Rank-1 lattice rules for multivariate integration in spaces of permutation-invariant functions. Error bounds and tractability (Q5965004)

The authors study high-dimensional integration problems of 1-periodic functions \(f:\,[0,\,1]^d \to {\mathbb C}\) that are invariant under permutations of their arguments. Using reproducing kernel Hilbert spaces and their permutation-invariant subspaces, they bound the minimal worst case error of quasi-Monte Carlo methods for the integration of \(f\). Under certain conditions, the minimal worst case error can be bounded independent of \(d\). Applying shifted rank-1 lattice rules, the authors derive conditions under which multivariate integration is (strongly) polynomially tractable. Finally, the authors show the existence of rank-1 lattice rules whose worst case error on the permutation- and shift-invariant spaces converge with (almost) optimal rate. This theory is motivated by problems from computational quantum physics.