Session A6 - Mathematical Foundations of Data Assimilation and Inverse Problems
July 11, 14:30 ~ 15:00 - Room T1
Observer strategies for inverse problems associated with wave-like equations
Inria - LMS, Ecole Polytechnique, CNRS - Université Paris-Saclay, France - email@example.com
We present a novel strategy to perform estimation for wave-like systems and more general evolution PDEs with uncertain initial conditions and parameters. We adopt a filtering approach on the dynamical system formulation to construct a joint state-parameter estimator that uses some measurements available in standard operating conditions. Namely, the aim is to obtain a modified dynamical system converging to the reference by incorporating correction terms using the data. First, in the case of known parameters, state estimation is performed using a Luenberger observer inspired from feedback control theory. This type of state estimator is chosen for its particular effectiveness and robustness. In particular, unlike the classical Kalman approach, this filter is computationally tractable for numerical systems arising from the discretization of PDEs and – although optimality is lost – the exponential stability of the corresponding error system gives exponential convergence of the estimator/observer. With uncertain parameters we extend the estimator by incorporating the parameters in an augmented dynamical system. The effect of the first stage state filter then consists in essence in circumscribing the uncertainty to the parameter space – which is usually much “smaller” than the parameter space – and allows for an $H^2$ type filter in the resulting low rank space. This second step is related to "reduced rank filtering" procedures in data assimilation. The convergence of the resulting joint state-parameter estimator can be mathematically established , and we demonstrate its effectiveness by identifying localized parameters. We propose to illustrate every aspect of this strategy from the observer formulation, analysis, robustness to noise, discretization up do the numerical implementation with various examples based on wave-like models in bounded but also unbounded domains.