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Multilevel Quasi-Monte Carlo methods for lognormal diffusion problems


Authors: Frances Y. Kuo, Robert Scheichl, Christoph Schwab, Ian H. Sloan and Elisabeth Ullmann
Journal: Math. Comp. 86 (2017), 2827-2860
MSC (2010): Primary 65D30, 65D32, 65N30
DOI: https://doi.org/10.1090/mcom/3207
Published electronically: March 31, 2017
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Abstract: In this paper we present a rigorous cost and error analysis of a multilevel estimator based on randomly shifted Quasi-Monte Carlo (QMC) lattice rules for lognormal diffusion problems. These problems are motivated by uncertainty quantification problems in subsurface flow. We extend the convergence analysis in [Graham et al., Numer. Math. 2014] to multilevel Quasi-Monte Carlo finite element discretisations and give a constructive proof of the dimension-independent convergence of the QMC rules. More precisely, we provide suitable parameters for the construction of such rules that yield the required variance reduction for the multilevel scheme to achieve an $ \varepsilon $-error with a cost of $ \mathcal {O}(\varepsilon ^{-\theta })$ with $ \theta < 2$, and in practice even $ \theta \approx 1$, for sufficiently fast decaying covariance kernels of the underlying Gaussian random field inputs. This confirms that the computational gains due to the application of multilevel sampling methods and the gains due to the application of QMC methods, both demonstrated in earlier works for the same model problem, are complementary. A series of numerical experiments confirms these gains. The results show that in practice the multilevel QMC method consistently outperforms both the multilevel MC method and the single-level variants even for nonsmooth problems.


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Additional Information

Frances Y. Kuo
Affiliation: School of Mathematics and Statistics, University of New South Wales, Sydney, New South Wales 2052, Australia
Email: f.kuo@unsw.edu.au

Robert Scheichl
Affiliation: Department of Mathematical Sciences, University of Bath, Bath BA2 7AY, United Kingdom
Email: R.Scheichl@bath.ac.uk

Christoph Schwab
Affiliation: Seminar für Angewandte Mathematik, ETH Zürich, Rämistrasse 101, 8092 Zürich, Switzerland
Email: christoph.schwab@sam.math.ethz.ch

Ian H. Sloan
Affiliation: School of Mathematics and Statistics, University of New South Wales, Sydney, New South Wales 2052, Australia
Email: i.sloan@unsw.edu.au

Elisabeth Ullmann
Affiliation: Department of Mathematics, Technische Universität München, Boltzmannstraße 3, 85748 Garching, Germany
Email: elisabeth.ullmann@ma.tum.de

DOI: https://doi.org/10.1090/mcom/3207
Received by editor(s): July 4, 2015
Received by editor(s) in revised form: July 6, 2016
Published electronically: March 31, 2017
Article copyright: © Copyright 2017 American Mathematical Society