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Multi-level higher order QMC Galerkin discretization for affine parametric operator equations

by J. Dick and F.Y. Kuo and Q.T. LeGia and Ch. Schwab

(Report number 2014-14)

Abstract
We develop a convergence analysis of a multi-level algorithm combining higher order quasi-Monte Carlo (QMC) quadratures with general Petrov-Galerkin discretizations of countably affine parametric operator equations of elliptic and parabolic type, extending both the multi-level first order analysis in [F.Y. Kuo, Ch. Schwab, and I.H. Sloan, Multi-level quasi-Monte Carlo finite element methods for a class of elliptic partial differential equations with random coefficient (in review)] and the single level higher order analysis in [J. Dick, F.Y. Kuo, Q.T. Le Gia, D. Nuyens, and Ch. Schwab, Higher order QMC Galerkin discretization for parametric operator equations (in review)]. We cover, in particular, both definite as well as indefinite, strongly elliptic systems of partial differential equations (PDEs) in non-smooth domains, and discuss in detail the impact of higher order derivatives of Karhunen-Loève eigenfunctions in the parametrization of random PDE inputs on the convergence results. Based on our a-priori error bounds, concrete choices of algorithm parameters are proposed in order to achieve a prescribed accuracy under minimal computational work. Problem classes and sufficient conditions on data are identified where multi-level higher order QMC Petrov- Galerkin algorithms outperform the corresponding single level versions of these algorithms.

Keywords: Quasi-Monte Carlo methods, multi-level methods, interlaced polynomial lattice rules, higher order digital nets, affine parametric operator equations, infinite dimensional quadrature, Petrov-Galerkin discretization

@Techreport{DKLS14_564,
  author = {J. Dick and F.Y. Kuo and Q.T. LeGia and Ch. Schwab},
  title = {Multi-level higher order QMC Galerkin discretization for affine parametric operator equations},
  institution = {Seminar for Applied Mathematics, ETH Z{\"u}rich},
  number = {2014-14},
  address = {Switzerland},
  url = {https://www.sam.math.ethz.ch/sam_reports/reports_final/reports2014/2014-14.pdf },
  year = {2014}
}

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