Function integration, reconstruction and approximation using rank-$1$ lattices
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- by Frances Y. Kuo, Giovanni Migliorati, Fabio Nobile and Dirk Nuyens;
- Math. Comp. 90 (2021), 1861-1897
- DOI: https://doi.org/10.1090/mcom/3595
- Published electronically: April 23, 2021
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Abstract:
We consider rank-$1$ lattices for integration and reconstruction of functions with series expansion supported on a finite index set. We explore the connection between the periodic Fourier space and the non-periodic cosine space and Chebyshev space, via tent transform and then cosine transform, to transfer known results from the periodic setting into new insights for the non-periodic settings. Fast discrete cosine transform can be applied for the reconstruction phase. To reduce the size of the auxiliary index set in the associated component-by-component (CBC) construction for the lattice generating vectors, we work with a bi-orthonormal set of basis functions, leading to three methods for function reconstruction in the non-periodic settings. We provide new theory and efficient algorithmic strategies for the CBC construction. We also interpret our results in the context of general function approximation and discrete least-squares approximation.References
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Bibliographic Information
- Frances Y. Kuo
- Affiliation: School of Mathematics and Statistics, University of New South Wales, Sydney NSW 2052, Australia
- MR Author ID: 703418
- Email: f.kuo@unsw.edu.au
- Giovanni Migliorati
- Affiliation: Laboratoire Jacques-Louis Lions, Sorbonne Université, Paris 75005, France
- MR Author ID: 941196
- ORCID: 0000-0002-6317-5663
- Email: migliorati@ljll.math.upmc.fr
- Fabio Nobile
- Affiliation: CSQI, Institute of Mathematics, École Polytechnique Fédérale de Lausanne, 1015 Lausanne, Switzerland
- MR Author ID: 650310
- ORCID: 0000-0002-8130-0114
- Email: fabio.nobile@epfl.ch
- Dirk Nuyens
- Affiliation: Department of Computer Science, KU Leuven, Celestijnenlaan 200A, 3001 Leuven, Belgium
- MR Author ID: 777310
- ORCID: 0000-0002-4555-2314
- Email: dirk.nuyens@cs.kuleuven.be
- Received by editor(s): August 3, 2019
- Received by editor(s) in revised form: January 20, 2020, and July 15, 2020
- Published electronically: April 23, 2021
- © Copyright 2021 American Mathematical Society
- Journal: Math. Comp. 90 (2021), 1861-1897
- MSC (2020): Primary 41A10, 42A10, 41A63, 65D30, 65D32, 65D15
- DOI: https://doi.org/10.1090/mcom/3595
- MathSciNet review: 4273118