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 $f:\Omega \to H$ by a k-dimensional subspace $U\subset H$ 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 $d_k^{\left(2\right)}$ 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 $\sqrt{k+1}\cdot d_k^{\left(2\right)}$. 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.