Optimal quasi-Monte Carlo rules on order 2 digital nets for the numerical integration of multivariate periodic functions

Understand

We investigate quasi-Monte Carlo rules for the numerical integration of multivariate periodic functions from Besov spaces $S^r_{p,q}B(\mathbb{T}^d)$ with dominating mixed smoothness $1/p<r<2$.

• We show that order 2 digital nets achieve the optimal rate of convergence $N^{-r} (\log N)^{(d-1)(1-1/q)}$.
• The logarithmic term does not depend on $r$ and hence improves the known bound provided by J.
• Dick for the special case of Sobolev spaces $H^r_{\text{mix}}(\mathbb{T}^d)$.