Matrices over a Hilbert space and their low-rank cross approximation

Motivated by applications in reduced-order modeling (ROM) of parametric partial differential equations, we investigate the algebraic properties of Bochner matrices -- matrices with entries in an abstract Hilbert space.

Low-rank cross approximation is extended to Bochner matrices and its approximation guarantees are derived.

The high-dimensional nature of the entries is shown to manifest itself in maximum-volume bounds, making them smaller than in the classical setting.

An analogue of adaptive cross approximation is proposed and validated as a non-intrusive ROM method in numerical experiments with parametric nonlinear Stokes equations.