Analysis of Probabilistic and Parametric Reduced Order Models

TL;DR

This paper explores the theoretical connections between probabilistic and parametric reduced order models, focusing on linear operators, correlation functions, and spectral decompositions to unify their analysis.

Contribution

It provides a unified theoretical framework linking stochastic and parametric models through linear operators and correlation operator factorizations.

Findings

Connections between stochastic and parametric models via linear maps.

Relation of correlation operator factorizations to spectral decompositions.

Unified perspective on reduced order modeling techniques.

Abstract

Stochastic models share many characteristics with generic parametric models. In some ways they can be regarded as a special case. But for stochastic models there is a notion of weak distribution or generalised random variable, and the same arguments can be used to analyse parametric models. Such models in vector spaces are connected to a linear map, and in infinite dimensional spaces are a true gener- alisation. Reproducing kernel Hilbert space and affine- / linear- representations in terms of tensor products are directly related to this linear operator. This linear map leads to a generalised correlation operator, and representations are connected with factorisations of the correlation operator. The fitting counterpart in the stochastic domain to make this point of view as simple as possible are algebras of random variables with a distinguished linear functional, the state, which is interpreted as expectation. The connections of factorisations of the generalised correlation to the spectral decomposition, as well as the associated Karhunen-Lo\`eve- or proper orthogonal decomposition will be sketched. The purpose of this short note is to show the common theoretical background and pull some lose ends together.