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Multilevel Monte Carlo methods and applications to elliptic PDEs with random coefficients

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Computing and Visualization in Science

Abstract

We consider the numerical solution of elliptic partial differential equations with random coefficients. Such problems arise, for example, in uncertainty quantification for groundwater flow. We describe a novel variance reduction technique for the standard Monte Carlo method, called the multilevel Monte Carlo method, and demonstrate numerically its superiority. The asymptotic cost of solving the stochastic problem with the multilevel method is always significantly lower than that of the standard method and grows only proportionally to the cost of solving the deterministic problem in certain circumstances. Numerical calculations demonstrating the effectiveness of the method for one- and two-dimensional model problems arising in groundwater flow are presented.

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References

  1. Barth, A., Schwab, C., Zollinger, N.: Multi–level Monte Carlo finite element method for elliptic PDE’s with stochastic coefficients. Numer. Math. Online First (2011)

  2. Brandt A., Galun M., Ron D.: Optimal multigrid algorithms for calculating thermodynamic limits. J. Stat. Phys. 74(1–2), 313–348 (1994)

    Article  MATH  Google Scholar 

  3. Brandt A., Ilyin V.: Multilevel Monte Carlo methods for studying large scale phenomena in fluids. J. Mol. Liq. 105(2-3), 245–248 (2003)

    Article  Google Scholar 

  4. Charrier, J.: Strong and weak error estimates for the solutions of elliptic partial differential equations with random coefficients. Tech. Rep. 7300, INRIA (2010). Available at http://hal.inria.fr/inria-00490045/en/

  5. 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. Tech. rep., University of Bath (2011). BICS Preprint 02/11

  6. Cliffe K.A., Graham I.G., Scheichl R., Stals L.: Parallel computation of flow in heterogeneous media using mixed finite elements. J. Comput. Phys. 164, 258–282 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  7. de Marsily G.: Quantitative Hydrogeology. Academic Press, London (1986)

    Google Scholar 

  8. de Marsily G., Delay F., Goncalves J., Renard P., Teles V., Violette S.: Dealing with spatial heterogeneity. Hydrogeol. J. 13, 161–183 (2005)

    Article  Google Scholar 

  9. Delhomme, J.P.: Spatial variability and uncertainty in groundwater flow parameters, a geostatistical approach. Water Resourc. Res. pp. 269–280 (1979)

  10. Eiermann, M., Ernst, O.G., Ullmann, E.: Computational aspects of the stochastic finite element method. In: Proceedings of ALGORITHMY 2005, pp. 1–10 (2005)

  11. Ernst O.G., Powell C.E., Silvester D.J., Ullmann E.: Efficient solvers for a linear stochastic Galerkin mixed formulation of diffusion problems with random data. SIAM J. Sci. Comput. 31(2), 1424–1447 (1979)

    Article  MathSciNet  Google Scholar 

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

    Book  MATH  Google Scholar 

  13. Giles, M.B.: Improved multilevel Monte Carlo convergence using the Milstein scheme. In: Monte Carlo and Quasi-Monte Carlo methods 2006, vol. 256, pp. 343–358. Springer, Berlin (2007)

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

    MathSciNet  Google Scholar 

  15. Giles, M.B., Waterhouse, B.J.: Multilevel quasi-Monte Carlo path simulation. In: Advanced financial modelling, Radon Ser. Comput. Appl. Math., vol. 8, pp. 165–181. Walter de Gruyter, Berlin (2009)

  16. 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 

  17. Heinrich, S.: Multilevel Monte Carlo Methods. Lecture Notes in Computer Science, vol. 2179, pp. 3624–3651. Springer, Philadelphia, PA (2001)

  18. Hoeksema R.J., Kitanidis P.K.: Analysis of the spatial structure of properties of selected aquifers. Water Resour. Res. 21, 536–572 (1985)

    Google Scholar 

  19. Le Maître O.P., Kino O.M.: Spectral Methods for Uncertainty Quantification, With Applications to Fluid Dynamics. Springer, Berlin (2010)

    Book  MATH  Google Scholar 

  20. Da Prato G., Zabczyk J.: Stochastic Equations in Infinite Dimensions, Encyclopedia of Mathematics and its Applications, vol. 44. Cambridge University Press, Cambridge (1992)

    Google Scholar 

  21. Schwab C., Todor R.A.: Karhunen-Loève approximation of random fields by generalized fast multipole methods. J. Comput. Phys. 217(1), 100–122 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  22. Vassilevski P.S.: Multilevel Block Factorization Preconditioners: Matrix-based Analysis and Algorithms for Solving Finite Element Equations. Springer, Berlin (2008)

    MATH  Google Scholar 

  23. Xiu D.: Numerical Methods for Stochastic Computations: A Spectral Method Approach. Princeton University Press, Princeton (2010)

    MATH  Google Scholar 

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Authors and Affiliations

  1. School of Mathematical Sciences, University of Nottingham, University Park, Nottingham, NG7 2RD, UK

    K. A. Cliffe

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

    M. B. Giles

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

    R. Scheichl & A. L. Teckentrup

Authors
  1. K. A. Cliffe
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  2. M. B. Giles
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  3. R. Scheichl
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  4. A. L. Teckentrup
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Correspondence to M. B. Giles.

Additional information

Communicated by: C. W. Oosterlee and A. Borzi.

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Cliffe, K.A., Giles, M.B., Scheichl, R. et al. Multilevel Monte Carlo methods and applications to elliptic PDEs with random coefficients. Comput. Visual Sci. 14, 3 (2011). https://doi.org/10.1007/s00791-011-0160-x

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  • DOI: https://doi.org/10.1007/s00791-011-0160-x

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