Uncertainty Quantification for Porous Media Flow Using Multilevel Monte Carlo

  • Jan MohringEmail author
  • René Milk
  • Adrian Ngo
  • Ole Klein
  • Oleg Iliev
  • Mario Ohlberger
  • Peter Bastian
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9374)


Uncertainty quantification (UQ) for porous media flow is of great importance for many societal, environmental and industrial problems. An obstacle for progress in this area is the extreme computational effort needed for solving realistic problems. It is expected that exa-scale computers will open the door for a significant progress in this area. We demonstrate how new features of the Distributed and Unified Numerics Environment DUNE [1] address these challenges. In the frame of the DFG funded project EXA-DUNE the software has been extended by multiscale finite element methods (MsFEM) and by a parallel framework for the multilevel Monte Carlo (MLMC) approach. This is a general concept for computing expected values of simulation results depending on random fields, e.g. the permeability of porous media. It belongs to the class of variance reduction methods and overcomes the slow convergence of classical Monte Carlo by combining cheap/inexact and expensive/accurate solutions in an optimal ratio.


Uncertainty quantification Multilevel Monte Carlo Multiscale finite elements Porous media Random permeability Exa-scale DUNE 



This research was funded by the DFG SPP 1648 Software for Exascale Computing.


  1. 1.
    Bastian, P., Blatt, M., Dedner, A., Engwer, C., Klöfkorn, R., Ohlberger, M., Sander, O.: A generic grid interface for parallel and adaptive scientific computing. Part I: abstract framework. Computing 82(2–3), 103–119 (2008)zbMATHMathSciNetCrossRefGoogle Scholar
  2. 2.
    Hoeksema, R.J., Kitanidis, P.K.: Analysis of the spatial structure of properties of selected aquifers. Water Resour. Res. 21(4), 563–572 (1985)CrossRefGoogle Scholar
  3. 3.
    Ghanem, R.G., Spanos, P.D.: Stochastic Finite Elements: A Spectral Approach. Springer, New York (1991)zbMATHCrossRefGoogle Scholar
  4. 4.
    Cliffe, K., Giles, M., Scheichl, R., Teckentrup, A.L.: Multilevel Monte Carlo methods and applications to elliptic PDEs with random coefficients. Comput. Vis. Sci. 14(1), 3–15 (2011)zbMATHMathSciNetCrossRefGoogle Scholar
  5. 5.
    Dietrich, C., Newsam, G.N.: Fast and exact simulation of stationary gaussian processes through circulant embedding of the covariance matrix. SIAM J. Sci. Comput. 18(4), 1088–1107 (1997)zbMATHMathSciNetCrossRefGoogle Scholar
  6. 6.
    Ngo, A.Q.: Discontinuous Galerkin based geostatistical inversion of stationary flow and transport processes in groundwater. Dissertation, University of Heidelberg (2015)Google Scholar
  7. 7.
    Milk, R., Mohring, J.: DUNE mlmc, March 2015.
  8. 8.
    Hou, T.Y., Wu, X.H.: A multiscale finite element method for elliptic problems in composite materials and porous media. J. Comput. Phys. 134(1), 169–189 (1997)zbMATHMathSciNetCrossRefGoogle Scholar
  9. 9.
    Efendiev, Y., Hou, T.Y.: Multiscale Finite Element Methods: Theory and Applications. Surveys and Tutorials in the Applied Mathematical Sciences, vol. 4. Springer, New York (2009)Google Scholar
  10. 10.
    Gloria, A.: An analytical framework for numerical homogenization. II. Windowing and oversampling. Multiscale Model. Simul. 7(1), 274–293 (2008)zbMATHMathSciNetCrossRefGoogle Scholar
  11. 11.
    Hou, T.Y., Wu, X.H., Zhang, Y.: Removing the cell resonance error in the multiscale finite element method via a Petrov-Galerkin formulation. Commun. Math. Sci. 2(2), 185–205 (2004)zbMATHMathSciNetCrossRefGoogle Scholar
  12. 12.
    Henning, P., Peterseim, D.: Oversampling for the multiscale finite element method. Multiscale Model. Simul. 11(4), 1149–1175 (2013)zbMATHMathSciNetCrossRefGoogle Scholar
  13. 13.
    Schindler, F., Milk, R.: DUNE Generic Discretization Toolbox, March 2015.
  14. 14.
    Milk, R., Schindler, F.: DUNE Stuff, March 2015.
  15. 15.
    Milk, R., Kaulmann, S.: DUNE Multiscale, March 2015.
  16. 16.
    Bastian, P., et al.: EXA-DUNE: flexible PDE solvers, numerical methods and applications. In: Lopes, L., et al. (eds.) Euro-Par 2014, Part II. LNCS, vol. 8806, pp. 530–541. Springer, Heidelberg (2014) Google Scholar
  17. 17.
    Giles, M.B.: Multilevel Monte Carlo path simulation. Oper. Res. 56(3), 607–617 (2008)zbMATHMathSciNetCrossRefGoogle Scholar
  18. 18.
    Efendiev, Y., Iliev, O., Kronsbein, C.: Multilevel Monte Carlo methods using ensemble level mixed MsFEM for two-phase flow and transport simulations. Comput. Geosci. 17(5), 833–850 (2013)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  • Jan Mohring
    • 1
    Email author
  • René Milk
    • 2
  • Adrian Ngo
    • 3
  • Ole Klein
    • 3
  • Oleg Iliev
    • 1
  • Mario Ohlberger
    • 2
  • Peter Bastian
    • 3
  1. 1.Fraunhofer ITWMKaiserslauternGermany
  2. 2.Institute for Computational and Applied MathematicsUniversity of MünsterMünsterGermany
  3. 3.Interdisciplinary Center for Scientific ComputingUniversity of HeidelbergHeidelbergGermany

Personalised recommendations