Collin, Annabelle; Chapelle, Dominique; Moireau, Philippe A Luenberger observer for reaction-diffusion models with front position data. (English) Zbl 1349.76445 J. Comput. Phys. 300, 288-307 (2015). Summary: 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. Cited in 5 Documents MSC: 76M20 Finite difference methods applied to problems in fluid mechanics 35K58 Semilinear parabolic equations 35K20 Initial-boundary value problems for second-order parabolic equations 35K57 Reaction-diffusion equations 65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs 94A08 Image processing (compression, reconstruction, etc.) in information and communication theory Keywords:reaction-diffusion model; data assimilation; image processing; front propagation; eikonal equation; cardiac electrophysiology Software:PETSc; EnKF; Verdandi PDFBibTeX XMLCite \textit{A. Collin} et al., J. Comput. Phys. 300, 288--307 (2015; Zbl 1349.76445) Full Text: DOI References: [1] Anthes, R. A., Data assimilation and initialization of hurricane prediction model, J. Atmos. Sci., 31, 702-719 (1974) [2] Aronson, D. G.; Weinberger, H. F., Multidimensional nonlinear diffusion arising in population genetics, Adv. 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