholar

6 A. Bourgeat and A. Piatnitski, Approximations of effective coefficients in stochastic homogenization, Ann. Inst. Henri Poincaré Probab. Stat. 40 (2004), 2, 153–165. 10.1016/S0246-0203(03)00065-7Search in Google Scholar

7 D. Cioranescu and P. Donato, An Introduction to Homogenization, Oxford Lecture Ser. Math. Appl. 17, Clarendon Press, Oxford, 1999. Search in Google Scholar

8 R. Costaouec, Asymptotic expansion of the homogenized matrix in two weakly stochastic homogenization settings, Appl. Math. Res. Express. AMRX 2012 (2012), 1, 76–104. 10.1093/amrx/abr011Search in Google Scholar

9 R. Costaouec, C. Le Bris and F. Legoll, Variance reduction in stochastic homogenization: Proof of concept, using antithetic variables, Bol. Soc. Esp. Mat. Apl. 50 (2010), 9–27. 10.1007/BF03322539Search in Google Scholar

10 A. Dembo and O. Zeitouni, Refinements of the Gibbs conditioning principle, Probab. Theory Related Fields 104 (1996), 1, 1–14. 10.1007/BF01303799Search in Google Scholar

11 B. Engquist and P. E. Souganidis, Asymptotic and numerical homogenization, Acta Numer. 17 (2008), 147–190. 10.1017/S0962492906360011Search in Google Scholar

12 A. Gloria, S. Neukamm and F. Otto, Quantification of ergodicity in stochastic homogenization: Optimal bounds via spectral gap on Glauber dynamics, Invent. Math. 199 (2015), 2, 455–515. 10.1007/s00222-014-0518-zSearch in Google Scholar

13 A. Gloria and F. Otto, Quantitative estimates on the periodic approximation of the corrector in stochastic homogenization, ESAIM Proc. 48 (2015), 80–97. 10.1051/proc/201448003Search in Google Scholar

14 V. V. Jikov, S. M. Kozlov and O. A. Oleĭnik, Homogenization of Differential Operators and Integral Functionals, Springer, Berlin, 1994. 10.1007/978-3-642-84659-5Search in Google Scholar

15 S. M. Kozlov, Averaging of random structures, USSR Dokl. 241 (1978), 5, 1016–1019. Search in Google Scholar

16 C. Le Bris, Some numerical approaches for “weakly” random homogenization, Numerical Mathematics and Advanced Applications (ENUMATH 2009), Springer, Berlin (2010), 29–45. 10.1007/978-3-642-11795-4_3Search in Google Scholar

17 F. Legoll and W. Minvielle, A control variate approach based on a defect-type theory for variance reduction in stochastic homogenization, Multiscale Model. Simul. 13 (2015), 2, 519–550. 10.1137/140980120Search in Google Scholar

18 F. Legoll and W. Minvielle, Variance reduction using antithetic variables for a nonlinear convex stochastic homogenization problem, Discrete Contin. Dyn. Syst. Ser. S 8 (2015), 1, 1–27. 10.3934/dcdss.2015.8.1Search in Google Scholar

19 W. Minvielle, Quelques problèmes liés à l'erreur statistique en homogénéisation stochastique, Ph.D. thesis, Université Paris-Est, 2015; Search in Google Scholar

20 J. Nolen, Normal approximation for a random elliptic equation, Probab. Theory Related Fields 159 (2014), 3–4, 661–700. 10.1007/s00440-013-0517-9Search in Google Scholar

21 G. C. Papanicolaou and S. R. S. Varadhan, Boundary value problems with rapidly oscillating random coefficients, Random Fields. Rigorous Results in Statistical Mechanics and Quantum Field Theory (Esztergom 1979), Colloq. Math. Soc. János Bolyai 10, North-Holland, Amsterdam (1981), 835–873. Search in Google Scholar

22 A. N. Shiryayev, Probability, Grad. Texts in Math. 95, Springer, New York, 1984. 10.1007/978-1-4899-0018-0Search in Google Scholar

23 J. von Pezold, A. Dick, M. Friák and J. Neugebauer, Generation and performance of special quasirandom structures for studying the elastic properties of random alloys: Application to Al–Ti, Phys. Rev. B 81 (2010), 9, Article ID 094203. 10.1103/PhysRevB.81.094203Search in Google Scholar

24 S.-H. Wei, L. G. Ferreira, J. E. Bernard and A. Zunger, Electronic properties of random alloys: Special quasirandom structures, Phys. Rev. B 42 (1990), 15, Article ID 9622. 10.1103/PhysRevB.42.9622Search in Google Scholar PubMed

25 A. Zunger, S.-H. Wei, L. G. Ferreira and J. E. Bernard, Special quasirandom structures, Phys. Rev. Lett. 65 (1990), 3, Article ID 353. 10.1103/PhysRevLett.65.353Search in Google Scholar PubMed

Received: 2015-9-4
Accepted: 2016-2-5
Published Online: 2016-2-17
Published in Print: 2016-3-1

© 2016 by De Gruyter

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