On the approximation of vector-valued functions by volume sampling

Given a Hilbert space Hand a finite measure space Omega, the approximation of a vector-valued function f : Omega -> H by a k-dimensional subspace U subset of 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 L-2(Omega; H), the best possible subspace approximation error d(k)((2)) 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 root k + 1 center dot d(k)((2)). Our work extends existing results by Binev et al. (2011) [3] on approximation in supremum norm and by Deshpande et al. (2006) [8] on column subset selection for matrices. (c) 2024 The Author(s). Published by Elsevier Inc. This is an open access article under the CC BY-NC license (http://creativecommons.org/licenses/by-nc/4.0/).