## Section: New Results

### Inverse Problems

#### Analysis of an observer strategy for initial state reconstruction of wave-like systems in unbounded domains

Participants : Sébastien Imperiale, Philippe Moireau [correspondant] .

In [29] we are interested in reconstructing the initial condition of a wave equation in an unbounded domain configuration from measurements available in time on a subdomain. To solve this problem, we adopt an iterative strategy of reconstruction based on observers and time reversal adjoint formulations. We prove the convergence of our reconstruction algorithm with perfect measurements and its robustness to noise. Moreover, we develop a complete strategy to practically solve this problem on a bounded domain using artificial transparent boundary conditions to account for the exterior domain. Our work then demonstrates that the consistency error introduced by the use of approximate transparent boundary conditions is compensated by the stabilisation properties obtained from the use of the available measurements, hence allowing to still be able to reconstruct the unknown initial condition.

#### Analysis and numerical simulation of an inverse problem for a structured cell population dynamics model

Participants : Frédérique Clément, Frédérique Robin [correspondant] .

We have studied (with Béatrice Laroche, INRA) a multiscale inverse problem associated with a multi-type model for age structured cell populations [20] (see also [21] for another application). In the single type case, the model is a McKendrick-VonFoerster like equation with a mitosis-dependent death rate and potential migration at birth. In the multi-type case, the migration term results in a unidirectional motion from one type to the next, so that the boundary condition at age 0 contains an additional extrinsic contribution from the previous type. We consider the inverse problem of retrieving microscopic information (the division rates and migration proportions) from the knowledge of macroscopic information (total number of cells per layer), given the initial condition. We have first shown the well-posedness of the inverse problem in the single type case using a Fredholm integral equation derived from the characteristic curves, and we have used a constructive approach to obtain the lattice division rate, considering either a synchronized or non-synchronized initial condition. We have taken advantage of the unidirectional motion to decompose the whole model into nested submodels corresponding to self-renewal equations with an additional extrinsic contribution. We have again derived a Fredholm integral equation for each submodel and deduced the well-posedness of the multi-type inverse problem. In each situation, we illustrate numerically our theoretical results.

#### Inverse problem based on data assimilation approaches for protein aggregation

Participants : Philippe Moireau [correspondant] , Cécile Della Valle [MAMBA] , Marie Doumic [MAMBA] .

Estimating reaction rates and size distributions of protein polymers is an important step for understanding the mechanisms of protein misfolding and aggregation. In a depolarization configuration, we here extend some previous results obtained during the PhD Thesis of A. Armiento. Now, the depolarization rate is time-dependent or in the presence of an additional vanishing viscosity term. We continue to develop our framework mixing inverse problems methodologies and optimal control approaches typically encountered in data assimilation, allowing to justify mathematically the methods but also to adopt efficient numerical strategies. Publications of this work will be soon submitted.

#### Front shape similarity measure for data-driven simulations of wildland fire spread based on state estimation: Application to the RxCADRE field-scale experiment

Participants : Annabelle Collin [MONC] , Philippe Moireau [correspondant] .

Data-driven wildfire spread modeling is emerging as a cornerstone for forecasting real-time fire behavior using thermal-infrared imaging data. One key challenge in data assimilation lies in the design of an adequate measure to represent the discrepancies between observed and simulated firelines (or “fronts”). A first approach consists in adopting a Lagrangian description of the flame front and in computing a Euclidean distance between simulated and observed fronts by pairing each observed marker with its closest neighbor along the simulated front. However, this front marker registration approach is difficult to generalize to complex front topology that can occur when fire propagation conditions are highly heterogeneous due to topography, biomass fuel and micrometeorology. To overcome this issue, we investigate in this paper an object-oriented approach derived from the Chan–Vese contour fitting functional used in image processing. The burning area is treated as a moving object that can undergo shape deformations and topological changes. We combine this non-Euclidean measure with a state estimation approach (a Luenberger observer) to perform simulations of the time-evolving fire front location driven by discrete observations of the fireline. We apply this object-oriented data assimilation method to the three-hectare RxCADRE S5 field-scale experiment. This collaboration with CERFACS (M. Rochoux) and University of Maryland (C. Zhang and A. Trouvé) led to a publication [34] in the Proceedings of the Combustion Institute.

#### Model assessment through data assimilation of realistic data in cardiac electrophysiology

Participants : Antoine Gerard [CARMEN] , Annabelle Collin [MONC] , Gautier Bureau, Philippe Moireau [correspondant] , Yves Coudière [CARMEN] .

We consider a model-based estimation procedure – namely a data assimilation algorithm – of the atrial depolarization state of a sub- ject using data corresponding to electro-anatomical maps. Our objective is to evaluate the sensitivity of such a model-based reconstruction with respect to model choices. The followed data assimilation approach is capable of using electrical activation times to adapt a monodomain model simulation, thanks to an ingenious model-data fitting term inspired from image processing. The resulting simulation smoothes and completes the activation maps when they are spatially incomplete. Moreover, conductivity parameters can also be inferred. The model sensitivity assessment is performed based on synthetic data generated with a validated realistic atria model and then inverted using simpler modeling ingredients. In particular, the impact of the muscle fibers definition and corresponding anisotropic conductivity parameters is studied. Finally, an application of the method to real data is presented, showing promising results. This collaborative work has been published, see [37].