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QMC integration for lognormal-parametric, elliptic PDEs: local supports and product weights

by L. Herrmann and Ch. Schwab

(Report number 2016-39)

Abstract
We analyze convergence rates of quasi-Monte Carlo (QMC) quadratures for countably-parametric solutions of linear, elliptic partial differential equations (PDE) in divergence form with log-Gaussian diffusion coefficient, based on the error bounds in [James A. Nichols and Frances Y. Kuo: Fast CBC construction of randomly shifted lattice rules achieving \(\mathcal{O}(N^{-1+\delta})\) convergence for unbounded integrands over \(\mathbb{R}^s\) in weighted spaces with POD weights. J. Complexity, 30(4):444-468, 2014]. We prove, for representations of the Gaussian random field PDE input with locally supported basis functions, and for continuous, piecewise polynomial Finite Element discretizations in the physical domain novel QMC error bounds in weighted spaces with \emph{product weights} that exploit localization of supports of the basis elements representing the input Gaussian random field. In this case, the cost of the fast component-by-component algorithm for constructing the QMC points scales linearly in terms of the integration dimension. The QMC convergence rate \(\mathcal{O}(N^{-1+\delta})\) (independent of the parameter space dimension \(s\) ) is achieved under weak summability conditions on the expansion coefficients.

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

@Techreport{HS16_676,
  author = {L. Herrmann and Ch. Schwab},
  title = {QMC integration for lognormal-parametric, elliptic PDEs: local supports and product weights},
  institution = {Seminar for Applied Mathematics, ETH Z{\"u}rich},
  number = {2016-39},
  address = {Switzerland},
  url = {https://www.sam.math.ethz.ch/sam_reports/reports_final/reports2016/2016-39.pdf },
  year = {2016}
}

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