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

TL;DR

This paper presents a convergence analysis of a multi-level higher order QMC Galerkin method for affine parametric operator equations, extending previous analyses and demonstrating improved efficiency for certain PDE problems.

Contribution

It develops a comprehensive convergence analysis for multi-level higher order QMC Galerkin discretizations, extending prior work and including non-smooth domains and indefinite systems.

Findings

Multi-level higher order QMC algorithms outperform single level methods in certain cases.

Theoretical error bounds guide optimal parameter choices for prescribed accuracy.

Numerical experiments confirm the theoretical convergence and efficiency improvements.

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 [\emph{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 [\emph{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 {\KL} eigenfunctions in the parametrization of random PDE inputs on the convergence results. Based on our \emph{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. Numerical experiments confirm the theoretical results.