This paper provides an a priori error analysis of a localized orthogonal decomposition method for the numerical stochastic homogenization of a model random diffusion problem. If the uniformly elliptic and bounded random coefficient field of the model problem is stationary and satisfies a quantitative decorrelation assumption in the form of the spectral gap inequality, then the expected $L^2$ error of the method can be estimated, up to logarithmic factors, by $H+(\varepsilon/H)^{d/2}$, $\varepsilon$ being the small correlation length of the random coefficient and $H$ the width of the coarse finite element mesh that determines the spatial resolution. The proof bridges recent results of numerical homogenization and quantitative stochastic homogenization.


  1. numerical homogenization
  2. stochastic homogenization
  3. quantitative theory
  4. a priori error estimates
  5. uncertainty
  6. model reduction

MSC codes

  1. 35R60
  2. 65N12
  3. 65N15
  4. 65N30
  5. 73B27
  6. 74Q05

Get full access to this article

View all available purchase options and get full access to this article.


R. Altmann, E. Chung, R. Maier, D. Peterseim, and S.-M. Pun, Computational multiscale methods for linear heterogeneous poroelasticity, J. Comput. Math., 38 (2020), pp. 41--57.
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 Multiscale Modeling and Analysis for Materials Simulation, Lect. Notes Ser. Inst. Math. Sci. Natl. Univ. Singap. 22, World Sci. Publ., Hackensack, NJ, 2012, pp. 197--272, https://doi.org/10.1142/9789814360906_0004.
S. Armstrong and P. Cardaliaguet, Stochastic homogenization of quasilinear Hamilton-Jacobi equations and geometric motions, J. Eur. Math. Soc. (JEMS), 20 (2018), pp. 797--864.
S. Armstrong, P. Cardaliaguet, and P. Souganidis, Error estimates and convergence rates for the stochastic homogenization of Hamilton-Jacobi equations, J. Amer. Math. Soc., 27 (2014), pp. 479--540.
S. Armstrong and J.-P. Daniel, Calderón-Zygmund estimates for stochastic homogenization, J. Funct. Anal., 270 (2016), pp. 312--329.
S. Armstrong, S. Ferguson, and T. Kuusi, Higher-Order Linearization and Regularity in Nonlinear Homogenization, preprint, arXiv:1910.03987, 2019.
S. Armstrong, A. Hannukainen, T. Kuusi, and J.-C. Mourrat, An Iterative Method for Elliptic Problems with Rapidly Oscillating Coefficients, preprint, arXiv:1803.03551, 2018.
S. Armstrong, T. Kuusi, and J.-C. Mourrat, The additive structure of elliptic homogenization, Invent. Math., 208 (2017), pp. 999--1154.
S. Armstrong and J.-C. Mourrat, Lipschitz regularity for elliptic equations with random coefficients, Arch. Ration. Mech. Anal., 219 (2016), pp. 255--348.
S. Armstrong and C. Smart, Quantitative stochastic homogenization of convex integral functionals, Ann. Sci. Éc. Norm. Supér. (4), 49 (2016), pp. 423--481.
P. Bella, J. Fischer, M. Josien, and C. Raithel, tba, in preparation, (2020).
X. Blanc, C. Le Bris, and F. Legoll, Some variance reduction methods for numerical stochastic homogenization, Philos. Trans. A, 374 (2016), 20150168.
A. Bourgeat and A. Piatnitski, Approximations of effective coefficients in stochastic homogenization, Ann. Inst. Henri Poincaré Probab. Stat., 40 (2004), pp. 153--165, https://doi.org/10.1016/S0246-0203(03)00065-7.
E. Cancès, V. Ehrlacher, F. Legoll, and B. Stamm, An embedded corrector problem to approximate the homogenized coefficients of an elliptic equation, C. R. Math. Acad. Sci. Paris, 353 (2015), pp. 801--806.
M. Duerinckx, A. Gloria, and F. Otto, The Structure of Fluctuations in Stochastic Homogenization, preprint, arXiv:1602.01717, 2016.
M. Duerinckx and F. Otto, Higher-Order Pathwise Theory of Fluctuations in Stochastic Homogenization, preprint, arXiv:1903.02329, 2019.
M. Feischl and D. Peterseim, Sparse Compression of Expected Solution Operators, preprint, arxiv:1807.01741, 2018.
J. Fischer, The choice of representative volumes in the approximation of effective properties of random materials, Arch. Ration. Mech. Anal., 234 (2019), pp. 635--726.
J. Fischer and S. Neukamm, Optimal Homogenization Rates in Stochastic Homogenization of Nonlinear Uniformly Elliptic Equations and Systems, preprint, arXiv:1908.02273, 2019.
J. Fischer and C. Raithel, Liouville principles and a large-scale regularity theory for random elliptic operators on the half-space, SIAM J. Math. Anal., 49 (2017), pp. 82--114.
D. Gallistl and D. Peterseim, Stable multiscale Petrov-Galerkin finite element method for high frequency acoustic scattering, Comput. Methods Appl. Mech. Engrg., 295 (2015), pp. 1--17.
D. Gallistl and D. Peterseim, Computation of quasilocal effective diffusion tensors and connections to the mathematical theory of homogenization, Multiscale Model. Simul., 15 (2017), pp. 1530--1552, https://doi.org/10.1137/16M1088533.
D. Gallistl and D. Peterseim, Numerical stochastic homogenization by quasilocal effective diffusion tensors, Commun. Math. Sci., 17 (2019), pp. 637--651.
A. Gloria, S. Neukamm, and F. Otto, A Regularity Theory for Random Elliptic Operators, preprint, arXiv:1409.2678, 2014.
A. Gloria, S. Neukamm, and F. Otto, Quantification of ergodicity in stochastic homogenization: Optimal bounds via spectral gap on Glauber dynamics, Invent. Math., 199 (2015), pp. 455--515, https://doi.org/10.1007/s00222-014-0518-z.
A. Gloria and F. Otto, An optimal variance estimate in stochastic homogenization of discrete elliptic equations, Ann. Probab., 39 (2011), pp. 779--856, https://doi.org/10.1214/10-AOP571.
A. Gloria and F. Otto, An optimal error estimate in stochastic homogenization of discrete elliptic equations, Ann. Appl. Probab., 22 (2012), pp. 1--28, https://doi.org/10.1214/10-AAP745.
A. Gloria and F. Otto, The Corrector in Stochastic Homogenization: Optimal Rates, Stochastic Integrability, and Fluctuations, preprint, arXiv:1510.08290, 2015.
C. Gu, An Efficient Algorithm for Solving Elliptic Problems on Percolation Clusters, preprint, arXiv:1907.13571, 2019.
P. Henning, A. M\aalqvist, and D. Peterseim, A localized orthogonal decomposition method for semi-linear elliptic problems, ESAIM Math. Model. Numer. Anal., 48 (2014), pp. 1331--1349, https://doi.org/10.1051/m2an/2013141.
P. Henning and D. Peterseim, Oversampling for the multiscale finite element method, Multiscale Model. Simul., 11 (2013), pp. 1149--1175, https://doi.org/10.1137/120900332.
V. Khoromskaia, B. Khoromskij, and F. Otto, Numerical Study in Stochastic Homogenization for Elliptic PDEs: Convergence Rate in the Size of Representative Volume Elements, preprint, arXiv:1903.12227, 2019.
R. Kornhuber, D. Peterseim, and H. Yserentant, An analysis of a class of variational multiscale methods based on subspace decomposition, Math. Comp., 87 (2018), pp. 2765--2774.
C. Le Bris, F. Legoll, and W. Minvielle, Special quasirandom structures: A selection approach for stochastic homogenization, Monte Carlo Methods Appl., 22 (2016), pp. 25--54.
A. M\aalqvist and D. Peterseim, Localization of elliptic multiscale problems, Math. Comp., 83 (2014), pp. 2583--2603, https://doi.org/10.1090/S0025-5718-2014-02868-8.
A. M\aalqvist and D. Peterseim, Generalized finite element methods for quadratic eigenvalue problems, ESAIM Math. Model. Numer. Anal., 51 (2017), pp. 147--163, https://doi.org/10.1051/m2an/2016019.
J.-C. Mourrat, Efficient methods for the estimation of homogenized coefficients, Found. Comput. Math., 19 (2019), pp. 435--483.
H. Owhadi, Bayesian numerical homogenization, Multiscale Model. Simul., 13 (2015), pp. 812--828, https://doi.org/10.1137/140974596.
H. Owhadi, Multigrid with rough coefficients and multiresolution operator decomposition from hierarchical information games, SIAM Rev., 59 (2017), pp. 99--149, https://arxiv.org/abs/1503.03467.
D. Peterseim, Variational multiscale stabilization and the exponential decay of fine-scale correctors, in Building Bridges: Connections and Challenges in Modern Approaches to Numerical Partial Differential Equations, G. R. Barrenechea, F. Brezzi, A. Cangiani, and E. H. Georgoulis, eds., Lect. Notes Comput. Sci. Eng. 114, Springer, Berlin, 2016, pp. 341--367.
D. Peterseim, Eliminating the pollution effect in Helmholtz problems by local subscale correction, Math. Comp., 86 (2017), pp. 1005--1036, https://doi.org/10.1090/mcom/3156.
D. Peterseim and M. Schedensack, Relaxing the CFL condition for the wave equation on adaptive meshes, J. Sci. Comput., 72 (2017), pp. 1196--1213, https://doi.org/10.1007/s10915-017-0394-y.
D. Peterseim, D. Varga, and B. Verfürth, From Domain Decomposition to Homogenization Theory, preprint, arXiv:1811.06319, 2018.
B. Verfürth, Heterogeneous multiscale method for the Maxwell equations with high contrast, ESAIM Math. Model. Numer. Anal., 53 (2019), pp. 35--61, https://doi.org/10.1051/m2an/2018064.

Information & Authors


Published In

cover image SIAM Journal on Numerical Analysis
SIAM Journal on Numerical Analysis
Pages: 660 - 674
ISSN (online): 1095-7170


Submitted: 27 December 2019
Accepted: 30 November 2020
Published online: 9 March 2021


  1. numerical homogenization
  2. stochastic homogenization
  3. quantitative theory
  4. a priori error estimates
  5. uncertainty
  6. model reduction

MSC codes

  1. 35R60
  2. 65N12
  3. 65N15
  4. 65N30
  5. 73B27
  6. 74Q05



Funding Information

H2020 European Research Council https://doi.org/10.13039/100010663 : 865751

Metrics & Citations



If you have the appropriate software installed, you can download article citation data to the citation manager of your choice. Simply select your manager software from the list below and click Download.

Cited By

View Options

View options


View PDF







Copy the content Link

Share with email

Email a colleague

Share on social media