Résumé : This presentation is set within the framework of inverse problems. The main objective is to recover initial conditions, states, or parameters of a system from available observations. We focus on sequential methods in data assimilation, where observations are incorporated as they become available.

The first part of this talk concerns the reconstruction of an unknown source term in a wave equation. In a deterministic infinite-dimensional setting, we introduce a Kalman estimator that sequentially reconstructs the source term. We show that this sequential estimator is equivalent to the minimization of a functional, which allows us to establish convergence results under appropriate observability conditions. These conditions are obtained using multiplier methods and Carleman estimates.

The second part addresses the estimation of states and parameters in a PDE model for the growth of ellipsoidal tumor spheroids. The general strategy is to extract relevant information from spheroid images, formulate a PDE model describing tumor evolution, and then reduce it to an ODE model for cost-effective data assimilation. We then use a reduced-order unscented Kalman filter combined with a Luenberger observer to jointly reconstruct the system state and identify physical parameters from biological measurements.