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

Annabelle Collin 1, 2 Dominique Chapelle 1 Philippe Moireau 1 
1 M3DISIM - Mathematical and Mechanical Modeling with Data Interaction in Simulations for Medicine
LMS - Laboratoire de mécanique des solides, Inria Saclay - Ile de France
2 MONC - Modélisation Mathématique pour l'Oncologie
IMB - Institut de Mathématiques de Bordeaux, Institut Bergonié [Bordeaux], Inria Bordeaux - Sud-Ouest
Abstract : We propose a Luenberger observer for reaction-diffusion models with propagating front features, and for data associated with the location of the front over time. Such models are considered in various application fields, such as electrophysiology, wild-land fire propagation and tumor growth modeling. Drawing our inspiration from image processing methods, we start by proposing an observer for the eikonal-curvature equation that can be derived from the reaction-diffusion model by an asymptotic expansion. We then carry over this observer to the underlying reaction-diffusion equation by an "inverse asymptotic analysis", and we show that the associated correction in the dynamics has a stabilizing effect for the linearized estimation error. 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 method with test problems pertaining to electrophysiology modeling, including with a realistic model of cardiac atria. Our numerical trials show that state estimation is directly very effective with the proposed Luenberger observer, while specific strategies are needed to accurately perform parameter estimation – as is usual with Kalman filtering used in a nonlinear setting – and we demonstrate two such successful strategies.
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Submitted on : Thursday, August 13, 2015 - 4:54:17 PM
Last modification on : Thursday, October 27, 2022 - 3:21:02 AM
Long-term archiving on: : Wednesday, April 26, 2017 - 9:55:47 AM


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Annabelle Collin, Dominique Chapelle, Philippe Moireau. A Luenberger observer for reaction-diffusion models with front position data. Journal of Computational Physics, 2015, 300, pp.20. ⟨10.1016/⟩. ⟨hal-01111675v2⟩



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