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.
Highlights
Many mathematical and computational models depend on parameters
As we have introduced the correlation’s spectral factorisation in Eq (13), some other factorisations come to mind, they may be mostly of theoretical value: C = BTB = (V Σ)(V Σ)T = (V ΣV T)(V ΣV T)T, (19)
It was shown that the associated linear map contains the full information present in the parametric entity
Summary
Many mathematical and computational models depend on parameters. These may be quantities which have to be optimised during a design, or controlled in a real-time setting, or these parameters may be uncertain and represent uncertainty present in the model. The correlation may be defined as before in Eq (43), and the kernel on Q = Q1 × Q2 is as in Eq (44), but the first diagonal entry is a function on M1 × M1 only, and analogous for the second diagonal entry Tensor fields This is similar to the case of vector fields in that the state space is W = U ⊗ A, where U is a space of scalar valued functions on some set; and A ⊂ B = E ⊗ E, where E is a finite-dimensional vector space [38], and A is a manifold of tensors in the full tensor product B of tensors of even degree. We have recovered formally the same situation as for orthogonal tensors just described, and the same procedures may be followed
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