On lower bounds for integration of multivariate permutation-invariant functions

Abstract

In this note we study multivariate integration for permutation-invariant functions from a certain Banach space E_{d,α} of Korobov type in the worst case setting. We present a lower error bound which particularly implies that in dimension d every cubature rule which reduces the initial error necessarily uses at least d+1 function values. Since this holds independently of the number of permutation-invariant coordinates, this shows that the integration problem can never be strongly polynomially tractable in this setting. Our assertions generalize results due to Sloan and Woźniakowski. Moreover, for large smoothness parameters αour bound can not be improved. Finally, we extend our results to the case of permutation-invariant functions from Korobov-type spaces equipped with product weights. Keywords: Permutation-invariance, Integration, Information complexity, Tractability, Lower bounds