Multilevel QMC with Product Weights for Affine-Parametric, Elliptic PDEs
METADATA ONLY
Loading...
Author / Producer
Date
2016-12
Publication Type
Report
ETH Bibliography
yes
Citations
Altmetric
METADATA ONLY
Data
Rights / License
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.
Permanent link
Publication status
published
Editor
Book title
Journal / series
Volume
2016-54
Pages / Article No.
Publisher
Seminar for Applied Mathematics, ETH Zurich
Event
Edition / version
Methods
Software
Geographic location
Date collected
Date created
Subject
Organisational unit
03435 - Schwab, Christoph / Schwab, Christoph