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Abstract

In this paper, we present a generalization of the multilevel Monte Carlo (MLMC) method to a setting where the level parameter is a continuous variable. This continuous level Monte Carlo (CLMC) estimator provides a natural framework in PDE applications to adapt the model hierarchy to each sample. In addition, it can be made unbiased with respect to the expected value of the true quantity of interest provided the quantity of interest converges sufficiently fast. The practical implementation of the CLMC estimator is based on interpolating actual evaluations of the quantity of interest at a finite number of resolutions. As our new level parameter, we use the logarithm of a goal-oriented finite element error estimator for the accuracy of the quantity of interest. We prove the unbiasedness, as well as a complexity theorem that shows the same rate of complexity for CLMC as for MLMC. Finally, we provide some numerical evidence to support our theoretical results, by successfully testing CLMC on a standard PDE test problem. The numerical experiments demonstrate clear gains for samplewise adaptive refinement strategies over uniform refinements.

Original languageEnglish
Pages (from-to)93-116
Number of pages24
JournalSIAM/ASA Journal on Uncertainty Quantification
Volume7
Issue number1
Early online date15 Jan 2019
DOIs
Publication statusPublished - 31 Dec 2019

Keywords

  • math.NA
  • CLMC
  • Adaptivity
  • Complexity theorem
  • Heterogeneous elliptic PDEs
  • Unbiased estimator
  • Multilevel Monte Carlo

ASJC Scopus subject areas

  • Applied Mathematics
  • Discrete Mathematics and Combinatorics
  • Statistics and Probability
  • Statistics, Probability and Uncertainty
  • Modelling and Simulation

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