Skip to main content
SpringerLink
Log in

Further analysis of multilevel Monte Carlo methods for elliptic PDEs with random coefficients

  • Published:
Numerische Mathematik Aims and scope Submit manuscript

Abstract

We consider the application of multilevel Monte Carlo methods to elliptic PDEs with random coefficients. We focus on models of the random coefficient that lack uniform ellipticity and boundedness with respect to the random parameter, and that only have limited spatial regularity. We extend the finite element error analysis for this type of equation, carried out in Charrier et al. (SIAM J Numer Anal, 2013), to more difficult problems, posed on non-smooth domains and with discontinuities in the coefficient. For this wider class of model problem, we prove convergence of the multilevel Monte Carlo algorithm for estimating any bounded, linear functional and any continuously Fréchet differentiable non-linear functional of the solution. We further improve the performance of the multilevel estimator by introducing level dependent truncations of the Karhunen–Loève expansion of the random coefficient. Numerical results complete the paper.

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

Access this article

Log in via an institution

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

References

  1. Barrett, J.W., Elliott, C.E.: Total flux estimates for a finite-element approximation of elliptic equations. IMA J. Numer. Anal. 7, 129–148 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  2. Barth, A., Schwab, Ch., Zollinger, N.: Multi-level Monte Carlo finite element method for elliptic PDE’s with stochastic coefficients. Numer. Math. 119(1), 123–161 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  3. Bourlard, M., Dauge, M., Lubama, M.S., Nicaise, S.: Coefficients of the singularities of elliptic boundary value problems in domains with conical points. III: Finite element methods on polygonal domains. SIAM J. Numer. Anal. 29(1), 136–155 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  4. Brenner, S.C., Scott, L.R.: The Mathematical Theory of Finite Element Methods. Texts in Applied Mathematics, 3rd edn, vol. 15. Springer, Berlin (2008)

  5. Charrier, J.: Strong and weak error estimates for the solutions of elliptic partial differential equations with random coefficients. SIAM J. Numer. Anal 50(1), 216–246 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  6. Charrier, J., Scheichl, R., Teckentrup, A.L.: Finite element error analysis of elliptic PDEs with random coefficients and its application to multilevel Monte Carlo methods. SIAM J. Numer. Anal. 51(1), 322–352 (2013)

    Google Scholar 

  7. Chen, X., Nashed, Z., Qi, L.: Smoothing methods and semismooth methods for nondifferentiable operator equations. SIAM J. Numer. Anal. 38(4), 1200–1216 (2001)

    Article  MathSciNet  Google Scholar 

  8. Cliffe, K.A., Giles, M.B., 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)

    Article  MathSciNet  MATH  Google Scholar 

  9. Costabel, M., Dauge, M., Nicaise, S.: Singularities of Maxwell interface problems. M2AN Math. Model. Numer. Anal. 33(3), 627–649 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  10. Dereich, S., Heidenreich, F.: A multilevel Monte Carlo algorithm for Lévy-driven stochastic differential equations. Stoch. Process. Appl. 121(7), 1565–1587 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  11. Dietrich, C.R., 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)

    Article  MathSciNet  MATH  Google Scholar 

  12. Douglas, J., Dupont, T., Wheeler, M.F.: A Galerkin procedure for approximating the flux on the boundary for elliptic and parabolic boundary value problems. RAIRO Modèl. Math. Anal. Numèr. 2, 47–59 (1974)

    MathSciNet  Google Scholar 

  13. Ghanem, R.G., Spanos, P.D.: Stochastic Finite Elements: A Spectral Approach. Springer, New York (1991)

    Book  MATH  Google Scholar 

  14. Gilbarg, D., Trudinger, N.S.: Elliptic partial differential equations of second order. In: Classics in Mathematics. Springer, Berlin, Reprint of the 1998 edition (2001)

  15. Giles, M.B.: Improved multilevel Monte Carlo convergence using the Milstein scheme. In: Monte Carlo and Quasi-Monte Carlo methods 2006, pp. 343–358. Springer, Berlin (2007)

  16. Giles, M.B.: Multilevel Monte Carlo path simulation. Oper. Res. 256, 981–986 (2008)

    MathSciNet  Google Scholar 

  17. Giles, M.B., Reisinger, C.: Stochastic finite differences and multilevel Monte Carlo for a class of SPDEs in finance. SIFIN 1(3), 575–592 (2012)

    Google Scholar 

  18. Giles, M.B., Süli, E.: Adjoint methods for PDEs: a posteriori error analysis and postprocessing by duality. Acta Numerica, vol., pp. 145–236. Cambridge University Press, Cambridge (2002)

  19. Gittelson, C.J., Könnö, J., Schwab, Ch., Stenberg, R.: The multilevel Monte Carlo finite element method for a stochastic Brinkman problem. SAM Research Report 2011–31, ETH Zurich (2011)

  20. 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)

    Article  MathSciNet  MATH  Google Scholar 

  21. Graham, I.G., Scheichl, R., Ullmann, E.: Mixed Finite Element analysis of lognormal diffusion and multilevel Monte Carlo methods, In preparation (2013)

  22. Graubner, S.: Multi-level Monte Carlo Methoden für stochastische partielle Differentialgleichungen. Diploma thesis, TU Darmstadt (2008). http://people.maths.ox.ac.uk/gilesm/files/Diplomarbeit.pdf

  23. Grisvard, P.: Elliptic Problems in Non-Smooth Domains. Pitman, Boston (1985)

  24. Grisvard, P.: Singularities in boundary value problems. In: Researh Notes in Mathematics. Springer, Berlin (1992)

  25. Hackbusch, W.: Elliptic differential equations. In: Springer Series in Computational Mathematics, vol. 18. Springer, Berlin (2010)

  26. Heinrich, S.: Multilevel Monte Carlo methods. In: Lecture Notes in Computer Science, vol. 2179, pp. 3624–3651. Springer, Berin (2001)

  27. Hoel, H., von Schwerin, E., Szepessy, A., Tempone, R.: Adaptive multilevel Monte Carlo simulation. In: Engquist, B., Runborg, O., Tsai, Y.-H. R. (eds.) Numerical Analysis of Multiscale Computations. Lecture Notes in Computational Science and Engineering, vol. 82, pp. 217–234. Springer, Berlin (2012)

  28. Kato, T.: Perturbation Theory for Linear Operators. Springer, Berlin (1966)

  29. Kloeden, P., Neuenkirch, A., Pavani, R.: Multilevel Monte Carlo for stochastic differential equations with additive fractional noise. Ann. App. Probab. 189, 255–276 (2011)

    MathSciNet  MATH  Google Scholar 

  30. Nicaise, S., Sändig, A.M.: General interface problems - I. Math. Methods Appl. Sci. 17(6), 395–429 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  31. Petzoldt, M.: Regularity and error estimators for elliptic problems with discontinuous coefficients. PhD thesis, WIAS Berlin (2001). http://webdoc.sub.gwdg.de/ebook/diss/2003/fu-berlin/2001/111/

  32. Pierce, N.A., Giles, M.B.: Adjoint and defect error bounding and correction for functional estimates. J. Comput. Phys. 200, 769–794 (2004)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematical Sciences, University of Bath, Claverton Down, Bath, BA2 7AY, UK

    A. L. Teckentrup, R. Scheichl & E. Ullmann

  2. Mathematical Institute, University of Oxford, 24-29 St Giles, Oxford, OX1 3LB, UK

    M. B. Giles

Authors
  1. A. L. Teckentrup
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. R. Scheichl
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. M. B. Giles
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. E. Ullmann
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to A. L. Teckentrup.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Teckentrup, A.L., Scheichl, R., Giles, M.B. et al. Further analysis of multilevel Monte Carlo methods for elliptic PDEs with random coefficients. Numer. Math. 125, 569–600 (2013). https://doi.org/10.1007/s00211-013-0546-4

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00211-013-0546-4

Mathematics Subject Classification

Search

Navigation