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

@article{Hinrichs2016OptimalQC,
  title={Optimal quasi-Monte Carlo rules on order 2 digital nets for the numerical integration of multivariate periodic functions},
  author={Aicke Hinrichs and Lev Markhasin and Jens Oettershagen and Tino Ullrich},
  journal={Numerische Mathematik},
  year={2016},
  volume={134},
  pages={163-196}
}
We investigate quasi-Monte Carlo rules for the numerical integration of multivariate periodic functions from Besov spaces Sp,q B(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 of Dick (SIAM J Numer Anal 45:2141– 2176, 2007) for the special case of Sobolev spaces Hr mix(T d). Secondly, the rate of convergence is… CONTINUE READING
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