Monsters populate mathematics : topologist's sine, Vitali set, Weierstrass function… These counter-examples to naive intuitions often have in common that they are defined either in a convoluted way, either with an oscillating function like the sine. The o-minimal paradigm allows us to forget those oddness and to make our first intuitions true, by considering only objects that have a "reasonable" definition in a way. What is an o-minimal structure? What examples do we know of? What is happening there? How do they act in complex geometry, in number theory, in optimization?
Reduced order models (ROMs) are parametric mathematical models derived from PDEs using previously computed solutions. In many applications, the solution space turns out to be low dimensional, so that one can trade a minimal loss of accuracy for speed and scalability of the numerical model. ROMs counteract the curse of dimensionality by significantly reducing the computational complexity. Overall, reduced order models have reached a certain level of maturity in the last decade, allowing their implementation in large-scale industrial codes, mainly in structural mechanics. Nevertheless, some hard points remain. Parametric problems governed by advection fields or solutions with a substantial compact support such as shock waves suffer from a limited possibility of dimensional reduction and, at the same time, from an insufficient generalization of the model (out-of-sample solutions). The main reason is that the solution space is usually approximated by an affine or linear representation. In this thesis, we aim to contribute to the use of non-intrusive model reduction methods by working on three axes: (i) Application to unsteady computations with non-intrusive interpolation methods; (ii) Use of hybrid models linking reduced models and numerical simulation models with a domain decomposition type approach; (iii) Application to complex industrial problems The flutter problem on a fin will be used as a first complex application case. Indeed, this fluid-structure problem presents very different behaviors according to the flow regimes and is very expensive to simulate without simplifying assumptions. Thus, a hybrid model could accelerate the computation time while remaining accurate in the complex areas. This CIFRE thesis financed by Ingeliance is part of the chaire PROVE financed by ONERA and the Nouvelle Aquitaine region.