2018/06/17 by Hermann G. Matthies, Matthies, Hermann G., Roger Ohayon +1
Computer Science · Mathematics · #35B30 #37M99 #41A05 #41A45 #41A63 #60G20 #60G60 #65J99 #93A30 #FOS: Mathematics #Matrix Theory and Algorithms #Numerical Analysis (math.NA) #Numerical methods for differential equations #Tensor decomposition and applications
paper · doi:10.48550/arxiv.1806.07255
openalex publication_date 2018/06/17 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In many instances one has to deal with parametric models. Such models in vector spaces are connected to a linear map. The 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, in fact it provides a factorisation of the correlation operator and of the reproducing kernel. The spectral decomposition of the correlation and kernel, as well as the associated Karhunen-Loève or proper orthogonal decomposition are a direct consequence. This formulation thus unifies many such constructions under a functional analytic view. Recursively applying factorisations in higher order tensor representations leads to hierarchical tensor decompositions. This format also allows refinements for cases when the parametric model has more structure. Examples are shown for vector- and tensor-fields with certain required properties. Another kind of structure is the parametric model of a coupled system. It is shown that this can also be reflected in the theoretical framework.