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Multilevel QMC with Product Weights for Affine-Parametric, Elliptic PDEs

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

(Report number 2016-54)

Abstract
We present an error analysis of higher order Quasi-Monte Carlo (QMC) integration and of randomly shifted QMC lattice rules for parametric operator equations with uncertain input data taking values in Banach spaces. Parametric expansions of these input data in locally supported bases such as splines or wavelets was shown in [R.N.\ Gantner, L.\ Herrmann, and Ch.\ Schwab: Quasi-Monte Carlo integration for affine-parametric, elliptic {PDE}s: local supports and product weights. SIAM J. Numer. Anal., (2017). to appear] to allow for dimension independent convergence rates of combined QMC-Galerkin approximations. In the present work, we review and refine the results in that reference to the multilevel setting, along the lines of [F.Y.\ Kuo, {\relax Ch}.\ Schwab, and I.H.\ Sloan: Multi-level Quasi-{M}onte {C}arlo Finite Element Methods for a Class of Elliptic {PDE}s with Random Coefficients. Found. Comput. Math. {\bf 15}(2), 441--449 (2015)] where randomly shifted lattice rules and globally supported representations were considered, and also the results of [J.\ Dick, F.Y.\ Kuo, Q.T.\ LeGia, and Ch.\ Schwab: Multilevel higher order QMC Petrov-Galerkin discretization for affine parametric operator equations. SIAM J. Numer. Anal. {\bf 54}(4), 2541--2568 (2016)] in the particular situation of locally supported bases in the parametrization of uncertain input data. In particular, we show that locally supported basis functions allow for multilevel QMC quadrature with product weights, and prove new error vs. work estimates superior to those in these references (albeit at stronger, mixed regularity assumptions on the parametric integrand functions than what was required in the single-level QMC error analysis in the first reference above). Numerical experiments on a model affine parametric elliptic problem confirm the analysis.

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

@Techreport{GHS16_691,
  author = {R. N. Gantner and L. Herrmann and Ch. Schwab},
  title = {Multilevel QMC with Product Weights for Affine-Parametric, Elliptic PDEs},
  institution = {Seminar for Applied Mathematics, ETH Z{\"u}rich},
  number = {2016-54},
  address = {Switzerland},
  url = {https://www.sam.math.ethz.ch/sam_reports/reports_final/reports2016/2016-54.pdf },
  year = {2016}
}

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