Parametric Models Analysed with Linear Maps

TL;DR

This paper presents a unified functional analytic framework for analyzing parametric models using linear maps, connecting various representations with reproducing kernel Hilbert spaces and spectral analysis to improve reduced order models.

Contribution

It introduces a general approach linking parametric entities to linear operators, unifying different representations and enabling spectral analysis for better reduced order models.

Findings

Linear maps associate parametric entities with operators.

Spectral analysis reveals approximation properties of ROMs.

Unified framework connects representations with RKHS and tensor products.

Abstract

Parametric entities appear in many contexts, be it in optimisation, control, modelling of random quantities, or uncertainty quantification. These are all fields where reduced order models (ROMs) have a place to alleviate the computational burden. Assuming that the parametric entity takes values in a linear space, we show how is is associated to a linear map or operator. This provides a general point of view on how to consider and analyse different representations of such entities. Analysis of the associated linear map in turn connects such representations with reproducing kernel Hilbert spaces and affine- / linear-representations in terms of tensor products. A generalised correlation operator is defined through the associated linear map, and its spectral analysis helps to shed light on the approximation properties of ROMs. This point of view thus unifies many such representations under a functional analytic roof, leading to a deeper understanding and making them available for appropriate analysis.