Foundations of Computational Mathematics

, Volume 19, Issue 2, pp 435–483 | Cite as

Efficient Methods for the Estimation of Homogenized Coefficients

  • J.-C. MourratEmail author


The main goal of this paper is to define and study new methods for the computation of effective coefficients in the homogenization of divergence-form operators with random coefficients. The methods introduced here are proved to have optimal computational complexity and are shown numerically to display small constant prefactors. In the spirit of multiscale methods, the main idea is to rely on a progressive coarsening of the problem, which we implement via a generalization of the Green–Kubo formula. The technique can be applied more generally to compute the effective diffusivity of any additive functional of a Markov process. In this broader context, we also discuss the alternative possibility of using Monte Carlo sampling and show how a simple one-step extrapolation can considerably improve the performance of this alternative method.


Homogenization Multiscale methods 

Mathematics Subject Classification

82B80 35B27 82B28 



I would like to thank Josselin Garnier for an inspiring talk which motivated me to revisit this problem, Tony Lelièvre for his helpful feedback and Harmen Stoppels for his precious help with the Julia language. This work has been partially supported by the ANR Grant LSD (ANR-15-CE40-0020-03).


  1. 1.
    A. Abdulle, W. E, B. Engquist, and E. Vanden-Eijnden. The heterogeneous multiscale method. Acta Numer., 21:1–87, 2012.Google Scholar
  2. 2.
    Y. Almog. Averaging of dilute random media: a rigorous proof of the Clausius-Mossotti formula. Arch. Ration. Mech. Anal., 207(3):785–812, 2013.MathSciNetzbMATHGoogle Scholar
  3. 3.
    Y. Almog. The Clausius-Mossotti formula in a dilute random medium with fixed volume fraction. Multiscale Model. Simul., 12(4):1777–1799, 2014.MathSciNetzbMATHGoogle Scholar
  4. 4.
    A. Anantharaman and C. Le Bris. A numerical approach related to defect-type theories for some weakly random problems in homogenization. Multiscale Model. Simul., 9(2):513–544, 2011.MathSciNetzbMATHGoogle Scholar
  5. 5.
    A. Anantharaman and C. Le Bris. Elements of mathematical foundations for numerical approaches for weakly random homogenization problems. Commun. Comput. Phys., 11(4):1103–1143, 2012.MathSciNetzbMATHGoogle Scholar
  6. 6.
    D. Arjmand and O. Runborg. A time dependent approach for removing the cell boundary error in elliptic homogenization problems. J. Comput. Phys., 314:206–227, 2016.MathSciNetzbMATHGoogle Scholar
  7. 7.
    S. Armstrong, A. Hannukainen, T. Kuusi, and J. C. Mourrat. An iterative method for elliptic problems with rapidly oscillating coefficients, preprint, arXiv:1803.03551.
  8. 8.
    S. Armstrong, T. Kuusi, and J.-C. Mourrat. Quantitative stochastic homogenization and large-scale regularity. Preliminary version available at (2018).
  9. 9.
    S. Armstrong, T. Kuusi, and J.-C. Mourrat. Mesoscopic higher regularity and subadditivity in elliptic homogenization. Comm. Math. Phys., 347(2):315–361, 2016.MathSciNetzbMATHGoogle Scholar
  10. 10.
    S. Armstrong, T. Kuusi, and J.-C. Mourrat. The additive structure of elliptic homogenization. Invent. Math., 208(3):999–1154, 2017.MathSciNetzbMATHGoogle Scholar
  11. 11.
    S. N. Armstrong and J.-C. Mourrat. Lipschitz regularity for elliptic equations with random coefficients. Arch. Ration. Mech. Anal., 219(1):255–348, 2016.MathSciNetzbMATHGoogle Scholar
  12. 12.
    S. N. Armstrong and C. K. Smart. Quantitative stochastic homogenization of convex integral functionals. Ann. Sci. Éc. Norm. Supér. (4), 49(2):423–481, 2016.Google Scholar
  13. 13.
    M. T. Barlow, A. A. Járai, T. Kumagai, and G. Slade. Random walk on the incipient infinite cluster for oriented percolation in high dimensions. Comm. Math. Phys., 278(2):385–431, 2008.MathSciNetzbMATHGoogle Scholar
  14. 14.
    G. Ben Arous, M. Cabezas, and A. Fribergh. Scaling limit for the ant in high-dimensional labyrinths, preprint, arXiv:1609.03977.
  15. 15.
    G. Ben Arous, M. Cabezas, and A. Fribergh. Scaling limit for the ant in a simple labyrinth, preprint, arXiv:1609.03980.
  16. 16.
    L. Berlyand and V. Mityushev. Generalized Clausius-Mossotti formula for random composite with circular fibers. J. Statist. Phys., 102(1-2):115–145, 2001.MathSciNetzbMATHGoogle Scholar
  17. 17.
    X. Blanc and C. Le Bris. Improving on computation of homogenized coefficients in the periodic and quasi-periodic settings. Netw. Heterog. Media, 5(1):1–29, 2010.MathSciNetzbMATHGoogle Scholar
  18. 18.
    A. Brandt. Multiscale scientific computation: review 2001. In Multiscale and multiresolution methods, volume 20 of Lect. Notes Comput. Sci. Eng., pages 3–95. Springer, Berlin, 2002.Google Scholar
  19. 19.
    M. Damron, J. Hanson, and P. Sosoe. Subdiffusivity of random walk on the 2D invasion percolation cluster. Stochastic Process. Appl., 123(9):3588–3621, 2013.MathSciNetzbMATHGoogle Scholar
  20. 20.
    D. Dolgopyat. Limit theorems for partially hyperbolic systems. Trans. Amer. Math. Soc., 356(4):1637–1689, 2004.MathSciNetzbMATHGoogle Scholar
  21. 21.
    M. Duerinckx and A. Gloria. Analyticity of homogenized coefficients under Bernoulli perturbations and the Clausius-Mossotti formulas. Arch. Ration. Mech. Anal., 220(1):297–361, 2016.MathSciNetzbMATHGoogle Scholar
  22. 22.
    Y. Efendiev and T. Y. Hou. Multiscale finite element methods, volume 4 of Surveys and Tutorials in the Applied Mathematical Sciences. Springer, New York, 2009.zbMATHGoogle Scholar
  23. 23.
    A.-C. Egloffe, A. Gloria, J.-C. Mourrat, and T. N. Nguyen. Random walk in random environment, corrector equation and homogenized coefficients: from theory to numerics, back and forth. IMA J. Numer. Anal., 35(2):499–545, 2015.MathSciNetzbMATHGoogle Scholar
  24. 24.
    A. Gloria. Numerical approximation of effective coefficients in stochastic homogenization of discrete elliptic equations. ESAIM Math. Model. Numer. Anal., 46(1):1–38, 2012.MathSciNetzbMATHGoogle Scholar
  25. 25.
    A. Gloria and Z. Habibi. Reduction in the resonance error in numerical homogenization II: Correctors and extrapolation. Found. Comput. Math., 16(1):217–296, 2016.MathSciNetzbMATHGoogle Scholar
  26. 26.
    A. Gloria and J.-C. Mourrat. Spectral measure and approximation of homogenized coefficients. Probab. Theory Related Fields, 154(1-2):287–326, 2012.MathSciNetzbMATHGoogle Scholar
  27. 27.
    A. Gloria and J.-C. Mourrat. Quantitative version of the Kipnis-Varadhan theorem and Monte Carlo approximation of homogenized coefficients. Ann. Appl. Probab., 23(4):1544–1583, 2013.MathSciNetzbMATHGoogle Scholar
  28. 28.
    A. Gloria, S. Neukamm, and F. Otto. Quantification of ergodicity in stochastic homogenization: optimal bounds via spectral gap on Glauber dynamics. Invent. Math., 199(2):455–515, 2015.MathSciNetzbMATHGoogle Scholar
  29. 29.
    A. Gloria, S. Neukamm, and F. Otto. A regularity theory for random elliptic operators, preprint, arXiv:1409.2678.
  30. 30.
    A. Gloria and F. Otto. An optimal variance estimate in stochastic homogenization of discrete elliptic equations. Ann. Probab., 39(3):779–856, 2011.MathSciNetzbMATHGoogle Scholar
  31. 31.
    A. Gloria and F. Otto. An optimal error estimate in stochastic homogenization of discrete elliptic equations. Ann. Appl. Probab., 22(1):1–28, 2012.MathSciNetzbMATHGoogle Scholar
  32. 32.
    A. Gloria and F. Otto. The corrector in stochastic homogenization: optimal rates, stochastic integrability, and fluctuations, preprint, arXiv:1510.08290.
  33. 33.
    P. Henning and D. Peterseim. Oversampling for the multiscale finite element method. Multiscale Model. Simul., 11(4):1149–1175, 2013.MathSciNetzbMATHGoogle Scholar
  34. 34.
    T. Y. Hou and X.-H. Wu. A multiscale finite element method for elliptic problems in composite materials and porous media. J. Comput. Phys., 134(1):169–189, 1997.MathSciNetzbMATHGoogle Scholar
  35. 35.
    T. Y. Hou, X.-H. Wu, and Z. Cai. Convergence of a multiscale finite element method for elliptic problems with rapidly oscillating coefficients. Math. Comp., 68(227):913–943, 1999.MathSciNetzbMATHGoogle Scholar
  36. 36.
    B. D. Hughes. Conduction and diffusion in percolating systems. In Encyclopedia of complexity and systems science, pages 1395–1424. Springer, 2009.Google Scholar
  37. 37.
    A. A. Járai and A. Nachmias. Electrical resistance of the low dimensional critical branching random walk. Comm. Math. Phys., 331(1):67–109, 2014.MathSciNetzbMATHGoogle Scholar
  38. 38.
    H. Kesten. Subdiffusive behavior of random walk on a random cluster. Ann. Inst. H. Poincaré Probab. Statist., 22(4):425–487, 1986.MathSciNetzbMATHGoogle Scholar
  39. 39.
    I. G. Kevrekidis, C. W. Gear, J. M. Hyman, P. G. Kevrekidis, O. Runborg, and C. Theodoropoulos. Equation-free, coarse-grained multiscale computation: enabling microscopic simulators to perform system-level analysis. Commun. Math. Sci., 1(4):715–762, 2003.MathSciNetzbMATHGoogle Scholar
  40. 40.
    C. Kipnis and S. R. S. Varadhan. Central limit theorem for additive functionals of reversible Markov processes and applications to simple exclusions. Comm. Math. Phys., 104(1):1–19, 1986.MathSciNetzbMATHGoogle Scholar
  41. 41.
    T. Komorowski, C. Landim, and S. Olla. Fluctuations in Markov processes, volume 345 of Grundlehren der Mathematischen Wissenschaften. Springer, Heidelberg, 2012.zbMATHGoogle Scholar
  42. 42.
    S. M. Kozlov. Geometric aspects of averaging. Uspekhi Mat. Nauk, 44(2(266)):79–120, 1989.Google Scholar
  43. 43.
    G. Kozma and A. Nachmias. The Alexander-Orbach conjecture holds in high dimensions. Invent. Math., 178(3):635–654, 2009.MathSciNetzbMATHGoogle Scholar
  44. 44.
    T. Kumagai. Random walks on disordered media and their scaling limits, volume 2101 of Lecture Notes in Mathematics. Springer, Cham, 2014.zbMATHGoogle Scholar
  45. 45.
    C. Le Bris and F. Legoll. Examples of computational approaches for elliptic, possibly multiscale PDEs with random inputs. J. Comput. Phys., 328:455–473, 2017.MathSciNetzbMATHGoogle Scholar
  46. 46.
    T. M. Liggett. Stochastic interacting systems: contact, voter and exclusion processes, volume 324 of Grundlehren der Mathematischen Wissenschaften. Springer-Verlag, Berlin, 1999.zbMATHGoogle Scholar
  47. 47.
    T. M. Liggett. Continuous time Markov processes, volume 113 of Graduate Studies in Mathematics. American Mathematical Society, Providence, RI, 2010.zbMATHGoogle Scholar
  48. 48.
    C. Liverani. Central limit theorem for deterministic systems. In International Conference on Dynamical Systems (Montevideo, 1995), volume 362 of Pitman Res. Notes Math. Ser., pages 56–75. Longman, Harlow, 1996.Google Scholar
  49. 49.
    A. Målqvist and D. Peterseim. Localization of elliptic multiscale problems. Math. Comp., 83(290):2583–2603, 2014.MathSciNetzbMATHGoogle Scholar
  50. 50.
    D. Marahrens and F. Otto. Annealed estimates on the Green function. Probab. Theory Related Fields, 163(3-4):527–573, 2015.MathSciNetzbMATHGoogle Scholar
  51. 51.
    J. C. Maxwell. Medium in which small spheres are uniformly disseminated. A treatise on electricity and magnetism, part II, chapter IX, article 314. Clarendon Press, 3d ed., 1891.Google Scholar
  52. 52.
    I. Melbourne and M. Nicol. Almost sure invariance principle for nonuniformly hyperbolic systems. Comm. Math. Phys., 260(1):131–146, 2005.MathSciNetzbMATHGoogle Scholar
  53. 53.
    J.-C. Mourrat. Variance decay for functionals of the environment viewed by the particle. Ann. Inst. Henri Poincaré Probab. Stat., 47(1):294–327, 2011.MathSciNetzbMATHGoogle Scholar
  54. 54.
    J.-C. Mourrat. First-order expansion of homogenized coefficients under Bernoulli perturbations. J. Math. Pures Appl. (9), 103(1):68–101, 2015.Google Scholar
  55. 55.
    G. Papanicolaou and S. R. S. Varadhan. Ornstein-Uhlenbeck process in a random potential. Comm. Pure Appl. Math., 38(6):819–834, 1985.MathSciNetzbMATHGoogle Scholar
  56. 56.
    G. C. Papanicolaou. Diffusion in random media. In Surveys in applied mathematics, Vol. 1, pages 205–253. Plenum, New York, 1995.Google Scholar
  57. 57.
    V. V. Petrov. Limit theorems of probability theory, volume 4 of Oxford Studies in Probability. The Clarendon Press, Oxford University Press, New York, 1995.zbMATHGoogle Scholar
  58. 58.
    A. Quarteroni, R. Sacco, and F. Saleri. Numerical mathematics, volume 37 of Texts in Applied Mathematics. Springer-Verlag, New York, 2000.zbMATHGoogle Scholar
  59. 59.
    J. W. Strutt, 3d Baron Rayleigh. On the influence of obstacles arranged in rectangular order upon the properties of a medium. Philos. mag., 34(211):481–502, 1892.Google Scholar
  60. 60.
    X. Yue and W. E. The local microscale problem in the multiscale modeling of strongly heterogeneous media: effects of boundary conditions and cell size. J. Comput. Phys., 222(2):556–572, 2007.Google Scholar

Copyright information

© SFoCM 2018

Authors and Affiliations

  1. 1.DMA, Ecole normale supérieure, CNRSPSL Research UniversityParisFrance

Personalised recommendations