On a variant of random homogenization theory: convergence of the residual process and approximation of the homogenized coefficients

Frédéric Legoll; Florian Thomines

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique (2014)

  • Volume: 48, Issue: 2, page 347-386
  • ISSN: 0764-583X

Abstract

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We consider the variant of stochastic homogenization theory introduced in [X. Blanc, C. Le Bris and P.-L. Lions, C. R. Acad. Sci. Série I 343 (2006) 717–724.; X. Blanc, C. Le Bris and P.-L. Lions, J. Math. Pures Appl. 88 (2007) 34–63.]. The equation under consideration is a standard linear elliptic equation in divergence form, where the highly oscillatory coefficient is the composition of a periodic matrix with a stochastic diffeomorphism. The homogenized limit of this problem has been identified in [X. Blanc, C. Le Bris and P.-L. Lions, C. R. Acad. Sci. Série I 343 (2006) 717–724.]. We first establish, in the one-dimensional case, a convergence result (with an explicit rate) on the residual process, defined as the difference between the solution to the highly oscillatory problem and the solution to the homogenized problem. We next return to the multidimensional situation. As often in random homogenization, the homogenized matrix is defined from a so-called corrector function, which is the solution to a problem set on the entire space. We describe and prove the almost sure convergence of an approximation strategy based on truncated versions of the corrector problem.

How to cite

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Legoll, Frédéric, and Thomines, Florian. "On a variant of random homogenization theory: convergence of the residual process and approximation of the homogenized coefficients." ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique 48.2 (2014): 347-386. <http://eudml.org/doc/273324>.

@article{Legoll2014,
abstract = {We consider the variant of stochastic homogenization theory introduced in [X. Blanc, C. Le Bris and P.-L. Lions, C. R. Acad. Sci. Série I 343 (2006) 717–724.; X. Blanc, C. Le Bris and P.-L. Lions, J. Math. Pures Appl. 88 (2007) 34–63.]. The equation under consideration is a standard linear elliptic equation in divergence form, where the highly oscillatory coefficient is the composition of a periodic matrix with a stochastic diffeomorphism. The homogenized limit of this problem has been identified in [X. Blanc, C. Le Bris and P.-L. Lions, C. R. Acad. Sci. Série I 343 (2006) 717–724.]. We first establish, in the one-dimensional case, a convergence result (with an explicit rate) on the residual process, defined as the difference between the solution to the highly oscillatory problem and the solution to the homogenized problem. We next return to the multidimensional situation. As often in random homogenization, the homogenized matrix is defined from a so-called corrector function, which is the solution to a problem set on the entire space. We describe and prove the almost sure convergence of an approximation strategy based on truncated versions of the corrector problem.},
author = {Legoll, Frédéric, Thomines, Florian},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique},
keywords = {stochastic homogenization; random stationary diffeomorphisms; central limit result; approximation of homogenized coefficients; central limit theorem},
language = {eng},
number = {2},
pages = {347-386},
publisher = {EDP-Sciences},
title = {On a variant of random homogenization theory: convergence of the residual process and approximation of the homogenized coefficients},
url = {http://eudml.org/doc/273324},
volume = {48},
year = {2014},
}

TY - JOUR
AU - Legoll, Frédéric
AU - Thomines, Florian
TI - On a variant of random homogenization theory: convergence of the residual process and approximation of the homogenized coefficients
JO - ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique
PY - 2014
PB - EDP-Sciences
VL - 48
IS - 2
SP - 347
EP - 386
AB - We consider the variant of stochastic homogenization theory introduced in [X. Blanc, C. Le Bris and P.-L. Lions, C. R. Acad. Sci. Série I 343 (2006) 717–724.; X. Blanc, C. Le Bris and P.-L. Lions, J. Math. Pures Appl. 88 (2007) 34–63.]. The equation under consideration is a standard linear elliptic equation in divergence form, where the highly oscillatory coefficient is the composition of a periodic matrix with a stochastic diffeomorphism. The homogenized limit of this problem has been identified in [X. Blanc, C. Le Bris and P.-L. Lions, C. R. Acad. Sci. Série I 343 (2006) 717–724.]. We first establish, in the one-dimensional case, a convergence result (with an explicit rate) on the residual process, defined as the difference between the solution to the highly oscillatory problem and the solution to the homogenized problem. We next return to the multidimensional situation. As often in random homogenization, the homogenized matrix is defined from a so-called corrector function, which is the solution to a problem set on the entire space. We describe and prove the almost sure convergence of an approximation strategy based on truncated versions of the corrector problem.
LA - eng
KW - stochastic homogenization; random stationary diffeomorphisms; central limit result; approximation of homogenized coefficients; central limit theorem
UR - http://eudml.org/doc/273324
ER -

References

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  1. [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. In vol. 22 of Lect. Not. Ser., edited by W. Bao and Q. Du. Institute for Mathematical Sciences, National University of Singapore, (2011) 197–272. MR2895600
  2. [2] G. Bal, J. Garnier, Y. Gu and W. Jing, Corrector theory for elliptic equations with long-range correlated random potential. Asymptot. Anal.77 (2012) 123–145. Zbl1259.35235MR2977330
  3. [3] G. Bal, J. Garnier, S. Motsch and V. Perrier, Random integrals and correctors in homogenization. Asymptot. Anal.59 (2008) 1–26. Zbl1157.34048MR2435670
  4. [4] J.M. Ball, Convexity conditions and existence theorems in nonlinear elasticity. Arch. Ration. Mech. Anal.63 (1977) 337–403. Zbl0368.73040MR475169
  5. [5] A. Bensoussan, J.-L. Lions and G. Papanicolaou, Asymptotic analysis for periodic structures, vol. 5 of Studi. Math. Appl. North-Holland Publishing Co., Amsterdam-New York (1978). Zbl0404.35001MR503330
  6. [6] P. Billingsley, Convergence of Probability Measures. John Wiley & Sons Inc (1968). Zbl0944.60003MR233396
  7. [7] X. Blanc, C. Le Bris and P.-L. Lions, Une variante de la théorie de l’homogénéisation stochastique des opérateurs elliptiques [A variant of stochastic homogenization theory for elliptic operators]. C. R. Acad. Sci. Série I343 (2006) 717–724. Zbl1103.35014MR2284699
  8. [8] X. Blanc, C. Le Bris and P.-L. Lions, Stochastic homogenization and random lattices. J. Math. Pures Appl.88 (2007) 34–63. Zbl1129.60055MR2334772
  9. [9] A. Bourgeat and A. Piatnitski, Estimates in probability of the residual between the random and the homogenized solutions of one-dimensional second-order operator. Asymptot. Anal.21 (1999) 303–315. Zbl0960.60057MR1728027
  10. [10] A. Bourgeat and A. Piatnitski, Approximation of effective coefficients in stochastic homogenization. Ann Inst. Henri Poincaré – PR40 (2004) 153–165. Zbl1058.35023MR2044813
  11. [11] D. Cioranescu and P. Donato, An introduction to homogenization, vol. 17 of Oxford Lect. Ser. Math. Appl. Oxford University Press, New York (1999). Zbl0939.35001MR1765047
  12. [12] R. Costaouec, C. Le Bris and F. Legoll, Approximation numérique d’une classe de problèmes en homogénéisation stochastique [Numerical approximation of a class of problems in stochastic homogenization]. C. R. Acad. Sci. Série I348 (2010) 99–103. Zbl1180.65166MR2586753
  13. [13] B. Engquist and P.E. Souganidis, Asymptotic and numerical homogenization. Acta Numerica17 (2008) 147–190. Zbl1179.65142MR2436011
  14. [14] D. Henao and C. Mora-Corral, Invertibility and weak continuity of the determinant for the modelling of cavitation and fracture in nonlinear elasticity. Arch. Ration. Mech. Anal.197 (2010) 619–655. Zbl1248.74006MR2660521
  15. [15] V.V. Jikov, S.M. Kozlov and O.A. Oleinik, Homogenization of differential operators and integral functionals. Springer-Verlag (1994). Zbl0801.35001MR1329546
  16. [16] U. Krengel, Ergodic theorems, vol. 6 of De Gruyter Studies in Mathematics. De Gruyter (1985). Zbl0575.28009MR797411
  17. [17] G.C. Papanicolaou and S.R.S. Varadhan, Boundary value problems with rapidly oscillating random coefficients, in Proc. Colloq. on Random Fields: Rigorous Results in Statistical Mechanics and Quantum Field Theory (1979). In vol. 10 of Colloquia Mathematica Societatis Janos Bolyai, edited by J. Fritz, J.L. Lebaritz and D. Szasz. North-Holland (1981) 835–873. Zbl0499.60059MR712714
  18. [18] A.N. Shiryaev, Probability, vol. 95 of Graduate Texts in Mathematics. Springer (1984). Zbl0536.60001
  19. [19] A.A. Tempel’man, Ergodic theorems for general dynamical systems. Trudy Moskov. Mat. Obsc.26 (1972) 94–132. Zbl0281.28008MR374388
  20. [20] V.V. Yurinskii, Averaging of symmetric diffusion in random medium. Sibirskii Mat. Zh.27 (1986) 167–180. Zbl0614.60051MR867870

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