Optimal order quasi-Monte Carlo integration in weighted Sobolev spaces of arbitrary smoothness
Optimal order quasi-Monte Carlo integration in weighted Sobolev spaces of arbitrary smoothness
We investigate quasi-Monte Carlo integration using higher order digital nets in weighted Sobolev spaces of arbitrary fixed smoothness $\alpha \in \mathbb{N}$, $\alpha \ge 2$, defined over the $s$-dimensional unit cube. We prove that randomly digitally shifted order $\beta$ digital nets can achieve the convergence of the root mean square worst-case …