On the reconstruction of functions from values at subsampled quadrature points

Bartel, Felix; Kämmerer, Lutz; Potts, Daniel; Ullrich, Tino

doi:10.1090/mcom/3896

On the reconstruction of functions from values at subsampled quadrature points
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by Felix Bartel, Lutz Kämmerer, Daniel Potts and Tino Ullrich

Math. Comp. 93 (2024), 785-809

DOI: https://doi.org/10.1090/mcom/3896

Published electronically: August 28, 2023

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Abstract:

This paper is concerned with function reconstruction from samples. The sampling points used in several approaches are (1) structured points connected with fast algorithms or (2) unstructured points coming from, e.g., an initial random draw to achieve an improved information complexity. We connect both approaches and propose a subsampling of structured points in an offline step. In particular, we start with quasi-Monte Carlo (QMC) points with inherent structure and stable $L_2$ reconstruction properties. The subsampling procedure consists of a computationally inexpensive random step followed by a deterministic procedure to further reduce the number of points while keeping its information. In these points functions (belonging to a reproducing kernel Hilbert space of bounded functions) will be sampled and reconstructed from whilst achieving state of the art error decay.

Our method is dimension-independent and is applicable as soon as we know some initial quadrature points. We apply our general findings on the $d$-dimensional torus to subsample rank-1 lattices, where it is known that full rank-1 lattices lose half the optimal order of convergence (expressed in terms of the size of the lattice). In contrast to that, our subsampled version regains the optimal rate since many of the lattice points are not needed. Moreover, we utilize fast and memory efficient Fourier algorithms in order to compute the approximation. Numerical experiments in several dimensions support our findings.

References

Felix Bartel, Martin Schäfer, and Tino Ullrich, Constructive subsampling of finite frames with applications in optimal function recovery, Appl. Comput. Harmon. Anal. 65 (2023), 209–248. MR 4566822, DOI 10.1016/j.acha.2023.02.004
Joshua D. Batson, Daniel A. Spielman, and Nikhil Srivastava, Twice-Ramanujan sparsifiers, STOC’09—Proceedings of the 2009 ACM International Symposium on Theory of Computing, ACM, New York, 2009, pp. 255–262. MR 2780071
Alain Berlinet and Christine Thomas-Agnan, Reproducing kernel Hilbert spaces in probability and statistics, Kluwer Academic Publishers, Boston, MA, 2004. With a preface by Persi Diaconis. MR 2239907, DOI 10.1007/978-1-4419-9096-9
Glenn Byrenheid, Lutz Kämmerer, Tino Ullrich, and Toni Volkmer, Tight error bounds for rank-1 lattice sampling in spaces of hybrid mixed smoothness, Numer. Math. 136 (2017), no. 4, 993–1034. MR 3671595, DOI 10.1007/s00211-016-0861-7
Glenn Byrenheid, Dinh Dũng, Winfried Sickel, and Tino Ullrich, Sampling on energy-norm based sparse grids for the optimal recovery of Sobolev type functions in $H^\gamma$, J. Approx. Theory 207 (2016), 207–231. MR 3494230, DOI 10.1016/j.jat.2016.02.012
A. Christmann and I. Steinwart, Support Vector Machines, Springer, 2008.
Albert Cohen, Mark A. Davenport, and Dany Leviatan, On the stability and accuracy of least squares approximations, Found. Comput. Math. 13 (2013), no. 5, 819–834. MR 3105946, DOI 10.1007/s10208-013-9142-3
Albert Cohen and Giovanni Migliorati, Optimal weighted least-squares methods, SMAI J. Comput. Math. 3 (2017), 181–203. MR 3716755, DOI 10.5802/smai-jcm.24
R. Cools and D. Nuyens, An overview of fast component-by-component constructions of lattice rules and lattice sequences, PAMM 7 (2007), 1022609–1022610.
Dinh Dũng, Vladimir Temlyakov, and Tino Ullrich, Hyperbolic cross approximation, Advanced Courses in Mathematics. CRM Barcelona, Birkhäuser/Springer, Cham, 2018. Edited and with a foreword by Sergey Tikhonov. MR 3887571, DOI 10.1007/978-3-319-92240-9
Josef Dick, Peter Kritzer, and Friedrich Pillichshammer, Lattice rules—numerical integration, approximation, and discrepancy, Springer Series in Computational Mathematics, vol. 58, Springer, Cham, [2022] ©2022. With an appendix by Adrian Ebert. MR 4472208, DOI 10.1007/978-3-031-09951-9
Josef Dick, Frances Y. Kuo, and Ian H. Sloan, High-dimensional integration: the quasi-Monte Carlo way, Acta Numer. 22 (2013), 133–288. MR 3038697, DOI 10.1017/S0962492913000044
Matthieu Dolbeault, David Krieg, and Mario Ullrich, A sharp upper bound for sampling numbers in $L_2$, Appl. Comput. Harmon. Anal. 63 (2023), 113–134. MR 4525968, DOI 10.1016/j.acha.2022.12.001
F. Filbir and H. N. Mhaskar, Marcinkiewicz-Zygmund measures on manifolds, J. Complexity 27 (2011), no. 6, 568–596. MR 2846706, DOI 10.1016/j.jco.2011.03.002
Anne Greenbaum, Iterative methods for solving linear systems, Frontiers in Applied Mathematics, vol. 17, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1997. MR 1474725, DOI 10.1137/1.9781611970937
Karlheinz Gröchenig, Sampling, Marcinkiewicz-Zygmund inequalities, approximation, and quadrature rules, J. Approx. Theory 257 (2020), 105455, 20. MR 4124840, DOI 10.1016/j.jat.2020.105455
Cécile Haberstich, Anthony Nouy, and Guillaume Perrin, Boosted optimal weighted least-squares, Math. Comp. 91 (2022), no. 335, 1281–1315. MR 4405496, DOI 10.1090/mcom/3710
Aicke Hinrichs, David Krieg, Erich Novak, and Jan Vybíral, Lower bounds for integration and recovery in $L_2$, J. Complexity 72 (2022), Paper No. 101662, 15. MR 4438175, DOI 10.1016/j.jco.2022.101662
T. Jahn, T. Ullrich, and F. Voigtlaender, Sampling numbers of smoothness classes via $\ell _1$-minimization, to appear in J. Complexity, arXiv:2212.00445 (2022).
Vesa Kaarnioja, Yoshihito Kazashi, Frances Y. Kuo, Fabio Nobile, and Ian H. Sloan, Fast approximation by periodic kernel-based lattice-point interpolation with application in uncertainty quantification, Numer. Math. 150 (2022), no. 1, 33–77. MR 4363819, DOI 10.1007/s00211-021-01242-3
L. Kämmerer, High Dimensional Fast Fourier Transform Based on Rank-1 Lattice Sampling, Dissertation, Universitätsverlag Chemnitz, 2014.
L. Kämmerer, A fast probabilistic component-by-component construction of exactly integrating rank-1 lattices and applications, arXiv:2012.14263 (2020).
Lutz Kämmerer, Daniel Potts, and Toni Volkmer, Approximation of multivariate periodic functions by trigonometric polynomials based on rank-1 lattice sampling, J. Complexity 31 (2015), no. 4, 543–576. MR 3348081, DOI 10.1016/j.jco.2015.02.004
Lutz Kämmerer, Daniel Potts, and Toni Volkmer, High-dimensional sparse FFT based on sampling along multiple rank-1 lattices, Appl. Comput. Harmon. Anal. 51 (2021), 225–257. MR 4185099, DOI 10.1016/j.acha.2020.11.002
Lutz Kämmerer, Tino Ullrich, and Toni Volkmer, Worst-case recovery guarantees for least squares approximation using random samples, Constr. Approx. 54 (2021), no. 2, 295–352. MR 4321776, DOI 10.1007/s00365-021-09555-0
B. Kashin, E. Kosov, I. Limonova, and V. Temlyakov, Sampling discretization and related problems, J. Complexity 71 (2022), Paper No. 101653, 55. MR 4420519, DOI 10.1016/j.jco.2022.101653
Jens Keiner, Stefan Kunis, and Daniel Potts, Efficient reconstruction of functions on the sphere from scattered data, J. Fourier Anal. Appl. 13 (2007), no. 4, 435–458. MR 2329012, DOI 10.1007/s00041-006-6915-y
David Krieg and Mario Ullrich, Function values are enough for $L_2$-approximation, Found. Comput. Math. 21 (2021), no. 4, 1141–1151. MR 4298242, DOI 10.1007/s10208-020-09481-w
David Krieg and Mario Ullrich, Function values are enough for $L_2$-approximation: Part II, J. Complexity 66 (2021), Paper No. 101569, 14. MR 4292366, DOI 10.1016/j.jco.2021.101569
Thomas Kühn, Winfried Sickel, and Tino Ullrich, Approximation of mixed order Sobolev functions on the $d$-torus: asymptotics, preasymptotics, and $d$-dependence, Constr. Approx. 42 (2015), no. 3, 353–398. MR 3416161, DOI 10.1007/s00365-015-9299-x
Stefan Kunis and Daniel Potts, Fast spherical Fourier algorithms, J. Comput. Appl. Math. 161 (2003), no. 1, 75–98. MR 2018576, DOI 10.1016/S0377-0427(03)00546-6
Frances Y. Kuo, Giovanni Migliorati, Fabio Nobile, and Dirk Nuyens, Function integration, reconstruction and approximation using rank-1 lattices, Math. Comp. 90 (2021), no. 330, 1861–1897. MR 4273118, DOI 10.1090/mcom/3595
I. Limonova and V. Temlyakov, On sampling discretization in $L_2$, J. Math. Anal. Appl. 515 (2022), no. 2, Paper No. 126457, 14. MR 4446186, DOI 10.1016/j.jmaa.2022.126457
J. Marcinkiewicz and A. Zygmund, Sur les fonctions indépendantes, Fund. Math. 29 (1937), no. 1, 60–90 (fre).
H. N. Mhaskar, F. J. Narcowich, and J. D. Ward, Spherical Marcinkiewicz-Zygmund inequalities and positive quadrature, Math. Comp. 70 (2001), no. 235, 1113–1130. MR 1710640, DOI 10.1090/S0025-5718-00-01240-0
Moritz Moeller and Tino Ullrich, $L_2$-norm sampling discretization and recovery of functions from RKHS with finite trace, Sampl. Theory Signal Process. Data Anal. 19 (2021), no. 2, Paper No. 13, 31. MR 4354442, DOI 10.1007/s43670-021-00013-3
Nicolas Nagel, Martin Schäfer, and Tino Ullrich, A new upper bound for sampling numbers, Found. Comput. Math. 22 (2022), no. 2, 445–468. MR 4407748, DOI 10.1007/s10208-021-09504-0
Akil Narayan, John D. Jakeman, and Tao Zhou, A Christoffel function weighted least squares algorithm for collocation approximations, Math. Comp. 86 (2017), no. 306, 1913–1947. MR 3626543, DOI 10.1090/mcom/3192
Gerlind Plonka, Daniel Potts, Gabriele Steidl, and Manfred Tasche, Numerical Fourier analysis, Applied and Numerical Harmonic Analysis, Birkhäuser/Springer, Cham, 2018. MR 3890075, DOI 10.1007/978-3-030-04306-3
Daniel Potts, Fast algorithms for discrete polynomial transforms on arbitrary grids, Linear Algebra Appl. 366 (2003), 353–370. Special issue on structured matrices: analysis, algorithms and applications (Cortona, 2000). MR 1987729, DOI 10.1016/S0024-3795(02)00592-X
Daniel Potts, Jürgen Prestin, and Antje Vollrath, A fast algorithm for nonequispaced Fourier transforms on the rotation group, Numer. Algorithms 52 (2009), no. 3, 355–384. MR 2563947, DOI 10.1007/s11075-009-9277-0
I. H. Sloan and S. Joe, Lattice methods for multiple integration, Oxford Science Publications, The Clarendon Press, Oxford University Press, New York, 1994. MR 1442955, DOI 10.1093/oso/9780198534723.001.0001
V. N. Temlyakov, Approximation of periodic functions, Computational Mathematics and Analysis Series, Nova Science Publishers, Inc., Commack, NY, 1993. MR 1373654
V. N. Temlyakov, The Marcinkiewicz-type discretization theorems, Constr. Approx. 48 (2018), no. 2, 337–369. MR 3848042, DOI 10.1007/s00365-018-9446-2
V. Temlyakov, On optimal recovery in $L_2$, J. Complexity 65 (2021), Paper No. 101545, 11. MR 4264807, DOI 10.1016/j.jco.2020.101545
V. Temlyakov and T. Ullrich, Bounds on Kolmogorov widths and sampling recovery for classes with small mixed smoothness, J. Complexity 67 (2021), Paper No. 101575, 19. MR 4311528, DOI 10.1016/j.jco.2021.101575
Vladimir N. Temlyakov and Tino Ullrich, Approximation of functions with small mixed smoothness in the uniform norm, J. Approx. Theory 277 (2022), Paper No. 105718, 23. MR 4395317, DOI 10.1016/j.jat.2022.105718
Lloyd N. Trefethen, Approximation theory and approximation practice, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2013. MR 3012510
Joel A. Tropp, User-friendly tail bounds for sums of random matrices, Found. Comput. Math. 12 (2012), no. 4, 389–434. MR 2946459, DOI 10.1007/s10208-011-9099-z