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

## 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## Notes

### Acknowledgments

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.

## References

- 1.E. Bach and J. Shallit,
*Algorithmic Number Theory (Volume I: Efficient Algorithms)*, MIT Press, Cambridge, 1966.Google Scholar - 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.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.P. G. Ciarlet,
*The Finite Element Method for Elliptic Problems*, Elsevier, Amsterdam 1978.Google Scholar - 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.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.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.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.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.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.J. Dick and F. Pillichshammer,
*Digital Nets and Sequences*, Cambridge University Press, 2010.Google Scholar - 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.J. Dick, I. H. Sloan, X. Wang, and H. Woźniakowski,
*Liberating the weights*, J. Complexity,**20**(2004), pp. 593–623.Google Scholar - 14.D. Gilbarg and N. S. Trudinger,
*Elliptic Partial Differential Equations of Second Order*, Springer-Verlag, New York, 2nd Ed., 2001.Google Scholar - 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.M.B. Giles,
*Multilevel Monte Carlo path simulation*, Oper. Res.**256**(2008), pp. 981–986.Google Scholar - 17.M. Gnewuch,
*Infinite-dimensional integration on weighted Hilbert spaces*, Math. Comp.,**81**(2012), pp. 2175–2205.Google Scholar - 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.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.S. Heinrich,
*Multilevel Monte Carlo methods*, Lecture notes in Compu. Sci. Vol. 2179, pp. 3624–3651, Springer, 2001.Google Scholar - 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.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.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.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.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.H. Nguyen,
*Finite element wavelets for solving partial differential equations*, Ph.D. Thesis, Department of Mathematics, Universiteit Utrecht, The Netherlands, 2005.Google Scholar - 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.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.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.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.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.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.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.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.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.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