The framework of this thesis is the approximation of functions for which the value is known at limited number of points. More precisely, we consider here the so-called kriging models from two points of view : the approximation in reproducing kernel Hilbert spaces and the Gaussian Process regression. When the function to approximate depends on many variables, the required number of points can become very large and the interpretation of the obtained models remains difficult because the model is still a high-dimensional function. In light of those remarks, the main part of our work adresses the issue of simplified models by studying a key concept of kriging models, the kernel. More precisely, the following aspects are adressed: additive kernels for additive models and kernel decomposition for sparse modeling. Finally, we propose a class of kernels that is well suited for functional ANOVA representation and global sensitivity analysis.
Étude de classes de noyaux adaptées à la simplification et à l'interprétation des modèles d'approximation. Une approche fonctionnelle et probabiliste.