Construction of quasi-Monte Carlo rules for multivariate integration in spaces of permutation-invariant functions

TL;DR

This paper develops new quasi-Monte Carlo methods for efficiently integrating permutation-invariant functions in high dimensions, achieving near-optimal convergence rates with improved dimension dependence.

Contribution

It introduces two explicit construction algorithms for lattice rules that attain near-optimal convergence rates for permutation-invariant functions, with enhanced dimension dependence.

Findings

Achieves convergence rate close to O(n^{-eta}) for smoothness b1>1/2.

Develops a component-by-component algorithm for generating vectors.

Provides a semi-constructive method with improved dimension dependence.

Abstract

We study multivariate integration of functions that are invariant under the permutation (of a subset) of their arguments. Recently, in Nuyens, Suryanarayana, and Weimar (Adv. Comput. Math. (2016), 42(1):55--84), the authors derived an upper estimate for the $n$th minimal worst case error for such problems, and showed that under certain conditions this upper bound only weakly depends on the dimension. We extend these results by proposing two (semi-) explicit construction schemes. We develop a component-by-component algorithm to find the generating vector for a shifted rank-$1$ lattice rule that obtains a rate of convergence arbitrarily close to $\mathcal{O}(n^{-\alpha})$, where $\alpha>1/2$ denotes the smoothness of our function space and $n$ is the number of cubature nodes. Further, we develop a semi-constructive algorithm that builds on point sets which can be used to approximate the integrands of interest with a small error; the cubature error is then bounded by the error of approximation. Here the same rate of convergence is achieved while the dependence of the error bounds on the dimension $d$ is significantly improved.