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A Multi Level Monte Carlo method with control variate for elliptic PDEs with log-normal coefficients

  • Fabio Nobile
  • Francesco TeseiEmail author
Article

Abstract

We consider the numerical approximation of the stochastic Darcy problem with log-normal permeability field and propose a novel Multi Level Monte Carlo (MLMC) approach with a control variate variance reduction technique on each level. We model the log-permeability as a stationary Gaussian random field with a covariance function belonging to the so called Matérn family, which includes both fields with very limited and very high spatial regularity. The control variate is obtained starting from the solution of an auxiliary problem with smoothed permeability coefficient and its expected value is effectively computed with a Stochastic Collocation method on the finest level in which the control variate is applied. We analyze the variance reduction induced by the control variate, and the total mean square error of the new estimator. To conclude we present some numerical examples and a comparison with the standard MLMC method, which shows the effectiveness of the proposed method.

Keywords

Log-normal random-fields Multi Level Monte Carlo Control variate Stochastic Collocation Matérn covariance  Stochastic Darcy Problem 

Mathematics Subject Classification

60H35 65C05 65N30 65N15 35R60 

Notes

Acknowledgments

F. Nobile and F. Tesei have been partially supported by the Swiss National Science Foundation under the Project No. 140574 “Efficient numerical methods for flow and transport phenomena in heterogeneous random porous media” and by the Center for ADvanced MOdeling Science (CADMOS).

References

  1. 1.
    Chilès, J.-P., Delfiner, P.: Geostatistics: Modeling Spatial Uncertainty. Wiley series in probability and statistics, 2nd edn. Wiley, Hoboken (2012)CrossRefGoogle Scholar
  2. 2.
    Babuška, I., Nobile, F., Tempone, R.: A stochastic collocation method for elliptic partial differential equations with random input data. SIAM J. Numer. Anal. 45(3), 1005–1034 (2007)zbMATHMathSciNetCrossRefGoogle Scholar
  3. 3.
    Babuška, I., Tempone, R., Zouraris, G.E.: Galerkin finite element approximations of stochastic elliptic partial differential equations. SIAM J. Numer. Anal. 42(2), 800–825 (2004)zbMATHMathSciNetCrossRefGoogle Scholar
  4. 4.
    Beck, J., Nobile, F., Tamellini, L., Tempone, R.: Stochastic spectral Galerkin and collocation methods for PDEs with random coefficients: a numerical comparison. In: Spectral and High Order Methods for Partial Differential Equations, Lecture Notes in Computational Science and Engineering, vol. 76, pp. 43–62 (2011)Google Scholar
  5. 5.
    Le Maitre, O.P., Knio, O.M.: Spectral Methods for Uncertainty Quantification. Springer, Berlin (2010)zbMATHCrossRefGoogle Scholar
  6. 6.
    Nobile, F., Tempone, R., Webster, C.G.: A sparse grid stochastic collocation method for partial differential equations with random input data. SIAM J. Numer. Anal. 46(5), 2309–2345 (2008)zbMATHMathSciNetCrossRefGoogle Scholar
  7. 7.
    Barth, A., Schwab, C., Zollinger, N.: Multi-level Monte Carlo finite element method for elliptic PDEs with stochastic coefficients. Numerische Mathematik 119(1), 123–161 (2011)zbMATHMathSciNetCrossRefGoogle Scholar
  8. 8.
    Xiu, D., Karniadakis, G.E.: Modeling uncertainty in flow simulations via generalized polynomial chaos. J. Comput. Phys. 187(1), 137–167 (2003)zbMATHMathSciNetCrossRefGoogle Scholar
  9. 9.
    Beck, J., Nobile, F., Tamellini, L., Tempone, R.: A quasi-optimal sparse grids procedure for groundwater flows, in spectral and high order methods for partial differential equations - ICOSAHOM 2012. In: Lecture Notes in Computational Science and Engineering, vol. 95, pp. 1–16 (2014)Google Scholar
  10. 10.
    Ernst, O.G., Sprungk, B.: Stochastic collocation for elliptic PDEs with random data: the lognormal case. In: Sparse Grids and Applications - Munich 2012, Lecture Notes in Computational Science and Engineering, vol. 97, pp. 29–53 (2014)Google Scholar
  11. 11.
    Charrier, J.: Analyse numérique d’équations aux dérivées partielles à coefficients aléatoires, applications à l’hydrologéologie, thèse de doctorat, ENS Cachan-Bretagne, (2011)Google Scholar
  12. 12.
    Cliffe, K.A., Giles, M.B., Scheichl, R., Teckentrup, A.L.: Multi Level Monte Carlo methods and applications to elliptic PDEs with random coefficients. Comput. Vis. Sci. 14(1), 3–15 (2011)zbMATHMathSciNetCrossRefGoogle Scholar
  13. 13.
    Graham, I.G., Kuo, F.Y., Nuyens, D., Scheichl, R., Sloan, I.H.: Quasi-Monte Carlo methods for elliptic PDEs with random coefficients and applications. J. Comput. Phys. 230(10), 3668–3694 (2011)zbMATHMathSciNetCrossRefGoogle Scholar
  14. 14.
    Charrier, J., Scheichl, R., Teckentrup, A.: Finite element error analysis of elliptic PDEs with random coefficients and its application to multilevel Monte Carlo methods. SIAM J. Numer. Anal. 51(1), 332–352 (2013)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Teckentrup, A.L., Scheichl, R., Giles, M.B., Ullmann, E.: Further analysis of multilevel Monte Carlo methods for elliptic PDEs with random coefficients. Numer. Math. 125(3), 569–600 (2013)zbMATHMathSciNetCrossRefGoogle Scholar
  16. 16.
    Gittelson, C.J.: Stochastic Galerkin discretization of the log-normal isotropic diffusion problem. Math. Models Methods Appl. Sci. 20(2), 237–263 (2010)zbMATHMathSciNetCrossRefGoogle Scholar
  17. 17.
    Galvis, J., Sarkis, M.: Approximating infinity-dimensional stochastic Darcy’s equations without uniform ellipticity. SIAM J. Numer. Anal. 47, 3624–3651 (2009)zbMATHMathSciNetCrossRefGoogle Scholar
  18. 18.
    Grisvard, P.: Elliptic Problems in Nonsmooth Domains, Classics in Applied Mathematics. SIAM, Philadelphia (2011)CrossRefGoogle Scholar
  19. 19.
    Charrier, J., Debussche, A.: Weak truncation error estimates for elliptic PDEs with lognormal coefficients. Stoch. Partial. Differ. Equ. 1(1), 63–93 (2013)zbMATHMathSciNetGoogle Scholar
  20. 20.
    Giles, M.B.: Multi Level Monte Carlo path simulation. Oper. Res. 56(3), 607–617 (2008)zbMATHMathSciNetCrossRefGoogle Scholar
  21. 21.
    Collier, N., Haji-Ali, A.L., Nobile, F., von Schwerin, E., Tempone, R.: A continuation Multi Level Monte Carlo algorithm. BIT Numer. Math. 55, 1–34 (2014)Google Scholar
  22. 22.
    Robert, Christian, Casella, George: Monte Carlo Statistical Methods. Springer texts in statistics. Springer, New York (2005)Google Scholar
  23. 23.
    Kronrod, A.S.: Nodes and weights of quadrature formulas. Sixteen-place tables. Authorized translation from the Russian. Consultants Bureau, New York (1965)zbMATHGoogle Scholar
  24. 24.
    Patterson, T.N.L.: The optimum addition of points to quadrature formulae. Math. Comput. 22, 847–856 (1968)CrossRefGoogle Scholar
  25. 25.
    Nobile, F., Tamellini, L., Tesei, F., Tempone, R.: An adaptive sparse grid algorithm for elliptic PDEs with lognormal diffusion coefficient, Mathicse report 04/2015, (2015)Google Scholar
  26. 26.
    Chkifa, A., Cohen, A., Schwab, C.: High-dimensional adaptive sparse polynomial interpolation and applications to parametric PDEs. Found. Comput. Math. 14(4), 601–633 (2014)zbMATHMathSciNetCrossRefGoogle Scholar
  27. 27.
    Lang, A., Potthoff, J.: Fast simulation of Gaussian random field. Monte Carlo Methods Appl. 17, 195–214 (2011)zbMATHMathSciNetCrossRefGoogle Scholar
  28. 28.
    Haji-Ali, A.L., Nobile, F., Tempone, R.: Multi Index Monte Carlo: when sparsity meets sampling, Mathicse Report (2014)Google Scholar
  29. 29.
    Da Prato, G.: Encyclopedia of Mathematics and Its Applications. Stochastic equations in infinite dimensions, vol. 44. Cambridge University Press, Cambridge (1992)Google Scholar

Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  1. 1.CSQI - MATHICSE, Ecole Politechnique Fédérale LausanneLausanneSwitzerland

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