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Abstract
In this paper we present a rigorous cost and error analysis of amultilevel estimator based on randomly shifted QuasiMonte 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 dimensionindependent 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 εerror 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 singlelevel variants even for nonsmoothproblems.
Original language  English 

Pages (fromto)  28272860 
Number of pages  34 
Journal  Mathematics of Computation (MCOM) 
Volume  86 
Issue number  308 
Early online date  31 Mar 2017 
DOIs  
Publication status  Published  1 Nov 2017 
Keywords
 math.NA
 65D30, 65D32, 65N30
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 1 Finished

Multilevel Monte Carlo Methods for Elliptic Problems
Scheichl, R.
Engineering and Physical Sciences Research Council
1/07/11 → 30/06/14
Project: Research council