Infinite-Dimensional Integration in Weighted Hilbert Spaces: Anchored Decompositions, Optimal Deterministic Algorithms, and Higher Order Convergence

TL;DR

This paper investigates infinite-dimensional numerical integration in weighted Hilbert spaces, establishing optimal algorithms and convergence rates using multilevel and changing dimension methods, especially with higher-order polynomial lattice rules.

Contribution

It introduces new optimal algorithms and convergence results for infinite-dimensional integration in weighted Hilbert spaces with various weight classes and cost models.

Findings

Multilevel algorithms achieve optimal convergence rates in the first cost model.

Changing dimension algorithms achieve optimal convergence in the second cost model.

New quasi-Monte Carlo algorithms based on polynomial lattice rules are developed for Sobolev spaces.

Abstract

We study numerical integration of functions depending on an infinite number of variables. We provide lower error bounds for general deterministic linear algorithms and provide matching upper error bounds with the help of suitable multilevel algorithms and changing dimension algorithms. More precisely, the spaces of integrands we consider are weighted reproducing kernel Hilbert spaces with norms induced by an underlying anchored function space decomposition. Here the weights model the relative importance of different groups of variables. The error criterion used is the deterministic worst case error. We study two cost models for function evaluation which depend on the number of active variables of the chosen sample points, and two classes of weights, namely product and order-dependent (POD) weights and the newly introduced weights with finite active dimension. We show for these classes of weights that multilevel algorithms achieve the optimal rate of convergence in the first cost model while changing dimension algorithms achieve the optimal convergence rate in the second model. As an illustrative example, we discuss the anchored Sobolev space with smoothness parameter $\alpha$ and provide new optimal quasi-Monte Carlo multilevel algorithms and quasi-Monte Carlo changing dimension algorithms based on higher-order polynomial lattice rules.