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

  • Fabio Nobile
  • Francesco TeseiEmail author


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.


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 



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).


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Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

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

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