A Localization Strategy for Data Assimilation; Application to State Estimation and Parameter Estimation

We present a localization procedure for addressing data assimilation tasks---state estimation and parameter estimation---in which the quantity of interest pertains to a subregion of the domain over which the mathematical model is properly defined. Given the domain $\Omega^{pb}$ associated with the full system, and the domain of interest $\Omega \subset \Omega^{pb}$, our localization procedure relies on the definition of an intermediate domain $\Omega^{bk}$ such that $\bar{\Omega} \subset \bar{\Omega}^{bk} \subset \bar{\Omega}^{pb}$. The domain ${\Omega}^{bk}$ is chosen to exclude many parameters associated with the parametrization of the mathematical model in $\Omega^{pb} \setminus \Omega$ and to thereby reduce the difficulty of the estimation problem. Our approach exploits a model-order-reduction (MOR) procedure to properly address (i) uncertainty in the value of the parameters in $\Omega$, and (ii) uncertainty in the boundary conditions at the interface between $\Omega^{bk}$ and $\Omega^{pb} \setminus \Omega^{bk}$. We present theoretical results to demonstrate the optimality of our construction. We further present two numerical synthetic examples in acoustics to demonstrate the effectiveness of our localization procedure in reducing uncertainty dimensionality, and thus in simplifying the data assimilation task.