Approximations of effective coefficients in stochastic homogenization
Annales de l'I.H.P. Probabilités et statistiques, Volume 40 (2004) no. 2, p. 153-165
@article{AIHPB_2004__40_2_153_0,
     author = {Bourgeat, Alain and Piatnitski, Andrey},
     title = {Approximations of effective coefficients in stochastic homogenization},
     journal = {Annales de l'I.H.P. Probabilit\'es et statistiques},
     publisher = {Elsevier},
     volume = {40},
     number = {2},
     year = {2004},
     pages = {153-165},
     doi = {10.1016/j.anihpb.2003.07.003},
     zbl = {1058.35023},
     mrnumber = {2044813},
     language = {en},
     url = {http://www.numdam.org/item/AIHPB_2004__40_2_153_0}
}
Bourgeat, Alain; Piatnitski, Andrey. Approximations of effective coefficients in stochastic homogenization. Annales de l'I.H.P. Probabilités et statistiques, Volume 40 (2004) no. 2, pp. 153-165. doi : 10.1016/j.anihpb.2003.07.003. http://www.numdam.org/item/AIHPB_2004__40_2_153_0/

[1] A. Badea, A. Bourgeat, Modeling and Computation in Environmental Sciences, in: Notes in Numerical Fluids Mechanics, vol. 59, 1997, pp. 13-24. | MR 1487420 | Zbl 0895.76064

[2] J. Bear, Dynamics of Fluids in Porous Media, Elsevier, New York, 1972.

[3] G. Dal Maso, L. Modica, Nonlinear stochastic homogenization and ergodic theory, J. Reine Angew. Math. 368 (1986) 28-42. | MR 850613 | Zbl 0582.60034

[4] A. Bensoussan, J.-L. Lions, G. Papanicolaou, Asymptotic Analysis of Periodic Structures, North Holland, Amsterdam, 1978. | MR 503330 | Zbl 0404.35001

[5] A. Bourgeat, M. Quintard, S. Whitaker, Eléments de comparaison entre la méthode d'homogénéisation et la méthode de prise de moyenne avec fermeture, C. R. Acad. Sci. Paris Sér. II 306 (1988) 463-466. | MR 979154 | Zbl 0637.45008

[6] A. Bourgeat, A. Mikelic, S. Wright, Stochastic two-scale convergence in the mean and applications, J. Reine Angew. Math. 456 (1994) 19-51. | MR 1301450 | Zbl 0808.60056

[7] A. Bourgeat, S.M. Kozlov, A. Mikelić, Effective equations of two-phase flow in random media, Cal. Var. PDE 8 (1998) 1-31.

[8] N. Dunford, J.T. Schwartz, Linear Operators, Part I, General Theory, Interscience, 1963. | MR 1009162 | Zbl 0084.10402

[9] L.J. Durlofski, Representation of grid block permeability in coarse scale models of randomly heterogeneous porous media, Water Resources Res. 28 (1992) 1791-1800.

[10] D. Gilbarg, N.S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer-Verlag, 1977. | MR 473443 | Zbl 0361.35003

[11] R. Hill, Elastic properties of reinforced solids: some theoretical principles, J. Mech. Phys. Solids 11 (1963) 357-372. | Zbl 0114.15804

[12] Hornung U. (Ed.), Homogenization and Porous Media, IAM, vol. 6, Springer-Verlag, Berlin, 1997. | MR 1434315 | Zbl 0872.35002

[13] V.V. Jikov, S.M. Kozlov, O.A. Oleinik, Homogenization of Differential Operators and Integral Functionals, Springer-Verlag, New York, 1994. | MR 1329546 | Zbl 0838.35001

[14] S.M. Kozlov, The averaging of random operators, Math. USSR Sb. 109 (1979) 188-202. | MR 542557 | Zbl 0415.60059

[15] S.M. Kozlov, Geometric aspects of homogenization, Russian Math. Surv. 44 (1989) 91-144. | MR 998362 | Zbl 0706.49029

[16] N. Krylov, Controlled Diffusion Processes, Application of Mathematics, vol. 14, Springer-Verlag, New York, 1980. | MR 601776 | Zbl 0459.93002

[17] T. Lewinski, J. Telega, Plates, Laminates and Shells. Asymptotic Analysis and Homogenization, Series on Advances in Mathematics for Applied Sciences, vol. 52, World Scientific, River Edge, NJ, 2000. | MR 1758600 | Zbl 0965.74003

[18] L. Modica, Stochastic Homogenization and Ergodic Theory, in: Conti , De Giorgi , Gianessi (Eds.), Proceedings ERICE 1984, Lecture Notes in Mathematics, vol. 1190, Springer-Verlag, 1986, Chapter 15. | MR 858358 | Zbl 0597.49010

[19] H. Owhadi, Approximation of effective conductivity of ergodic media by periodization, Probab. Theory Related Fields 125 (2003) 225-258. | MR 1961343 | Zbl 1040.60025

[20] G. Papanicolaou, 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. Janos Bolyai 27 (1981) 835-873. | MR 712714 | Zbl 0499.60059

[21] K. Sab, On the homogenization and the simulation of random materials, Eur. J. Mech., A/Solids 11 (5) (1992) 585-607. | MR 1184222 | Zbl 0766.73008

[22] A.V. Pozhidaev, V.V. Yurinskii, On the error of averaging of symmetric elliptic systems, Math. USSR-Izv. 35 (1) (1990) 183-201. | MR 1018751 | Zbl 0702.35254

[23] Ph. Renard, G. Demarsily, Calculating equivalent permeability: A review, Adv. Water Resour. 20 (1997) 253-278.

[24] V.V. Yurinski, Averaging of symmetric diffusion in random medium, Sibirskii Mat. Zh. 27 (4) (1986) 167-180. | MR 867870 | Zbl 0614.60051