Foundations of Computational Mathematics

, Volume 15, Issue 2, pp 411–449 | Cite as

Multi-level Quasi-Monte Carlo Finite Element Methods for a Class of Elliptic PDEs with Random Coefficients



This paper is a sequel to our previous work (Kuo et al. in SIAM J Numer Anal, 2012) where quasi-Monte Carlo (QMC) methods (specifically, randomly shifted lattice rules) are applied to finite element (FE) discretizations of elliptic partial differential equations (PDEs) with a random coefficient represented by a countably infinite number of terms. We estimate the expected value of some linear functional of the solution, as an infinite-dimensional integral in the parameter space. Here, the (single-level) error analysis of our previous work is generalized to a multi-level scheme, with the number of QMC points depending on the discretization level and with a level-dependent dimension truncation strategy. In some scenarios, it is shown that the overall error (i.e., the root-mean-square error averaged over all shifts) is of order \({\fancyscript{O}}(h^2)\), where \(h\) is the finest FE mesh width, or \({\fancyscript{O}}(N^{-1+\delta })\) for arbitrary \(\delta >0\), where \(N\) denotes the maximal number of QMC sampling points in the parameter space. For these scenarios, the total work for all PDE solves in the multi-level QMC FE method is essentially of the order of one single PDE solve at the finest FE discretization level, for spatial dimension \(d=2\) with linear elements. The analysis exploits regularity of the parametric solution with respect to both the physical variables (the variables in the physical domain) and the parametric variables (the parameters corresponding to randomness). As in our previous work, families of QMC rules with “POD weights” (“product and order dependent weights”) which quantify the relative importance of subsets of the variables are found to be natural for proving convergence rates of QMC errors that are independent of the number of parametric variables.


Multi-level Quasi-Monte Carlo methods Infinite-dimensional integration Elliptic partial differential equations with random coefficients  Finite element methods 

Mathematics Subject Classification

65D30 65D32 65N30 



The authors thank Mike Giles and Robert Scheichl for valuable discussions. Frances Kuo was supported by an Australian Research Council QEII Fellowship, an Australian Research Council Discovery Project, and the Vice-Chancellor’s Childcare Support Fund for Women Researchers at the University of New South Wales. Christoph Schwab was supported by the Swiss National Science Foundation under Grant No. 200021-120290/1, and by the European Research Council under FP7 Grant AdG247277. Ian Sloan was supported by the Australian Research Council. Part of this work was completed during the Hausdorff Research Institute for Mathematics Trimester Program on Analysis and Numerics for High-Dimensional Problems in 2011.


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Copyright information

© SFoCM 2015

Authors and Affiliations

  • Frances Y. Kuo
    • 1
  • Christoph Schwab
    • 2
  • Ian H. Sloan
    • 1
  1. 1.School of Mathematics and StatisticsUniversity of New South WalesSydneyAustralia
  2. 2.Seminar for Applied MathematicsETH Zürich, ETH ZentrumZurichSwitzerland

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