A Component-by-Component Construction for the Trigonometric Degree

Conference paper
First Online:
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 23)

Abstract

We propose an alternative to the algorithm from Cools, Kuo, Nuyens (Computing 87(1–2):63–89, 2010), for constructing lattice rules with good trigonometric degree. The original algorithm has construction cost \(O(\vert {\mathcal{A}}_{d}(m)\vert + dN\log N)\) for an N-point lattice rule in d dimensions having trigonometric degree m, where the set \({\mathcal{A}}_{d}(m)\) has exponential size in both d and m (in the “unweighted degree” case, which is what we consider here). We reduce the cost to \(O(dN{(\log N)}^{2})\) with an implicit constant governing the needed precision (which is dependent on N and d).

Keywords

Fourier Coefficient Reproduce Kernel Hilbert Space Dual Lattice Lattice Rule Tensor Product Form 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
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Notes

Acknowledgements

The authors would like to thank the two anonymous referees for useful comments on the manuscript.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  1. 1.Department of Computer ScienceK.U.LeuvenHeverleeBelgium

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