Observers for wave equations and associated discretization

par Tiphaine Delaunay

mardi 10 févr. 2026, 11:00 → 12:00 Europe/Paris

The context of this talk is the study of inverse problems for wave propagation phenomena. The goal is to formalize, analyze, and discretize sequential data assimilation strategies, in which measurements are incorporated as they become available.
The resulting system, called an observer, stabilizes on the observed trajectory and hence reconstructs the state and possibly some unknown parameters of the system.
In the first part of the talk, I focus on source reconstruction, an estimation problem intermediate in complexity between state estimation and the more general problem of parameter identification. In this setting, we define, in a deterministic infinite-dimensional framework, a so-called Kalman estimator that sequentially estimates the source term to be identified. Using tools from dynamic programming, we show that this sequential estimator is equivalent to the minimization of a functional. This equivalence allows us to establish convergence results under observability conditions. We demonstrate these observability inequalities for different source types by combining functional analysis, multiplier methods, and Carleman estimates.
The second part concerns discretization issues and their analysis. We propose a discretize-then-optimize approach, where the observability inequalities need to be extended to discretized systems. In particular, in this context, we extend uniform exponential stabilization results to the discretization for high-order finite element discretizations of a 1D wave equation. Joint work with S. Imperiale and P. Moireau.