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

Dirk Nuyens, Gowri Suryanarayana, Markus Weimar

Published 2015-07-29Version 1

We study multivariate integration of functions that are invariant under the permutation (of a subset) of their arguments. Recently an upper estimate for the $n$th minimal worst case error for such problems was derived and it was shown that under certain conditions this upper bound only weakly depends on the number of dimensions. 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 O(n^{-\alpha}), where $\alpha>1/2$ denotes the smoothness of our function space. Further, we develop a semi-constructive algorithm that builds on point sets which allow 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.