Our objective is to address a large class of identification problems for PDEs with original methods coming from observer - also called sequential estimator - methods. In fact, in identification the most widely used method consists in minimizing a least-square functional involving a data discrepancy term and some regularization terms. From a data assimilation point of view, this strategy is equivalent to a classical 4D-Var method applied to an augmented variable gathering the PDE state variables and the parameters. By contrast, our strategy here relies on the definition of a new system, called observer, which corrects in time the state and the parameters by using the computed data discrepancy as an instantaneous correction. The most widely-known observer is the Kalman estimator in linear systems, and its various extensions to handle non-linearities. In practice, these observers can be applied to a large class of systems, but they suffer from a "curse of dimensionality" that makes them untractable for PDEs. More precisely, a standard finite element discretization of the PDE state variable would require the solution of a Riccati equation in a very large space. Our idea is therefore to rely on better-suited observer strategies for state estimation, while keeping the optimal Kalman strategy on the parameters which can be assumed to be discretized in a much coarser manner than the state. This method allows to identify the parameters of a system as soon as we already have a state observer for the system with exactly known parameters. Our approach - first published in 2008 - combined and unified two categories of approaches: Reduced-Order Filtering and Adaptive Observers. We demonstrated its convergence in linear configurations, extended it to non-linear evolution equations, and successfully demonstrated its applicability with real data for cardiovascular estimation problems.
Here are our contributions on this topic: