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Quasi-Monte Carlo integration for affine-parametric, elliptic PDEs: local supports and product weights

by R. N. Gantner and L. Herrmann and Ch. Schwab

(Report number 2016-32)

Abstract
We analyze convergence rates of first order quasi-Monte Carlo (QMC) integration with randomly shifted lattice rules and for higher order, interlaced polynomial lattice rules, for a class of countably parametric integrands that result from linear functionals of solutions of linear, elliptic diffusion equations with affine-parametric, uncertain coefficient function \(a(x,\boldsymbol{y}) = \bar{a}(x) + \sum_{j\geq 1} y_j \psi_j(x)\) in a bounded domain \(D\subset \mathbb{R}^d\). Extending the result in [F. Y. Kuo, Ch. Schwab, and I. H. Sloan, SIAM J. Numer. Anal., 50 (2012), pp. 3351-3374], where \(\psi_j\) was assumed to have global support in the domain \(D\), we assume in the present paper that \({\rm supp}(\psi_j)\) is localized in \(D\), and that we have control on the overlaps of these supports. Under these conditions we prove \emph{dimension independent convergence rates} in \([1/2,1)\) of randomly shifted lattice rules with \emph{product weights} and corresponding higher order convergence rates by higher order, interlaced polynomial lattice rules with product weights. The product structure of the QMC weights facilitates work bounds for the fast, component-by-component constructions of [D. Nuyens and R. Cools, Math. Comp., 75 (2006), pp. 903-920] which scale linearly with respect to the parameter dimension \(s\). The dimension independent convergence rates are only limited by the degree of digit interlacing used in the construction of the higher order QMC quadrature rule and, for locally supported coefficient functions, by the summability of the locally supported coefficient sequence in the affine-parametric coefficient.

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

@Techreport{GHS16_669,
  author = {R. N. Gantner and L. Herrmann and Ch. Schwab},
  title = {Quasi-Monte Carlo integration for affine-parametric, elliptic PDEs: local supports and product weights},
  institution = {Seminar for Applied Mathematics, ETH Z{\"u}rich},
  number = {2016-32},
  address = {Switzerland},
  url = {https://www.sam.math.ethz.ch/sam_reports/reports_final/reports2016/2016-32.pdf },
  year = {2016}
}

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