Higher order Quasi-Monte Carlo integration for holomorphic, parametric operator equations

TL;DR

This paper investigates the convergence of higher order Quasi-Monte Carlo methods for solving parametric nonlinear operator equations with uncertain inputs, extending previous regularity results to more general problem classes.

Contribution

It extends regularity analysis and QMC convergence results to countably-parametric, nonlinear operator equations with general Banach space inputs, using SPOD weights and hybrid CBC construction.

Findings

Established convergence rates for higher order QMC quadratures in this setting.

Extended regularity results to nonlinear, countably-parametric operator equations.

Provided a hybridized CBC construction for efficient QMC point set generation.

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 SPOD weights recently introduced in [F.Y.~Kuo, Ch.~Schwab, I.H.~Sloan, Quasi-Monte Carlo finite element methods for a class of elliptic partial differential equations with random coefficients. SIAM J. Numer. Anal., 50, 3351--3374, 2012.] can be used to describe the solution regularity. The regularity results in the present paper extend those in [J. Dick, F.Y.~Kuo, Q.T.~Le Gia, D.~Nuyens, Ch.~Schwab, Higher order QMC (Petrov-)Galerkin discretization for parametric operator equations. SIAM J. Numer. Anal., 52, 2676 -- 2702, 2014.] established for affine parametric, linear operator families; they imply, in particular, efficient constructions of (sequences of) QMC quadrature methods there, which are applicable to these problem classes. We present a hybridized version of the fast component-by-component (CBC for short) construction of a certain type of higher order digital net.