The construction of good lattice rules and polynomial lattice rules

A comprehensive overview of lattice rules and polynomial lattice rules is given for function spaces based on $\ell_p$ semi-norms. Good lattice rules and polynomial lattice rules are defined as those obtaining worst-case errors bounded by the optimal rate of convergence for the function space. The focus is on algebraic rates of convergence $O(N^{-\alpha+\epsilon})$ for $\alpha \ge 1$ and any $\epsilon > 0$, where $\alpha$ is the decay of a series representation of the integrand function. The dependence of the implied constant on the dimension can be controlled by weights which determine the influence of the different dimensions. Different types of weights are discussed. The construction of good lattice rules, and polynomial lattice rules, can be done using the same method for all $1 < p \le \infty$; but the case $p=1$ is special from the construction point of view. For $1 < p \le \infty$ the component-by-component construction and its fast algorithm for different weighted function spaces is then discussed

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

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.Comment: 26 pages; minor changes due to reviewer's comments; the final publication is available at link.springer.co

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

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