• Corpus ID: 246240716

Construction-free median quasi-Monte Carlo rules for function spaces with unspecified smoothness and general weights

@article{Goda2022ConstructionfreeMQ,
  title={Construction-free median quasi-Monte Carlo rules for function spaces with unspecified smoothness and general weights},
  author={Takashi Goda and Pierre L’Ecuyer},
  journal={ArXiv},
  year={2022},
  volume={abs/2201.09413}
}

• T. Goda, P. L’Ecuyer
• Published 24 January 2022
• Mathematics, Computer Science
• ArXiv

. We study quasi-Monte Carlo (QMC) integration of smooth functions defined over the multi-dimensional unit cube. Inspired by a recent work of Pan and Owen, we study a new construction-free median QMC rule which can exploit the smoothness and the weights of function spaces adaptively. For weighted Korobov spaces, we draw a sample of r independent generating vectors of rank-1 lattice rules, compute the integral estimate for each, and approximate the true integral by the median of these r estimates. For weighted Sobolev spaces, we use the same approach but with the rank-1 lattice rules replaced by high-order polynomial lattice rules. A major advantage over the existing approaches is that we do not need to construct good generating vectors by a computer search algorithm, while our median QMC rule achieves almost the optimal worst-case error rate for the respective function space with any smoothness and weights , with a probability that converges to 1 exponentially fast as r increases. Numerical experiments illustrate and support our theoretical findings.