Skip to main content
Log in

Efficient Methods for the Estimation of Homogenized Coefficients

  • Published:
Foundations of Computational Mathematics Aims and scope Submit manuscript

Abstract

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.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6

Similar content being viewed by others

Notes

  1. In order to estimate the slope of the regression line of the set of blue dots in [24, Figure 14], I first drew the straight line going through the extremal blue dots and observed that all the other dots are very close to this line. I then measured the coordinates of the extremal dots to be \((3.59, -\,0.62)\) and \((9.89,-\,3.06)\), respectively, which yields that the dx / dy slope of the line is about \(-\,2.58\). Moreover, the seven dots closest to the leftmost blue point are below this line, while the three dots closest to the rightmost blue point are above this line, and thus essentially any other reasonable choice of pair of points to draw a line and measure the slope from would yield a larger absolute slope.

References

  1. A. Abdulle, W. E, B. Engquist, and E. Vanden-Eijnden. The heterogeneous multiscale method. Acta Numer., 21:1–87, 2012.

  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.

    Article  MathSciNet  MATH  Google Scholar 

  3. Y. Almog. The Clausius-Mossotti formula in a dilute random medium with fixed volume fraction. Multiscale Model. Simul., 12(4):1777–1799, 2014.

    Article  MathSciNet  MATH  Google Scholar 

  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.

    Article  MathSciNet  MATH  Google Scholar 

  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.

    Article  MathSciNet  MATH  Google Scholar 

  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.

    Article  MathSciNet  MATH  Google Scholar 

  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. S. Armstrong, T. Kuusi, and J.-C. Mourrat. Quantitative stochastic homogenization and large-scale regularity. Preliminary version available at www.math.ens.fr/~mourrat/lecturenotes.pdf (2018).

  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.

    Article  MathSciNet  MATH  Google Scholar 

  10. S. Armstrong, T. Kuusi, and J.-C. Mourrat. The additive structure of elliptic homogenization. Invent. Math., 208(3):999–1154, 2017.

    Article  MathSciNet  MATH  Google Scholar 

  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.

    Article  MathSciNet  MATH  Google Scholar 

  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.

  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.

    Article  MathSciNet  MATH  Google Scholar 

  14. G. Ben Arous, M. Cabezas, and A. Fribergh. Scaling limit for the ant in high-dimensional labyrinths, preprint, arXiv:1609.03977.

  15. G. Ben Arous, M. Cabezas, and A. Fribergh. Scaling limit for the ant in a simple labyrinth, preprint, arXiv:1609.03980.

  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.

    Article  MathSciNet  MATH  Google Scholar 

  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.

    Article  MathSciNet  MATH  Google Scholar 

  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.

  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.

    Article  MathSciNet  MATH  Google Scholar 

  20. D. Dolgopyat. Limit theorems for partially hyperbolic systems. Trans. Amer. Math. Soc., 356(4):1637–1689, 2004.

    Article  MathSciNet  MATH  Google Scholar 

  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.

    Article  MathSciNet  MATH  Google Scholar 

  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.

    MATH  Google Scholar 

  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.

    Article  MathSciNet  MATH  Google Scholar 

  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.

    Article  MathSciNet  MATH  Google Scholar 

  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.

    Article  MathSciNet  MATH  Google Scholar 

  26. A. Gloria and J.-C. Mourrat. Spectral measure and approximation of homogenized coefficients. Probab. Theory Related Fields, 154(1-2):287–326, 2012.

    Article  MathSciNet  MATH  Google Scholar 

  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.

    Article  MathSciNet  MATH  Google Scholar 

  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.

    Article  MathSciNet  MATH  Google Scholar 

  29. A. Gloria, S. Neukamm, and F. Otto. A regularity theory for random elliptic operators, preprint, arXiv:1409.2678.

  30. A. Gloria and F. Otto. An optimal variance estimate in stochastic homogenization of discrete elliptic equations. Ann. Probab., 39(3):779–856, 2011.

    Article  MathSciNet  MATH  Google Scholar 

  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.

    Article  MathSciNet  MATH  Google Scholar 

  32. A. Gloria and F. Otto. The corrector in stochastic homogenization: optimal rates, stochastic integrability, and fluctuations, preprint, arXiv:1510.08290.

  33. P. Henning and D. Peterseim. Oversampling for the multiscale finite element method. Multiscale Model. Simul., 11(4):1149–1175, 2013.

    Article  MathSciNet  MATH  Google Scholar 

  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.

    Article  MathSciNet  MATH  Google Scholar 

  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.

    Article  MathSciNet  MATH  Google Scholar 

  36. B. D. Hughes. Conduction and diffusion in percolating systems. In Encyclopedia of complexity and systems science, pages 1395–1424. Springer, 2009.

  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.

    Article  MathSciNet  MATH  Google Scholar 

  38. H. Kesten. Subdiffusive behavior of random walk on a random cluster. Ann. Inst. H. Poincaré Probab. Statist., 22(4):425–487, 1986.

    MathSciNet  MATH  Google Scholar 

  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.

    Article  MathSciNet  MATH  Google Scholar 

  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.

    Article  MathSciNet  MATH  Google Scholar 

  41. T. Komorowski, C. Landim, and S. Olla. Fluctuations in Markov processes, volume 345 of Grundlehren der Mathematischen Wissenschaften. Springer, Heidelberg, 2012.

    Book  MATH  Google Scholar 

  42. S. M. Kozlov. Geometric aspects of averaging. Uspekhi Mat. Nauk, 44(2(266)):79–120, 1989.

  43. G. Kozma and A. Nachmias. The Alexander-Orbach conjecture holds in high dimensions. Invent. Math., 178(3):635–654, 2009.

    Article  MathSciNet  MATH  Google Scholar 

  44. T. Kumagai. Random walks on disordered media and their scaling limits, volume 2101 of Lecture Notes in Mathematics. Springer, Cham, 2014.

    Book  MATH  Google Scholar 

  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.

    Article  MathSciNet  MATH  Google Scholar 

  46. T. M. Liggett. Stochastic interacting systems: contact, voter and exclusion processes, volume 324 of Grundlehren der Mathematischen Wissenschaften. Springer-Verlag, Berlin, 1999.

    Book  MATH  Google Scholar 

  47. T. M. Liggett. Continuous time Markov processes, volume 113 of Graduate Studies in Mathematics. American Mathematical Society, Providence, RI, 2010.

    MATH  Google Scholar 

  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.

  49. A. Målqvist and D. Peterseim. Localization of elliptic multiscale problems. Math. Comp., 83(290):2583–2603, 2014.

    Article  MathSciNet  MATH  Google Scholar 

  50. D. Marahrens and F. Otto. Annealed estimates on the Green function. Probab. Theory Related Fields, 163(3-4):527–573, 2015.

    Article  MathSciNet  MATH  Google Scholar 

  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.

  52. I. Melbourne and M. Nicol. Almost sure invariance principle for nonuniformly hyperbolic systems. Comm. Math. Phys., 260(1):131–146, 2005.

    Article  MathSciNet  MATH  Google Scholar 

  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.

    Article  MathSciNet  MATH  Google Scholar 

  54. J.-C. Mourrat. First-order expansion of homogenized coefficients under Bernoulli perturbations. J. Math. Pures Appl. (9), 103(1):68–101, 2015.

  55. G. Papanicolaou and S. R. S. Varadhan. Ornstein-Uhlenbeck process in a random potential. Comm. Pure Appl. Math., 38(6):819–834, 1985.

    Article  MathSciNet  MATH  Google Scholar 

  56. G. C. Papanicolaou. Diffusion in random media. In Surveys in applied mathematics, Vol. 1, pages 205–253. Plenum, New York, 1995.

  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.

    MATH  Google Scholar 

  58. A. Quarteroni, R. Sacco, and F. Saleri. Numerical mathematics, volume 37 of Texts in Applied Mathematics. Springer-Verlag, New York, 2000.

    MATH  Google Scholar 

  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.

    Article  Google Scholar 

  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.

Download references

Acknowledgements

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).

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to J.-C. Mourrat.

Additional information

Communicated by Endre Suli.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Mourrat, JC. Efficient Methods for the Estimation of Homogenized Coefficients. Found Comput Math 19, 435–483 (2019). https://doi.org/10.1007/s10208-018-9389-9

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10208-018-9389-9

Keywords

Mathematics Subject Classification

Navigation