Matrices over a Hilbert space and their low-rank approximation

TL;DR

This paper develops a theory for matrices with entries in a Hilbert space, extending low-rank approximation techniques like cross approximation to these matrices, with promising applications in differential equations and data modeling.

Contribution

It introduces a novel framework for Hilbert space-valued matrices and adapts low-rank approximation algorithms for this setting, bridging a gap in existing matrix theory.

Findings

Achieves quasioptimal approximation of Hilbert space matrices.

Integrates the new approach with existing PDE computational software.

Demonstrates effectiveness through numerical experiments.

Abstract

Matrices are typically considered over fields or rings. Motivated by applications in parametric differential equations and data-driven modeling, we suggest to study matrices with entries from a Hilbert space and present an elementary theory of them: from basic properties to low-rank approximation. Specifically, we extend the idea of cross approximation to such matrices and propose an analogue of the adaptive cross approximation algorithm. Our numerical experiments show that this approach can achieve quasioptimal approximation and be integrated with the existing computational software for partial differential equations.