On the approximation of vector-valued functions by volume sampling

  • Given a Hilbert space H and a finite measure space Ω, the approximation of a vector-valued function f : Ω → H by a k-dimensional subspace U ⊂ H plays an important role in dimension reduction techniques, such as reduced basis methods for solving parameter dependent partial differential equations. For functions in the Lebesgue–Bochner space L2 (Ω; H), the best possible subspace approximation error d (2) k is characterized by the singular values of f. However, for practical reasons, U is often restricted to be spanned by point samples of f. We show that this restriction only has a mild impact on the attainable error; there always exist k samples such that the resulting error is not larger than √ k + 1 · d (2) k . Our work extends existing results by Binev at al. (SIAM J. Math. Anal., 43(3):1457–1472, 2011) on approximation in supremum norm and by Deshpande et al. (Theory Comput., 2:225–247, 2006) on column subset selection for matrices.

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Metadaten
Author:Daniel Kressner, Tingting Ni, André UschmajewGND
URN:urn:nbn:de:bvb:384-opus4-1147066
Frontdoor URLhttps://opus.bibliothek.uni-augsburg.de/opus4/114706
ISSN:0885-064XOPAC
Parent Title (English):Journal of Complexity
Publisher:Elsevier BV
Type:Article
Language:English
Year of first Publication:2025
Publishing Institution:Universität Augsburg
Release Date:2024/08/26
Volume:86
First Page:101887
DOI:https://doi.org/10.1016/j.jco.2024.101887
Institutes:Mathematisch-Naturwissenschaftlich-Technische Fakultät
Mathematisch-Naturwissenschaftlich-Technische Fakultät / Institut für Mathematik
Mathematisch-Naturwissenschaftlich-Technische Fakultät / Institut für Mathematik / Lehrstuhl für Mathematical Data Science
Dewey Decimal Classification:5 Naturwissenschaften und Mathematik / 51 Mathematik / 510 Mathematik
Licence (German):CC-BY-NC 4.0: Creative Commons: Namensnennung - Nicht kommerziell (mit Print on Demand)