A Luenberger observer for reaction-diffusion models with front position data

par Annabelle Collin (Univ. Bordeaux)

mardi 14 mars 2017, 14:00 → 15:00 Europe/Paris

4e étage, salle 435 (UMPA, ENS Lyon - Site Monod)

Cardiac electrophysiology describes and models chemical and electrical phenomena taking place in the cardiac tissue. Given the large number of related pathologies, there is an important need for understanding these phenomena. The electric wave propagating in the cardiac tissue can be represented by a nonlinear reaction-diffusion partial differential equation "coupled with an ordinary differential equation representing cellular activity" called the bidomain model. The complex bidomain model must be adapted to each individual case in order to produce predictive simulations for a given patient. In this context, we can use the abundant available medical data, especially the patient electrical activation maps - which correspond to the location of the front over time - in order to adapt the bidomain model. In this presentation, we present a Luenberger observer for reaction-diffusion models with propagating front features, and for data associated with the location of the front over time. We start by proposing an observer for the eikonal equation that can be derived from the reaction-diffusion model by an asymptotic expansion, drawing our inspiration from image processing methods. We then carry over this observer to the underlying reaction-diffusion equation by an "inverse asymptotic analysis". We also discuss the extension to joint state-parameter estimation by using the earlier-proposed ROUKF strategy. We then illustrate and assess our proposed observer strategy with test problems associated with electrophysiology modeling and also with atrial real data. Our numerical trials show that state estimation is directly very effective.