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

TL;DR

This paper develops optimal randomized multilevel algorithms for infinite-dimensional integration in weighted reproducing kernel Hilbert spaces with ANOVA decomposition, providing both upper and lower error bounds.

Contribution

It introduces new multilevel algorithms and establishes their optimality through matching upper and lower error bounds, extending previous analyses.

Findings

Proposed algorithms achieve optimal error bounds.

Established the first non-trivial lower bounds for randomized algorithms.

Demonstrated effectiveness with Sobolev space example using scrambled polynomial lattice rules.

Abstract

In this paper, we consider the infinite-dimensional integration problem on weighted reproducing kernel Hilbert spaces with norms induced by an underlying function space decomposition of ANOVA-type. The weights model the relative importance of different groups of variables. We present new randomized multilevel algorithms to tackle this integration problem and prove upper bounds for their randomized error. Furthermore, we provide in this setting the first non-trivial lower error bounds for general randomized algorithms, which, in particular, may be adaptive or non-linear. These lower bounds show that our multilevel algorithms are optimal. Our analysis refines and extends the analysis provided in [F. J. Hickernell, T. M\"uller-Gronbach, B. Niu, K. Ritter, J. Complexity 26 (2010), 229-254], and our error bounds improve substantially on the error bounds presented there. As an illustrative example, we discuss the unanchored Sobolev space and employ randomized quasi-Monte Carlo multilevel algorithms based on scrambled polynomial lattice rules.