Benoît Perthame

Marie Doumic Jauffret, Benoît Perthame, Jorge P. Zubelli

We consider a size-structured model for cell division and address the question of determining the division (birth) rate from the measured stable size distribution of the population. We propose a new regularization technique based on a filtering approach. We prove convergence of the algorithm and validate the theoretical results by implementing numerical simulations, based on classical techniques. We compare the results for direct and inverse problems, for the filtering method and for the quasi-reversibility method proposed in [Perthame-Zubelli].

Luís Neves de Almeida, Federica Bubba, Benoît Perthame et al.

The parabolic-elliptic Keller-Segel equation with sensitivity saturation, because of its pattern formation ability, is a challenge for numerical simulations. We provide two finite-volume schemes whose goals are to preserve, at the discrete level, the fundamental properties of the solutions, namely energy dissipation, steady states, positivity and conservation of total mass. These requirements happen to be critical when it comes to distinguishing between discrete steady states, Turing unstable transient states, numerical artifacts or approximate steady states as obtained by a simple upwind approach. These schemes are obtained either by following closely the gradient flow structure or by a proper exponential rewriting inspired by the Scharfetter-Gummel discretization. An interesting feature is that upwind is also necessary for all the expected properties to be preserved at the semi-discrete level. These schemes are extended to the fully discrete level and this leads us to tune precisely the terms according to explicit or implicit discretizations. Using some appropriate monotony properties (reminiscent of the maximum principle), we prove well-posedness for the scheme as well as all the other requirements. Numerical implementations and simulations illustrate the respective advantages of the three methods we compare.

Benjamin Gess, Benoît Perthame, Panagiotis E. Souganidis

We develop a semi-discretization approximation for scalar conservation laws with multiple rough time dependence in inhomogeneous fluxes. The method is based on Brenier's transport-collapse algorithm and uses characteristics defined in the setting of rough paths. We prove strong $L^1$-convergence for inhomogeneous fluxes and provide a rate of convergence for homogeneous one's. The approximation scheme as well as the proofs are based on the recently developed theory of pathwise entropy solutions and uses the kinetic formulation which allows to define globally the (rough) characteristics.

Edouard Ribes, Karim Touahri, Benoît Perthame

This paper illustrates the similarities between the problems of customer churn and employee turnover. An example of employee turnover prediction model leveraging classical machine learning techniques is developed. Model outputs are then discussed to design \& test employee retention policies. This type of retention discussion is, to our knowledge, innovative and constitutes the main value of this paper.

Marie Doumic Jauffret, Peter S. Kim, Benoît Perthame

We analyze the asymptotic behavior of a partial differential equation (PDE) model for hematopoiesis. This PDE model is derived from the original agent-based model formulated by (Roeder et al., Nat. Med., 2006), and it describes the progression of blood cell development from the stem cell to the terminally differentiated state. To conduct our analysis, we start with the PDE model of (Kim et al, JTB, 2007), which coincides very well with the simulation results obtained by Roeder et al. We simplify the PDE model to make it amenable to analysis and justify our approximations using numerical simulations. An analysis of the simplified PDE model proves to exhibit very similar properties to those of the original agent-based model, even if for slightly different parameters. Hence, the simplified model is of value in understanding the dynamics of hematopoiesis and of chronic myelogenous leukemia, and it presents the advantage of having fewer parameters, which makes comparison with both experimental data and alternative models much easier.