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QMC Algorithms with Product Weights for Lognormal-Parametric, Elliptic PDEs

by L. Herrmann and Ch. Schwab

(Report number 2017-04)

Abstract
We survey recent convergence rate bounds for single-level and multilevel QMC Finite Element (FE for short) algorithms for the numerical approximation of linear, second order elliptic PDEs in divergence form in a bounded, polygonal domain $D$ . The diffusion coefficient $a$ is assumed to be an isotropic, log-Gaussian random field (GRF for short) in $D$ . The representation of the GRF $Z = \log a$ is assumed affine-parametric with i.i.d. standard normal random variables, and with \emph{locally supported} functions $\psi_j$ characterizing the spatial variation of the GRF $Z$ . The goal of computation is the evaluation of expectations (i.e., of so-called ``ensemble averages'') of (linear functionals of) the random solution. The QMC rules employed are randomly shifted lattice rules proposed in [J.A. Nichols and F.Y. Kuo: Fast CBC construction of randomly shifted lattice rules achieving O(N−1+δ) convergence for unbounded integrands over Rs in weighted spaces with POD weights. J. Complexity, 30(4):444-468, 2014] as used and analyzed previously in a similar setting (albeit for globally in $D$ supported spatial representation functions $\psi_j$ as arise in Karhunen Lo\`eve expansions) in [I.G. Graham, F.Y. Kuo, J.A. Nichols, R. Scheichl, Ch. Schwab, and I.H. Sloan: Quasi-Monte Carlo finite element methods for elliptic PDEs with lognormal random coefficients. Numer. Math. 131(2), 329-368 (2015)], [F.Y. Kuo, R. Scheichl, Ch. Schwab, I.H. Sloan, and E. Ullmann: Multilevel Quasi-Monte Carlo methods for lognormal diffusion problems, Math. Comp. 86(308), 2827–2860 (2017)]. The multilevel QMC-FE algorithm $Q^*_L$ analyzed here for locally supported $\psi_j$ was proposed first in [F.Y. Kuo, Ch. Schwab, and I.H. Sloan: Multi-level Quasi-Monte Carlo Finite Element Methods for a Class of Elliptic PDEs with Random Coefficients, Journ. Found. Comp. Math., 15(2), 411-449 (2015)] for affine-parametric operator equations. As shown in [R.N. Gantner, L. Herrmann, and Ch. Schwab: Quasi-Monte Carlo integration for affine-parametric, elliptic PDEs: local supports and product weights, Tech. Rep. 2016-32, Seminar for Applied Mathematics, ETH Z\"urich], [R.N. Gantner, L. Herrmann, and Ch. Schwab: Multilevel QMC with product weights for affine-parametric, elliptic PDEs, Tech. Rep. 2016-54, Seminar for Applied Mathematics, ETH Z\"urich], [L. Herrmann and Ch. Schwab: QMC integration for lognormal-parametric, elliptic PDEs: local supports and product weights, Tech. Rep. 2016-39, Seminar for Applied Mathematics, ETH Z\"urich], [L. Herrmann and Ch. Schwab: Multilevel quasi-Monte Carlo integration with product weights for elliptic PDEs with lognormal coefficients, Tech. Rep. 2017-19, Seminar for Applied Mathematics, ETH Z\"urich] localized supports of the $\psi_j$ (which appear in multiresolution representations of GRFs $Z$ of L\'{e}vy-Cieselski type in $D$ ) allow for the use of product weights, originally proposed in construction of QMC rules in [I.H. Sloan and H. Wo{\'z}niakowski: When are quasi-{M}onte {C}arlo algorithms efficient for high-dimensional integrals?, J. Complexity 14(1), 1-33 (1998)] (cp.~the survey [J. Dick, F.Y. Kuo: High-dimensional integration: the quasi-{M}onte {C}arlo way, 22, 133-288 (2013)] and references there). The present results from [L. Herrmann and Ch. Schwab: Multilevel quasi-Monte Carlo integration with product weights for elliptic PDEs with lognormal coefficients, Tech. Rep. 2017-19, Seminar for Applied Mathematics, ETH Z\"urich] on convergence rates for the MLQMC FE algorithm allow for general polygonal domains $D$ and for GRFs $Z$ whose realizations take values in weighted spaces containing $W^{1,\infty}(D)$ . Localized support assumptions on $\psi_j$ are shown to allow QMC rule generation by the fast, FFT based CBC constructions in [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(1), 4-28 (2006)] and [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(254), 903-920, (2006)] which scale linearly in the integration dimension which, for multiresolution representations of GRFs, is proportional to the number of degrees of freedom used in the FE discretization in the physical domain $D$ . We show numerical experiments based on public domain QMC rule generating software in [F.Y. Kuo and D. Nuyens: Application of Quasi-Monte Carlo Methods to Elliptic PDEs with Random Diffusion Coefficients: A Survey of Analysis and Implementation, Found. Comput. Math., 16(6), 1631-1696, (2016)] and [R.N. Gantner: A Generic C++ Library for Multilevel Quasi-Monte Carlo, In: Proceedings of the Platform for Advanced Scientific Computing Conference, PASC'16. pp. 112:1-11:12. ACM, New York, NY, USA (2016)].

Keywords: Quasi-Monte Carlo methods, multilevel quasi-Monte Carlo, uncertainty quantification, error estimates, high-dimensional quadrature, elliptic partial differential equations with lognormal input

BibTeX

@Techreport{HS17_700,
  author = {L. Herrmann and Ch. Schwab},
  title = {QMC Algorithms with Product Weights for Lognormal-Parametric, Elliptic PDEs},
  institution = {Seminar for Applied Mathematics, ETH Z{\"u}rich},
  number = {2017-04},
  address = {Switzerland},
  url = {https://www.sam.math.ethz.ch/sam_reports/reports_final/reports2017/2017-04.pdf },
  year = {2017}
}

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