Optimal order quasi-Monte Carlo integration in weighted Sobolev spaces of arbitrary smoothness

Abstract

We investigate quasi-Monte Carlo integration using higher order digital nets in weighted Sobolev spaces of arbitrary fixed smoothness $α\in \mathbb{N}$, $α\ge 2$, defined over the $s$-dimensional unit cube. We prove that randomly digitally shifted order $β$ digital nets can achieve the convergence of the root mean square worst-case error of order $N^{-α}(\log N)^{(s-1)/2}$ when $β\ge 2α$. The exponent of the logarithmic term, i.e., $(s-1)/2$, is improved compared to the known result by Baldeaux and Dick, in which the exponent is $sα/2$. Our result implies the existence of a digitally shifted order $β$ digital net achieving the convergence of the worst-case error of order $N^{-α}(\log N)^{(s-1)/2}$, which matches a lower bound on the convergence rate of the worst-case error for any cubature rule using $N$ function evaluations and thus is best possible.