Higher Order Quasi--Monte Carlo Integration for Holomorphic, Parametric Operator Equations

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 $X$ admitting an unconditional Schauder basis. 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, deterministic. 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 smoothness-driven product-and-order dependent (SPOD) weights recently introduced in [F. Y. Kuo, Ch. Schwab, I. H. Sloan, SIAM J. Numer. Anal., 50 (2012), pp. 3351--3374] can be us...