Optimal randomized changing dimension algorithms for infinite-dimensional integration on function spaces with ANOVA-type decomposition

TL;DR

This paper develops and analyzes optimal randomized changing dimension algorithms for infinite-dimensional integration in function spaces with ANOVA decomposition, achieving near-best convergence rates under certain weight conditions.

Contribution

It introduces new randomized changing dimension algorithms, provides matching lower and upper error bounds, and demonstrates their optimality for specific weight classes.

Findings

Matching lower and upper error bounds for product and finite-intersection weights.

Changing dimension algorithms achieve near-optimal convergence rates.

Analysis extends previous work with a different cost model.

Abstract

We study the numerical integration problem for functions with infinitely many variables. The function spaces of integrands we consider are weighted reproducing kernel Hilbert spaces with norms related to the ANOVA decomposition of the integrands. The weights model the relative importance of different groups of variables. We investigate randomized quadrature algorithms and measure their quality by estimating the randomized worst-case integration error. In particular, we provide lower error bounds for a very general class of randomized algorithms that includes non-linear and adaptive algorithms. Furthermore, we propose new randomized changing dimension algorithms and present favorable upper error bounds. For product weights and finite-intersection weights our lower and upper error bounds match and show that our changing dimension algorithms are optimal in the sense that they achieve convergence rates arbitrarily close to the best possible convergence rate. As more specific examples, we discuss unanchored Sobolev spaces of different degrees of smoothness and randomized changing dimension algorithms that use as building blocks scrambled polynomial lattice rules. Our analysis extends the analysis given in [J. Baldeaux, M. Gnewuch. Optimal randomized multilevel algorithms for infinite-dimensional integration on function spaces with ANOVA-type decomposition. arXiv:1209.0882v1 [math.NA], Preprint 2012]. In contrast to the previous article we now investigate a different cost model for algorithms. With respect to that cost model, randomized multilevel algorithms cannot, in general, achieve optimal convergence rates, but, as we prove for important classes of weights, changing dimension algorithms actually can.