Multilevel quasi-Monte Carlo methods for lognormal diffusion problems

Frances Y. Kuo, Robert Scheichl, Christoph Schwab, Ian H. Sloan, Elisabeth Ullmann

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In this paper we present a rigorous cost and error analysis of amultilevel estimator based on randomly shifted Quasi-Monte Carlo (QMC)lattice rules for lognormal diffusion problems. These problems are motivatedby uncertainty quantification problems in subsurface flow. We extend the convergenceanalysis in [Graham et al., Numer. Math. 2014] to multilevel QuasiMonteCarlo finite element discretisations and give a constructive proof ofthe dimension-independent convergence of the QMC rules. More precisely, weprovide suitable parameters for the construction of such rules that yield therequired variance reduction for the multilevel scheme to achieve an ε-error witha cost of O(ε−θ) with θ < 2, and in practice even θ ≈ 1, for sufficiently fastdecaying covariance kernels of the underlying Gaussian random field inputs.This confirms that the computational gains due to the application of multilevelsampling methods and the gains due to the application of QMC methods, bothdemonstrated in earlier works for the same model problem, are complementary.A series of numerical experiments confirms these gains. The results showthat in practice the multilevel QMC method consistently outperforms boththe multilevel MC method and the single-level variants even for nonsmoothproblems.
Original languageEnglish
Pages (from-to)2827-2860
Number of pages34
JournalMathematics of Computation (MCOM)
Issue number308
Early online date31 Mar 2017
Publication statusPublished - 1 Nov 2017


  • math.NA
  • 65D30, 65D32, 65N30


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