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.
Similar content being viewed by others
References
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)
Brandt A., Galun M., Ron D.: Optimal multigrid algorithms for calculating thermodynamic limits. J. Stat. Phys. 74(1–2), 313–348 (1994)
Brandt A., Ilyin V.: Multilevel Monte Carlo methods for studying large scale phenomena in fluids. J. Mol. Liq. 105(2-3), 245–248 (2003)
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/
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
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)
de Marsily G.: Quantitative Hydrogeology. Academic Press, London (1986)
de Marsily G., Delay F., Goncalves J., Renard P., Teles V., Violette S.: Dealing with spatial heterogeneity. Hydrogeol. J. 13, 161–183 (2005)
Delhomme, J.P.: Spatial variability and uncertainty in groundwater flow parameters, a geostatistical approach. Water Resourc. Res. pp. 269–280 (1979)
Eiermann, M., Ernst, O.G., Ullmann, E.: Computational aspects of the stochastic finite element method. In: Proceedings of ALGORITHMY 2005, pp. 1–10 (2005)
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)
Ghanem R.G., Spanos P.D.: Stochastic Finite Elements: A Spectral Approach. Springer, New York (1991)
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)
Giles M.B.: Multilevel Monte Carlo path simulation. Oper. Res. 256, 981–986 (2008)
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)
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)
Heinrich, S.: Multilevel Monte Carlo Methods. Lecture Notes in Computer Science, vol. 2179, pp. 3624–3651. Springer, Philadelphia, PA (2001)
Hoeksema R.J., Kitanidis P.K.: Analysis of the spatial structure of properties of selected aquifers. Water Resour. Res. 21, 536–572 (1985)
Le Maître O.P., Kino O.M.: Spectral Methods for Uncertainty Quantification, With Applications to Fluid Dynamics. Springer, Berlin (2010)
Da Prato G., Zabczyk J.: Stochastic Equations in Infinite Dimensions, Encyclopedia of Mathematics and its Applications, vol. 44. Cambridge University Press, Cambridge (1992)
Schwab C., Todor R.A.: Karhunen-Loève approximation of random fields by generalized fast multipole methods. J. Comput. Phys. 217(1), 100–122 (2006)
Vassilevski P.S.: Multilevel Block Factorization Preconditioners: Matrix-based Analysis and Algorithms for Solving Finite Element Equations. Springer, Berlin (2008)
Xiu D.: Numerical Methods for Stochastic Computations: A Spectral Method Approach. Princeton University Press, Princeton (2010)
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by: C. W. Oosterlee and A. Borzi.
Rights and permissions
About this article
Cite this article
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
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s00791-011-0160-x