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Foundations of Computational Mathematics

, Volume 16, Issue 6, pp 1631–1696 | Cite as

Application of Quasi-Monte Carlo Methods to Elliptic PDEs with Random Diffusion Coefficients: A Survey of Analysis and Implementation

  • Frances Y. KuoEmail author
  • Dirk Nuyens
Article

Abstract

This article provides a survey of recent research efforts on the application of quasi-Monte Carlo (QMC) methods to elliptic partial differential equations (PDEs) with random diffusion coefficients. It considers and contrasts the uniform case versus the lognormal case, single-level algorithms versus multi-level algorithms, first-order QMC rules versus higher-order QMC rules, and deterministic QMC methods versus randomized QMC methods. It gives a summary of the error analysis and proof techniques in a unified view, and provides a practical guide to the software for constructing and generating QMC points tailored to the PDE problems. The analysis for the uniform case can be generalized to cover a range of affine parametric operator equations.

Keywords

Quasi-Monte Carlo methods Infinite-dimensional integration Partial differential equations with random coefficients Uniform Lognormal Single-level Multi-level First order Higher order Deterministic Randomized 

Mathematics Subject Classification

65D30 65D32 65N30 

Notes

Acknowledgments

We graciously acknowledge many insightful discussions and valuable comments from our collaborators Josef Dick, Mahadevan Ganesh, Thong Le Gia, Alexander Gilbert, Ivan Graham, Yoshihito Kazashi, James Nichols, Pieterjan Robbe, Robert Scheichl, Christoph Schwab and Ian Sloan. We especially thank Mahadevan Ganesh for suggesting an alternative proof strategy to improve some existing estimates. We are also grateful for the financial supports from the Australian Research Council (FT130100655 and DP150101770) and the KU Leuven research fund (OT:3E130287 and C3:3E150478).

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Copyright information

© SFoCM 2016

Authors and Affiliations

  1. 1.School of Mathematics and StatisticsUniversity of New South WalesSydneyAustralia
  2. 2.Department of Computer ScienceKU LeuvenLeuvenBelgium

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