Analysis of Probabilistic and Parametric Reduced Order Models
For researchers in reduced order modeling, this provides a conceptual unification but is primarily a theoretical synthesis without new empirical results.
This paper unifies the theoretical foundations of probabilistic and parametric reduced order models by showing that stochastic models can be viewed as a special case of parametric models via a generalized correlation operator, connecting concepts like Karhunen-Loève decomposition and spectral factorization.
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-Loève- 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.