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

Stanislav Budzinskiy

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arXiv · 2505.05134

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

Stanislav Budzinskiy

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Abstract

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.

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Stanislav Budzinskiy. 2025-05-08. Matrices over a Hilbert space and their low-rank cross approximation. https://arxiv.org/abs/2505.05134

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