Multilevel higher order QMC Petrov-Galerkin discretization for affine parametric operator equations

Abstract

We develop a convergence analysis of a multilevel algorithm combining higher order quasi--Monte Carlo (QMC) quadratures with general Petrov--Galerkin discretizations of countably affine parametric operator equations of elliptic and parabolic types, extending both the multilevel first order analysis in [F. Y. Kuo, Ch. Schwab, and I. H. Sloan, Found. Comput. Math., 15 (2015), pp. 411--449] and the single level higher order analysis in [J. Dick et al., SIAM J. Numer. Anal., 52 (2014), pp. 2676--2702]. We cover, in particular, both definite as well as indefinite strongly elliptic systems of partial differential equations (PDEs) in nonsmooth domains, and we 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 multilevel higher order QMC Petrov--Galerkin algorithms outperform the corresponding single level versions of these algorithms. Numerical experiments confirm the theoretical results.ISSN:0036-1429ISSN:1095-717