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

Article

Abstract

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.

Keywords

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 

References

  1. 1.
    E. Bach and J. Shallit, Algorithmic Number Theory (Volume I: Efficient Algorithms), MIT Press, Cambridge, 1966.Google Scholar
  2. 2.
    A. Barth, Ch. Schwab, and N. Zollinger, Multi-level Monte Carlo finite element method for elliptic PDEs with stochastic coefficients, Numer. Math., 119 (2011), pp. 123–161.Google Scholar
  3. 3.
    J. Charrier, R. Scheichl, and A. L. Teckentrup, Finite element error analysis of elliptic PDEs with random coefficients and its application to multilevel Monte Carlo methods, SIAM J. Numer. Anal., 51 (2013), pp. 322–352.Google Scholar
  4. 4.
    P. G. Ciarlet, The Finite Element Method for Elliptic Problems, Elsevier, Amsterdam 1978.Google Scholar
  5. 5.
    K. A. Cliffe, M. B. Giles, R. Scheichl, and A. L. Teckentrup, Multilevel Monte Carlo methods and applications to elliptic PDEs with random coefficients, Computing and Visualization in Science Science 14 (2011), pp. 3–15.Google Scholar
  6. 6.
    A. Cohen, R. DeVore and Ch. Schwab, Convergence rates of best N-term Galerkin approximations for a class of elliptics PDEs, Found. Comp. Math., 10 (2010), pp. 615–646.Google Scholar
  7. 7.
    R. Cools, F. Y. Kuo, and D. Nuyens, Constructing embedded lattice rules for multivariate integration, SIAM J. Sci. Comput., 28 (2006), pp. 2162–2188.Google Scholar
  8. 8.
    W. Dahmen, A. Kunoth, and K. Urban, Biorthogonal spline wavelets on the interval – stability and moment conditions, Appl. Comput. Harmon. Anal. 6 (1999), pp. 132–196.Google Scholar
  9. 9.
    J. Dick, On the convergence rate of the component-by-component construction of good lattice rules, J. Complexity, 20 (2004), pp. 493–522.Google Scholar
  10. 10.
    J. Dick, F. Y. Kuo, I. H. Sloan, High-dimensional integration: the Quasi-Monte Carlo way, Acta Numer. 22(2013), pp. 133–288.Google Scholar
  11. 11.
    J. Dick and F. Pillichshammer, Digital Nets and Sequences, Cambridge University Press, 2010.Google Scholar
  12. 12.
    J. Dick, F. Pillichshammer, and B. J. Waterhouse, The construction of good extensible rank-1 lattices, Math. Comp., 77 (2008), pp. 2345–2374.Google Scholar
  13. 13.
    J. Dick, I. H. Sloan, X. Wang, and H. Woźniakowski, Liberating the weights, J. Complexity, 20 (2004), pp. 593–623.Google Scholar
  14. 14.
    D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer-Verlag, New York, 2nd Ed., 2001.Google Scholar
  15. 15.
    M.B. Giles, Improved multilevel Monte Carlo convergence using the Milstein scheme, Monte Carlo and Quasi-Monte Carlo methods 2006, pp 343–358, Springer, 2007.Google Scholar
  16. 16.
    M.B. Giles, Multilevel Monte Carlo path simulation, Oper. Res. 256 (2008), pp. 981–986.Google Scholar
  17. 17.
    M. Gnewuch, Infinite-dimensional integration on weighted Hilbert spaces, Math. Comp., 81 (2012), pp. 2175–2205.Google Scholar
  18. 18.
    I. G. Graham, F. Y. Kuo, J. Nichols, R. Scheichl, Ch. Schwab, and I. H. Sloan, QMC FE methods for PDEs with log-normal random coefficients, Numer. Math. (2014). doi:10.1007/s00211-014-0689-y.
  19. 19.
    I. G. Graham, F. Y. Kuo, D. Nuyens, R. Scheichl, and I. H. Sloan, Quasi-Monte Carlo methods for elliptic PDEs with random coefficients and applications, J. Comput. Phys., 230 (2011), pp. 3668–3694.Google Scholar
  20. 20.
    S. Heinrich, Multilevel Monte Carlo methods, Lecture notes in Compu. Sci. Vol. 2179, pp. 3624–3651, Springer, 2001.Google Scholar
  21. 21.
    F. J. Hickernell, T. Müller-Gronbach, B. Niu, and K. Ritter, Multi-level Monte Carlo algorithms for infinite-dimensional integration on \({{\mathbb{R}}}^{{\mathbb{N}}}\). J. Complexity, 26 (2010), pp. 229–254. Google Scholar
  22. 22.
    F. Y. Kuo, Component-by-component constructions achieve the optimal rate of convergence for multivariate integration in weighted Korobov and Sobolev spaces, J. Complexity, 19 (2003), pp. 301–320.Google Scholar
  23. 23.
    F. Y. Kuo, Ch. Schwab, and I. H. Sloan, Quasi-Monte Carlo methods for high dimensional integration: the standard weighted-space setting and beyond, ANZIAM J. 53 (2011), pp 1–37.Google Scholar
  24. 24.
    F. Y. Kuo, Ch. Schwab, and I. H. Sloan, Quasi-Monte Carlo finite element methods for a class of elliptic partial differential equations with random coefficient, SIAM J. Numer. Anal., 50 (2012), pp. 3351–3374.Google Scholar
  25. 25.
    F. Y. Kuo, I. H. Sloan, G. W. Wasilkowski, and H. Woźniakowski, Liberating the dimension, J. Complexity, 26 (2010), pp. 422–454.Google Scholar
  26. 26.
    H. Nguyen, Finite element wavelets for solving partial differential equations, Ph.D. Thesis, Department of Mathematics, Universiteit Utrecht, The Netherlands, 2005.Google Scholar
  27. 27.
    B. Niu, F.J. Hickernell, T. Müller-Gronbach, and K. Ritter, Deterministic multi-level algorithms for infinite-dimensional integration on \({\mathbb{R}}^{\mathbb{N}}\), J. Complexity, 27 (2011), pp. 331–351. Google Scholar
  28. 28.
    D. Nuyens and R. Cools, Fast algorithms for component-by-component construction of rank-1 lattice rules in shift-invariant reproducing kernel Hilbert spaces, Math. Comp., 75 (2006), pp. 903–920.Google Scholar
  29. 29.
    D. Nuyens and R. Cools, Fast component-by-component construction of rank-1 lattice rules with a non-prime number of points, J. Complexity, 22 (2006), pp. 4–28.Google Scholar
  30. 30.
    L. Plaskota and G. W. Wasilkowski, Tractability of infinite-dimensional integration in the worst case and randomized settings, J. Complexity, 27 (2011), pp. 505–518.Google Scholar
  31. 31.
    Ch. Schwab and C. J. Gittelson, Sparse tensor discretizations of high-dimensional parametric and stochastic PDEs, Acta Numer., 20 (2011), pp. 291–467.Google Scholar
  32. 32.
    Ch. Schwab and R. A. Todor, Karhunen–Loève approximation of random fields by generalized fast multipole methods, J. Comput. Phy., 217 (2006), pp. 100–122.Google Scholar
  33. 33.
    I. H. Sloan, F. Y. Kuo, and S. Joe, Constructing randomly shifted lattice rules in weighted Sobolev spaces, SIAM J. Numer. Anal., 40 (2002), pp. 1650–1665.Google Scholar
  34. 34.
    I. H. Sloan and H. Woźniakowski, When are quasi-Monte Carlo algorithms efficient for high-dimensional integrals?, J. Complexity, 14 (1998), pp. 1–33.Google Scholar
  35. 35.
    I. H. Sloan, X. Wang, and H. Woźniakowski, Finite-order weights imply tractability of multivariate integration, J. Complexity, 20 (2004), pp. 46–74.Google Scholar
  36. 36.
    A. L. Teckentrup, R. Scheichl, M. B. Giles, and E. Ullmann Further analysis of multilevel Monte Carlo methods for elliptic PDEs with random coefficient, Numer. Math., 125 (2013), pp. 569–600.Google Scholar

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

Personalised recommendations