A continuation multilevel Monte Carlo algorithm

Nathan Collier, Abdul Lateef Haji Ali, Fabio Nobile, Erik von Schwerin, Raul Tempone

Research output: Contribution to journalArticlepeer-review

72 Scopus citations


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. © 2014, Springer Science+Business Media Dordrecht.
Original languageEnglish (US)
Pages (from-to)399-432
Number of pages34
JournalBIT Numerical Mathematics
Issue number2
StatePublished - Sep 5 2014

Bibliographical note

KAUST Repository Item: Exported on 2020-10-01
Acknowledgements: Raul 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].

ASJC Scopus subject areas

  • Computational Mathematics
  • Software
  • Applied Mathematics
  • Computer Networks and Communications


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