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Higher order Quasi Monte Carlo integration for holomorphic, parametric operator equations

by J. Dick and Q. T. Le Gia and Ch. Schwab

(Report number 2014-23)

Abstract
We analyze the convergence of higher order Quasi-Monte Carlo (QMC) quadratures of solution-functionals to countably-parametric, nonlinear operator equations with distributed uncertain parameters taking values in a separable Banach space \(X\). Such equations arise in numerical uncertainty quantification with random field inputs. Unconditional bases of \(X\) render the random inputs and the solutions of the forward problem countably parametric. We show that these parametric solutions belong to a class of weighted Bochner spaces of functions of countably many variables, with a particular structure of the QMC quadrature weights: up to a (problem-dependent, and possibly large) finite dimension, product weights can be used, and beyond this dimension, weighted spaces with so-called SPOD weights recently introduced in [F.Y. Kuo, Ch. Schwab, I.H. Sloan, Quasi-Monte Carlo finite element methods for a class of elliptic partial differential equations with random coefficients. SIAM J. Numer. Anal. 50 (2012), 3351--3374] can be used to describe the solution regularity. The regularity results in the present paper extend those in [J. Dick, F.Y. Kuo, Q.T. Le Gia, D.~Nuyens, Ch. Schwab, Higher order QMC (Petrov-)Galerkin discretization for parametric operator equations. To appear in SIAM J. Numer. Anal., 2015. Available at arXiv:1309.4624] established for affine parametric, linear operator families; they imply, in particular, efficient constructions of (sequences of) QMC quadrature methods there, which are applicable to these problem classes. We present a hybridized version of the fast component-by-component (CBC for short) construction of a certain type of higher order digital net. We prove that this construction exploits the product nature of the QMC weights with linear scaling with respect to the integration dimension up to a possibly large, problem dependent finite dimension, and the SPOD structure of the weights with quadratic scaling with respect to the weights beyond this dimension.

Keywords: Quasi-Monte Carlo, lattice rules, digital nets, parametric operator equations, infinite-dimensional quadrature, Uncertainty Quantification, CBC construction, SPOD weights

@Techreport{DLS14_573,
  author = {J. Dick and Q. T. Le Gia and Ch. Schwab},
  title = {Higher order Quasi Monte Carlo integration for holomorphic, parametric operator equations},
  institution = {Seminar for Applied Mathematics, ETH Z{\"u}rich},
  number = {2014-23},
  address = {Switzerland},
  url = {https://www.sam.math.ethz.ch/sam_reports/reports_final/reports2014/2014-23.pdf },
  year = {2014}
}

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