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Numerische Mathematik

, Volume 136, Issue 4, pp 993–1034 | Cite as

Tight error bounds for rank-1 lattice sampling in spaces of hybrid mixed smoothness

  • Glenn Byrenheid
  • Lutz KämmererEmail author
  • Tino Ullrich
  • Toni Volkmer
Article

Abstract

We consider the approximate recovery of multivariate periodic functions from a discrete set of function values taken on a rank-1 lattice. Moreover, the main result is the fact that any (non-)linear reconstruction algorithm taking function values on any integration lattice of size M has a dimension-independent lower bound of \(2^{-(\alpha +1)/2} M^{-\alpha /2}\) when considering the optimal worst-case error with respect to function spaces of (hybrid) mixed smoothness \(\alpha >0\) on the d-torus. We complement this lower bound with upper bounds that coincide up to logarithmic terms. These upper bounds are obtained by a detailed analysis of a rank-1 lattice sampling strategy, where the rank-1 lattices are constructed by a component–by–component method. The lattice (group) structure allows for an efficient approximation of the underlying function from its sampled values using a single one-dimensional fast Fourier transform. This is one reason why these algorithms keep attracting significant interest. We compare our results to recent (almost) optimal methods based upon samples on sparse grids.

Mathematics Subject Classification

65T40 42A10 65D30 65D32 68Q17 68Q25 42B35 65T50 65Y20 

Notes

Acknowledgements

The authors acknowledge the fruitful discussions with A. Hinrichs, M. Ullrich and R. Bergmann on this topic, especially at the conference “Approximationsmethoden und schnelle Algorithmen” in Hasenwinkel, 2014. Furthermore, the authors thank V.N. Temlyakov for his valuable comments and historical hints on that topic. Especially, the authors thank A. Hinrichs for pointing out an alternative proof argument for the non-optimality of rank-1 lattice sampling, cf. Remark 5. The authors thank the referees for their valuable suggestions and remarks. Moreover, LK and TV gratefully acknowledge the support by the German Research Foundation (DFG) within the Priority Program 1324, project PO 711/10-2. Additionally, TV acknowledges the funding by the European Union and the Free State of Saxony (EFRE/ESF NBest-SF). GB and TU acknowledge the support by the DFG Emmy-Noether programme (UL403/1-1) and the Hausdorff-Center for Mathematics, University of Bonn.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2017

Authors and Affiliations

  • Glenn Byrenheid
    • 1
  • Lutz Kämmerer
    • 2
    Email author
  • Tino Ullrich
    • 1
  • Toni Volkmer
    • 2
  1. 1.Institute for Numerical SimulationUniversity of BonnBonnGermany
  2. 2.Faculty of MathematicsTechnische Universität ChemnitzChemnitzGermany

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