Rank-1 lattice rules for multivariate integration in spaces of permutation-invariant functions: Error bounds and tractability

TL;DR

This paper investigates the use of rank-1 lattice rules for multivariate integration of permutation-invariant functions, establishing error bounds, tractability conditions, and the necessity of shifts for optimal convergence.

Contribution

It provides new error bounds and tractability results for lattice rules in permutation-invariant spaces, including the importance of shifts and existence of optimal lattice rules.

Findings

Error bounds independent of dimensions under certain conditions

Polynomial and strong polynomial tractability with Monte Carlo rate

Existence of lattice rules with near-optimal convergence rates

Abstract

We study multivariate integration of functions that are invariant under permutations (of subsets) of their arguments. We find an upper bound for the $n$th minimal worst case error and show that under certain conditions, it can be bounded independent of the number of dimensions. In particular, we study the application of unshifted and randomly shifted rank-$1$ lattice rules in such a problem setting. We derive conditions under which multivariate integration is polynomially or strongly polynomially tractable with the Monte Carlo rate of convergence $O(n^{-1/2})$. Furthermore, we prove that those tractability results can be achieved with shifted lattice rules and that the shifts are indeed necessary. Finally, we show the existence of rank-$1$ lattice rules whose worst case error on the permutation- and shift-invariant spaces converge with (almost) optimal rate. That is, we derive error bounds of the form $O(n^{-\lambda/2})$ for all $1 \leq \lambda < 2 \alpha$, where $\alpha$ denotes the smoothness of the spaces. Keywords: Numerical integration, Quadrature, Cubature, Quasi-Monte Carlo methods, Rank-1 lattice rules.