MULTILEVEL HIGHER ORDER QMC PETROV-GALERKIN DISCRETIZATION FOR AFFINE PARAMETRIC OPERATOR EQUATIONS

作者：Dick Josef; Kuo Frances Y; Le Gia Quoc T; Schwab Christoph

来源：SIAM Journal on Numerical Analysis, 2016, 54(4): 2541-2568.

DOI：10.1137/16M1078690

• quasi-Monte Carlo methods
• multilevel methods
• interlaced polynomial lattice rules
• higher order digital nets
• affine parametric operator equations
• Petrov-Galerkin discretization

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-Loeve 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 con firm the theoretical results.

• 出版日期2016