Approche spectrale pour l'interpolation à noyaux et positivité conditionnelle

Abstract : We propose a spectral approach for the resolution of kernel-based interpolation problems of which numerical solution can not be directly computed. Such a situation occurs in particular when the number of data is infinite. We first consider optimal interpolation in Hilbert subspaces. For a given problem, an integral operator is defined from the underlying kernel and a parameterization of the data set based on a measurable space. The spectral decomposition of the operator is used in order to obtain a representation formula for the optimal interpolator and spectral truncation allows its approximation. The choice of the measure on the parameters space introduces a hierarchy onto the data set which allows a tunable precision of the approximation. As an example, we show how this methodology can be used in order to enforce boundary conditions in kernel-based interpolation models. The Gaussian processes conditioning problem is also studied in this context. The last part of this thesis is devoted to the notion of conditionally positive kernels. We propose a general definition of symmetric conditionally positive kernels relative to a given space and exposed the associated theory of semi-Hilbert subspaces. We finally study the optimal interpolation problem in such spaces.
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Bertrand Gauthier. Approche spectrale pour l'interpolation à noyaux et positivité conditionnelle. Mathématiques [math]. École Nationale Supérieure des Mines de Saint-Étienne, 2011. Français. ⟨NNT : 2011 EMSE 0615⟩. ⟨tel-00631252⟩

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