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.
Author: | Daniel Kressner, Tingting Ni, André UschmajewGND |
---|---|
URN: | urn:nbn:de:bvb:384-opus4-1147066 |
Frontdoor URL | https://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) |