Observateurs optimaux dans des variétés - des formulations aux applications

Data assimilation is the process of optimally combining dynamical models with observational data in order to estimate the evolving state and parameters of a physical system. In the deterministic optimal observer framework, this is achieved by defining an observer: a modified version of the original dynamics, augmented with a gain term acting on the discrepancy between the observations and the model output. The optimal observer is then designed to minimize a criterion that penalizes deviations from the model dynamics and discrepancies with the observations. The aim of this thesis is to develop an optimal observer theory from a deterministic perspective, in the context of constrained physical systems. More specifically, we focus on cases where the constrained space admits a formulation as a Riemannian manifold. In the first part, we derive an exact formulation of the deterministic optimal observer on compact manifolds, extending the Mortensen observer originally defined in the Euclidean setting. Since such a formulation inherits the high computational cost and the curse of dimensionality of its Euclidean counterpart, we then develop approximation strategies that preserve the theoretical foundations of the optimal observer in Riemannian manifolds while reducing its computational burden. In particular, we propose and theoretically justify extensions of the Extended Kalman Filter (EKF) and the Unscented Kalman Filter (UKF) to the manifold setting, by approximating the value functions of the Mortensen observer with quadratic functions. We also develop an approximation of the fully space and time discretized Mortensen observer using dimension-reduction techniques, and in particular low-rank matrix representations of the associated expansion coefficients. For each proposed observer, we provide numerical illustrations on submanifolds of $\R^3$ and Cartesian products thereof. Finally, we assess the established framework in a real-world application: the monitoring of wildland fire propagation. By endowing the abstract space of shapes with a Riemannian structure, we are able to design an observer for this front-propagation problem using observational data of shape type. This work thus bridges rigorous theoretical advances in observer design with practical strategies for data assimilation in a complex, constrained physical systems.