BIT Numerical Mathematics

, Volume 55, Issue 2, pp 399–432 | Cite as

A continuation multilevel Monte Carlo algorithm

  • Nathan Collier
  • Abdul-Lateef Haji-AliEmail author
  • Fabio Nobile
  • Erik von Schwerin
  • Raúl Tempone


We propose a novel Continuation Multi Level Monte Carlo (CMLMC) algorithm for weak approximation of stochastic models. The CMLMC algorithm solves the given approximation problem for a sequence of decreasing tolerances, ending when the required error tolerance is satisfied. CMLMC assumes discretization hierarchies that are defined a priori for each level and are geometrically refined across levels. The actual choice of computational work across levels is based on parametric models for the average cost per sample and the corresponding variance and weak error. These parameters are calibrated using Bayesian estimation, taking particular notice of the deepest levels of the discretization hierarchy, where only few realizations are available to produce the estimates. The resulting CMLMC estimator exhibits a non-trivial splitting between bias and statistical contributions. We also show the asymptotic normality of the statistical error in the MLMC estimator and justify in this way our error estimate that allows prescribing both required accuracy and confidence in the final result. Numerical results substantiate the above results and illustrate the corresponding computational savings in examples that are described in terms of differential equations either driven by random measures or with random coefficients.


Multilevel Monte Carlo Monte Carlo Partial differential equations with random data Stochastic differential equations Bayesian inference 

Mathematics Subject Classification

65C05 65N22 



Raúl Tempone is a member of the Strategic Research Initiative on Uncertainty Quantification in Computational Science and Engineering at KAUST (SRI-UQ). The authors would like to recognize the support of King Abdullah University of Science and Technology (KAUST) AEA project “Predictability and Uncertainty Quantification for Models of Porous Media” and University of Texas at Austin AEA Round 3 “Uncertainty quantification for predictive modeling of the dissolution of porous and fractured media”. We would also like to acknowledge the use of the following open source software packages: PETSc [4], PetIGA [8], NumPy, matplotlib [21].


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Copyright information

© Springer Science+Business Media Dordrecht 2014

Authors and Affiliations

  • Nathan Collier
    • 1
  • Abdul-Lateef Haji-Ali
    • 2
    Email author
  • Fabio Nobile
    • 3
  • Erik von Schwerin
    • 4
  • Raúl Tempone
    • 2
  1. 1.Environmental Sciences Division, Oak Ridge National LabClimate Change Science Institute (CCSI)Oak RidgeUSA
  2. 2.Applied Mathematics and Computational SciencesKAUSTThuwalSaudi Arabia
  3. 3.MATHICSE-CSQI, EPF de LausanneLausanneSwitzerland
  4. 4.Department of MathematicsKungliga Tekniska HögskolanStockholmSweden

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