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

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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.

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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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Authors and Affiliations

  1. Institute for Numerical Simulation, University of Bonn, Endenicher Allee 62, 53115, Bonn, Germany

    Glenn Byrenheid & Tino Ullrich

  2. Faculty of Mathematics, Technische Universität Chemnitz, 09107, Chemnitz, Germany

    Lutz Kämmerer & Toni Volkmer

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  1. Glenn Byrenheid
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  2. Lutz Kämmerer
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  3. Tino Ullrich
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Correspondence to Lutz Kämmerer.

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Byrenheid, G., Kämmerer, L., Ullrich, T. et al. Tight error bounds for rank-1 lattice sampling in spaces of hybrid mixed smoothness. Numer. Math. 136, 993–1034 (2017). https://doi.org/10.1007/s00211-016-0861-7

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  • DOI: https://doi.org/10.1007/s00211-016-0861-7

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