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Analysis of parametric models. Linear methods and approximations. (English) Zbl 1444.47080
This is a high lever discussion of parametric models that have many applications such as optimal control, uncertainty quantification, model reduction, etc. The main purpose of the paper is to provide a survey and a unifying approach. In the abstract setting of the paper, the problem takes the form of a function \(r(p)\) in some Hilbert space \(\mathcal{U}\) depending on parameters \(p\) (e.g., \(r(p)\) is describing the state) and it is the unique solution of some equation \(F(r(p),p)=0\). The Hilbert space is essential because taking an inner product in \(\mathcal{U}\) allows to define a linear map \(R:u\mapsto \langle r(p),u\rangle \in\mathbb{R}\). Then a reproducing kernel or an orthonormal basis for \(\mathcal{U}\) can be defined. The operator \(C=R^*R\) is the correlation, which has a spectral decomposition, and can have several differential factorizations, giving rise to different (known) methods and techniques. The paper analyses the equivalence and relations between all these approaches and links these to known, seemingly different, methods and applications.

MSC:
47N70 Applications of operator theory in systems, signals, circuits, and control theory
93C30 Control/observation systems governed by functional relations other than differential equations (such as hybrid and switching systems)
00A71 General theory of mathematical modeling
47A68 Factorization theory (including Wiener-Hopf and spectral factorizations) of linear operators
46E22 Hilbert spaces with reproducing kernels (= (proper) functional Hilbert spaces, including de Branges-Rovnyak and other structured spaces)
35B30 Dependence of solutions to PDEs on initial and/or boundary data and/or on parameters of PDEs
37M99 Approximation methods and numerical treatment of dynamical systems
41A45 Approximation by arbitrary linear expressions
41A63 Multidimensional problems (should also be assigned at least one other classification number from Section 41-XX)
60G20 Generalized stochastic processes
60G60 Random fields
65J99 Numerical analysis in abstract spaces
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