Construction of Quasi-Monte Carlo Rules for Multivariate Integration in Spaces of Permutation-Invariant Functions

Abstract

We study multivariate integration of functions that are invariant under the permutation (of a subset) of their arguments. Recently, in Nuyens et al. (Adv Comput Math 42(1):55–84, 2016), the authors derived an upper estimate for the nth 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 that 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.