Approximations of effective coefficients in stochastic homogenization

Alain Bourgeat; Andrey Piatnitski

Annales de l'I.H.P. Probabilités et statistiques (2004)

  • Volume: 40, Issue: 2, page 153-165
  • ISSN: 0246-0203

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Bourgeat, Alain, and Piatnitski, Andrey. "Approximations of effective coefficients in stochastic homogenization." Annales de l'I.H.P. Probabilités et statistiques 40.2 (2004): 153-165. <http://eudml.org/doc/77804>.

@article{Bourgeat2004,
author = {Bourgeat, Alain, Piatnitski, Andrey},
journal = {Annales de l'I.H.P. Probabilités et statistiques},
keywords = {Random operator; Volume averaging},
language = {eng},
number = {2},
pages = {153-165},
publisher = {Elsevier},
title = {Approximations of effective coefficients in stochastic homogenization},
url = {http://eudml.org/doc/77804},
volume = {40},
year = {2004},
}

TY - JOUR
AU - Bourgeat, Alain
AU - Piatnitski, Andrey
TI - Approximations of effective coefficients in stochastic homogenization
JO - Annales de l'I.H.P. Probabilités et statistiques
PY - 2004
PB - Elsevier
VL - 40
IS - 2
SP - 153
EP - 165
LA - eng
KW - Random operator; Volume averaging
UR - http://eudml.org/doc/77804
ER -

References

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  1. [1] A. Badea, A. Bourgeat, Modeling and Computation in Environmental Sciences, in: Notes in Numerical Fluids Mechanics, vol. 59, 1997, pp. 13-24. Zbl0895.76064MR1487420
  2. [2] J. Bear, Dynamics of Fluids in Porous Media, Elsevier, New York, 1972. Zbl1191.76001
  3. [3] G. Dal Maso, L. Modica, Nonlinear stochastic homogenization and ergodic theory, J. Reine Angew. Math.368 (1986) 28-42. Zbl0582.60034MR850613
  4. [4] A. Bensoussan, J.-L. Lions, G. Papanicolaou, Asymptotic Analysis of Periodic Structures, North Holland, Amsterdam, 1978. Zbl0404.35001MR503330
  5. [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. II306 (1988) 463-466. Zbl0637.45008MR979154
  6. [6] A. Bourgeat, A. Mikelic, S. Wright, Stochastic two-scale convergence in the mean and applications, J. Reine Angew. Math.456 (1994) 19-51. Zbl0808.60056MR1301450
  7. [7] A. Bourgeat, S.M. Kozlov, A. Mikelić, Effective equations of two-phase flow in random media, Cal. Var. PDE8 (1998) 1-31. Zbl0824.35099
  8. [8] N. Dunford, J.T. Schwartz, Linear Operators, Part I, General Theory, Interscience, 1963. Zbl0084.10402MR1009162
  9. [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. [10] D. Gilbarg, N.S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer-Verlag, 1977. Zbl0361.35003MR473443
  11. [11] R. Hill, Elastic properties of reinforced solids: some theoretical principles, J. Mech. Phys. Solids11 (1963) 357-372. Zbl0114.15804
  12. [12] Hornung U. (Ed.), Homogenization and Porous Media, IAM, vol. 6, Springer-Verlag, Berlin, 1997. Zbl0872.35002MR1434315
  13. [13] V.V. Jikov, S.M. Kozlov, O.A. Oleinik, Homogenization of Differential Operators and Integral Functionals, Springer-Verlag, New York, 1994. Zbl0838.35001MR1329546
  14. [14] S.M. Kozlov, The averaging of random operators, Math. USSR Sb.109 (1979) 188-202. Zbl0415.60059MR542557
  15. [15] S.M. Kozlov, Geometric aspects of homogenization, Russian Math. Surv.44 (1989) 91-144. Zbl0706.49029MR998362
  16. [16] N. Krylov, Controlled Diffusion Processes, Application of Mathematics, vol. 14, Springer-Verlag, New York, 1980. Zbl0459.93002MR601776
  17. [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. Zbl0965.74003MR1758600
  18. [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. Zbl0597.49010MR858358
  19. [19] H. Owhadi, Approximation of effective conductivity of ergodic media by periodization, Probab. Theory Related Fields125 (2003) 225-258. Zbl1040.60025MR1961343
  20. [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 Bolyai27 (1981) 835-873. Zbl0499.60059MR712714
  21. [21] K. Sab, On the homogenization and the simulation of random materials, Eur. J. Mech., A/Solids11 (5) (1992) 585-607. Zbl0766.73008MR1184222
  22. [22] A.V. Pozhidaev, V.V. Yurinskii, On the error of averaging of symmetric elliptic systems, Math. USSR-Izv.35 (1) (1990) 183-201. Zbl0702.35254MR1018751
  23. [23] Ph. Renard, G. DeMarsily, Calculating equivalent permeability: A review, Adv. Water Resour.20 (1997) 253-278. 
  24. [24] V.V. Yurinski, Averaging of symmetric diffusion in random medium, Sibirskii Mat. Zh.27 (4) (1986) 167-180. Zbl0614.60051MR867870

Citations in EuDML Documents

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  1. Frédéric Legoll, Florian Thomines, On a variant of random homogenization theory: convergence of the residual process and approximation of the homogenized coefficients
  2. Pierre-Louis Lions, Panagiotis E. Souganidis, Homogenization of degenerate second-order PDE in periodic and almost periodic environments and applications
  3. Antoine Gloria, Numerical approximation of effective coefficients in stochastic homogenization of discrete elliptic equations
  4. Daniel Boivin, Tail estimates for homogenization theorems in random media
  5. Antoine Gloria, Numerical approximation of effective coefficients in stochastic homogenization of discrete elliptic equations
  6. Jean-Christophe Mourrat, Variance decay for functionals of the environment viewed by the particle

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