Infinite-Dimensional Quadrature and Quantization

TL;DR

This paper investigates numerical integration of Lipschitz functionals on Banach spaces, establishing bounds and algorithms for Gaussian measures and diffusion processes, linking quadrature to measure quantization.

Contribution

It introduces a comprehensive analysis connecting quadrature and measure quantization, providing bounds and near-optimal algorithms for integration in infinite-dimensional spaces.

Findings

Derived lower bounds for worst-case errors based on computational cost.

Presented matching upper bounds and almost optimal algorithms.

Analyzed asymptotic behavior of quantization numbers and Kolmogorov widths.

Abstract

We study numerical integration of Lipschitz functionals on a Banach space by means of deterministic and randomized (Monte Carlo) algorithms. This quadrature problem is shown to be closely related to the problem of quantization of the underlying probability measure. In addition to the general setting we analyze in particular integration w.r.t. Gaussian measures and distributions of diffusion processes. We derive lower bounds for the worst case error of every algorithm in terms of its computational cost, and we present matching upper bounds, up to logarithms, and corresponding almost optimal algorithms. As auxiliary results we determine the asymptotic behaviour of quantization numbers and Kolmogorov widths for diffusion processes.