Publications

2019

• On a Wasserstein-type distance between solutions to stochastic differential equations
  
  • Bion-Nadal Jocelyne
  • Talay Denis
  The Annals of Applied Probability, Institute of Mathematical Statistics (IMS), 2019, 29 (3), pp.1609-1639. In this paper, we introduce a Wasserstein-type distance on the set of the probability distributions of strong solutions to stochastic differential equations. This new distance is defined by restricting the set of possible coupling measures. We prove that it may also be defined by means of the value function of a stochastic control problem whose Hamilton–Jacobi–Bellman equation has a smooth solution, which allows one to deduce a priori estimates or to obtain numerical evaluations. We exhibit an optimal coupling measure and characterize it as a weak solution to an explicit stochastic differential equation, and we finally describe procedures to approximate this optimal coupling measure. A notable application concerns the following modeling issue: given an exact diffusion model, how to select a simplified diffusion model within a class of admissible models under the constraint that the probability distribution of the exact model is preserved as much as possible? (10.1214/18-AAP1423)
  DOI : 10.1214/18-AAP1423
• Avis en réponse à la saisine HCB - habilitation agents 2019. Paris, le 4 juillet 2019
  
  • Comité Scientifique Du Haut Conseil Des Biotechnologies .
  • Angevin Frédérique
  • Bagnis Claude
  • Bar-Hen Avner
  • Barny Marie-Anne
  • Boireau Pascal
  • Brévault Thierry
  • Chauvel Bruno B.
  • Collonnier Cécile
  • Couvet Denis
  • Dassa Elie
  • de Verneuil Hubert
  • Demeneix Barbara
  • Franche Claudine
  • Guerche Philippe
  • Guillemain Joël
  • Hernandez Raquet Guillermina
  • Khalife Jamal
  • Klonjkowski Bernard
  • Lavielle Marc
  • Le Corre Valérie
  • Lefèvre François
  • Lemaire Olivier
  • Lereclus Didier D.
  • Maximilien Rémy
  • Meurs Eliane
  • Naffakh Nadia
  • Négre Didier
  • Noyer Jean-Louis
  • Ochatt Sergio
  • Pages Jean-Christophe
  • Raynaud Xavier
  • Regnault-Roger Catherine
  • Renard Michel M.
  • Renault Tristan
  • Saindrenan Patrick
  • Simonet Pascal
  • Troadec Marie-Bérengère
  • Vaissière Bernard
  • Vilotte Jean-Luc
  , 2019, pp.2 p..
• Avis en réponse à la saisine HCB - EFSA-GMO-ES-2018-154. Paris, le 5 avril 2019
  
  • Du Haut Conseil Des Biotechnologies Comité Scientifique
  • Angevin Frédérique
  • Bagnis Claude
  • Bar-Hen Avner
  • Barny Marie-Anne
  • Boireau Pascal
  • Brévault Thierry
  • Chauvel Bruno B.
  • Collonnier Cécile
  • Couvet Denis
  • Dassa Elie
  • de Verneuil Hubert
  • Demeneix Barbara
  • Franche Claudine
  • Guerche Philippe
  • Guillemain Joël
  • Hernandez Raquet Guillermina
  • Khalife Jamal
  • Klonjkowski Bernard
  • Lavielle Marc
  • Le Corre Valérie
  • Lefèvre François
  • Lemaire Olivier
  • Lereclus Didier D.
  • Maximilien Rémy
  • Meurs Eliane
  • Naffakh Nadia
  • Négre Didier
  • Noyer Jean-Louis
  • Ochatt Sergio
  • Pages Jean-Christophe
  • Raynaud Xavier
  • Regnault-Roger Catherine
  • Renard Michel M.
  • Renault Tristan
  • Saindrenan Patrick
  • Simonet Pascal
  • Troadec Marie-Bérengère
  • Vaissière Bernard
  • Vilotte Jean-Luc
  , 2019.
• Avis en réponse à la saisine HCB - dossier EFSA-GMO-RX-013. Paris, le 30 janvier 2019
  
  • Comité Scientifique Du Haut Conseil Des Biotechnologies .
  • Angevin Frédérique
  • Bagnis Claude
  • Bar-Hen Avner
  • Barny Marie-Anne
  • Boireau Pascal
  • Brévault Thierry
  • Chauvel Bruno B.
  • Collonnier Cécile
  • Couvet Denis
  • Dassa Elie
  • de Verneuil Hubert
  • Demeneix Barbara
  • Franche Claudine
  • Guerche Philippe
  • Guillemain Joël
  • Hernandez Raquet Guillermina
  • Khalife Jamal
  • Klonjkowski Bernard
  • Lavielle Marc
  • Le Corre Valérie
  • Lefèvre François
  • Lemaire Olivier
  • Lereclus Didier D.
  • Maximilien Rémy
  • Meurs Eliane
  • Naffakh Nadia
  • Négre Didier
  • Noyer Jean-Louis
  • Ochatt Sergio
  • Pages Jean-Christophe
  • Raynaud Xavier
  • Regnault-Roger Catherine
  • Renard Michel M.
  • Renault Tristan
  • Saindrenan Patrick
  • Simonet Pascal
  • Troadec Marie-Bérengère
  • Vaissière Bernard
  • Vilotte Jean-Luc
  , 2019.
• Initiation of a validation strategy of reduced-order two-fluid flow models using direct numerical simulations in the context of jet atomization
  
  • Cordesse Pierre
  • Murrone A.
  • Ménard T.
  • Massot Marc
  NASA Technical Memorandum, National Aeronautics and Space Administration, 2019, pp.1-11. In industrial applications, developing predictive tools relying on numerical simulations using reduced-order models nourish the need of building a validation strategy. In the context of cryogenic atomization, we propose to build a hierarchy of direct numerical simulation test cases to assess qualitatively and quantitatively diffuse interface models. The present work proposes an initiation of the validation strategy with an air-assisted water atomization using a coaxial injector.
• Derivation of a two-phase flow model with two-scale kinematics, geometric variables and surface tension using variational calculus
  
  • Cordesse Pierre
  • Kokh Samuel
  • Di Battista Ruben
  • Drui Florence
  • Massot Marc
  NASA Technical Memorandum, National Aeronautics and Space Administration, 2019. The present paper proposes a two-phase flow model that is able to account for two-scale kinematics and two-scale surface tension effects based on geometric variables at small scale. At large scale, the flow and the full geometry of the interface may be retrieved thanks to the bulk variables, while at small scale the interface is accurately described by volume fraction, interfacial area density and mean curvature, called the geometric variables. Our work mainly relies on the Least Action Principle. The resulting system is an extension of a previous work modeling small scale pulsation in which surface tension was not taken into account at large or small scale. Whereas the original derivation assumes a cloud of monodispersed spherical bubbles, the present context allows for polydispersed, non-spherical bubbles. The resulting system of equations solely involves small scale geometric variables, thus contributing in the construction of a unified model describing both large and small scales.
• On the link between infinite horizon control and quasi-stationary distributions
  
  • Champagnat Nicolas
  • Claisse Julien
  Stochastic Processes and their Applications, Elsevier, 2019, 129 (3), pp.771-798. We study infinite horizon control of continuous-time non-linear branching processes with almost sure extinction for general (positive or negative) discount. Our main goal is to study the link between infinite horizon control of these processes and an optimization problem involving their quasi-stationary distributions and the corresponding extinction rates. More precisely, we obtain an equivalent of the value function when the discount parameter is close to the threshold where the value function becomes infinite , and we characterize the optimal Markov control in this limit. To achieve this, we present a new proof of the dynamic programming principle based upon a pseudo-Markov property for controlled jump processes. We also prove the convergence to a unique quasi-stationary distribution of non-linear branching processes controlled by a Markov control conditioned on non-extinction. (10.1016/j.spa.2018.03.018)
  DOI : 10.1016/j.spa.2018.03.018
• Optimal control of PDEs in a complex space setting; application to the Schrödinger equation
  
  • Aronna Maria Soledad
  • Bonnans Joseph Frédéric
  • Kröner Axel
  SIAM Journal on Control and Optimization, Society for Industrial and Applied Mathematics, 2019, 57 (2), pp.1390-1412. In this paper we discuss optimality conditions for abstract optimization problems over complex spaces. We then apply these results to optimal control problems with a semigroup structure. As an application we detail the case when the state equation is the Schrödinger one, with pointwise constraints on the "bilinear'" control. We derive first and second order optimality conditions and address in particular the case that the control enters the state equation and cost function linearly. (10.1137/17M1117653)
  DOI : 10.1137/17M1117653
• Comparison of methods for early-readmission prediction in a high-dimensional heterogeneous covariates and time-to-event outcome framework
  
  • Bussy Simon
  • Veil Raphaël
  • Looten Vincent
  • Burgun Anita
  • Gaiffas Stéphane
  • Guilloux Agathe
  • Ranque Brigitte
  • Jannot Anne-Sophie
  BMC Medical Research Methodology, BioMed Central, 2019, 19 (1), pp.50. Background: Choosing the most performing method in terms of outcome prediction or variables selection is a recurring problem in prognosis studies, leading to many publications on methods comparison. But some aspects have received little attention. First, most comparison studies treat prediction performance and variable selection aspects separately. Second, methods are either compared within a binary outcome setting (where we want to predict whether the readmission will occur within an arbitrarily chosen delay or not) or within a survival analysis setting (where the outcomes are directly the censored times), but not both. In this paper, we propose a comparison methodology to weight up those different settings both in terms of prediction and variables selection, while incorporating advanced machine learning strategies. Methods: Using a high-dimensional case study on a sickle-cell disease (SCD) cohort, we compare 8 statistical methods. In the binary outcome setting, we consider logistic regression (LR), support vector machine (SVM), random forest (RF), gradient boosting (GB) and neural network (NN); while on the survival analysis setting, we consider the Cox Proportional Hazards (PH), the CURE and the C-mix models. We also propose a method using Gaussian Processes to extract meaningfull structured covariates from longitudinal data. Results: Among all assessed statistical methods, the survival analysis ones obtain the best results. In particular the C-mix model yields the better performances in both the two considered settings (AUC =0.94 in the binary outcome setting), as well as interesting interpretation aspects. There is some consistency in selected covariates across methods within a setting, but not much across the two settings. Conclusions: It appears that learning withing the survival analysis setting first (so using all the temporal information), and then going back to a binary prediction using the survival estimates gives significantly better prediction performances than the ones obtained by models trained “directly” within the binary outcome setting. (10.1186/s12874-019-0673-4)
  DOI : 10.1186/s12874-019-0673-4
• A phase-field approximation of the Steiner problem in dimension two
  
  • Chambolle Antonin
  • Ferrari Luca Alberto Davide
  • Merlet Benoît
  Advances in Calculus of Variation, Walter de Gruyter GmbH, 2019, 12 (2), pp.157–179. In this paper we consider the branched transportation problem in 2D associated with a cost per unit length of the form $1 + αm$ where $m$ denotes the amount of transported mass and $α > 0$ is a fixed parameter (notice that the limit case $α = 0$ corresponds to the classical Steiner problem). Motivated by the numerical approximation of this problem, we introduce a family of func-tionals $({F ε } ε>0)$ which approximate the above branched transport energy. We justify rigorously the approximation by establishing the equicoercivity and the $Γ$-convergence of ${F ε } as ε ↓ 0$. Our functionals are modeled on the Ambrosio-Tortorelli functional and are easy to optimize in practice. We present numerical evidences of the efficiency of the method. (10.1515/acv-2016-0034)
  DOI : 10.1515/acv-2016-0034
• A mathematical model to predict BNP levels in hemodialysis patients
  
  • Touzot Maxime
  • Seris Pascal
  • Maheas Catherine
  • Vanmassenhove Jill
  • Langlois Anne-Lyse
  • Moubakir Kamal
  • Laplanche Sophie
  • Petitclerc Thierry
  • Ridel Christophe
  • Lavielle Marc
  Nephrology, Wiley, 2019. Aim: Clinical interpretation of B-Type Natriuretic Peptide (BNP) levels in hemodialysis patients (HD) for fluid management remains elusive. Method: We conducted a retrospective observational monocentric study. We built a mathematical model to predict BNP levels, using multiple linear regressions. Fifteen clinical/biological associated with BNP variation were selected. A first cohort of 150 prevalent HD (from September 2015 to march 2016) was used to build several models. The best model proposed was internally validated in an independent cohort of 62 incidents HD (from March 2016 to September 2017). Results: In cohort 1, mean BNP Level was 630±717 ng/ml. Cardiac disease (CD = Stable Coronary Artery Disease and/or Atrial Fibrillation) was present in 45% of patient. The final model includes: Age, systolic Blood Pressure (sBP), Albumin, CD, Normo-hydrated Weight (NHW) and the Fluid Overload (FO) assessed by bio-impedancemetry. The correlation between the measured and the predicted log-BNP was 0.567 and 0.543 in cohort-1 and 2 respectively. Age (β=3.175e-2, p<0.00), CD (β=5.243e-1, p<0.001) and FO (β=1.227e-1, p<0.001) contribute the most significantly to the BNP level, respectively, but within a certain range. We observed a logistic relationship between BNP and age between 30 to 60 years, after which this relationship was lost. BNP level was inversely correlated with NHW independently of CD. Finally, our model allows us to predict the BNP level according to the FO. Conclusion: We developed a mathematical model capable of predicting the BNP level in HD. Our results show the complex contribution of age, CD and FO on BNP level. (10.1111/nep.13586)
  DOI : 10.1111/nep.13586
• Variational approximation of size-mass energies for k-dimensional currents
  
  • Chambolle Antonin
  • Ferrari Luca A.D.
  • Merlet Benoit
  ESAIM: Control, Optimisation and Calculus of Variations, EDP Sciences, 2019, 25, pp.43. In this paper we produce a Γ-convergence result for a class of energies Fε,ak modeled on the Ambrosio-Tortorelli functional. For the choice k = 1 we show that Fε,a1 Γ-converges to a branched transportation energy whose cost per unit length is a function fan−1 depending on a parameter a > 0 and on the codimension n − 1. The limit cost fa(m) is bounded from below by 1 + m so that the limit functional controls the mass and the length of the limit object. In the limit a ↓ 0 we recover the Steiner energy. We then generalize the approach to any dimension and codimension. The limit objects are now k-currents with prescribed boundary, the limit functional controls both their masses and sizes. In the limit a ↓ 0, we recover the Plateau energy defined on k-currents, k < n. The energies Fε,ak then could be used for the numerical treatment of the k-Plateau problem. (10.1051/cocv/2018027)
  DOI : 10.1051/cocv/2018027
• Trajectories in random minimal transposition factorizations
  
  • Féray Valentin
  • Kortchemski Igor
  ALEA : Latin American Journal of Probability and Mathematical Statistics, Instituto Nacional de Matemática Pura e Aplicada (Rio de Janeiro, Brasil) [2006-....], 2019, 16 (1), pp.759. We study random typical minimal factorizations of the n-cycle, which are factorizations of (1, . . . , n) as a product of n−1 transpositions, chosen uniformly at random. Our main result is, roughly speaking, a local convergence theorem for the trajectories of finitely many points in the factorization. The main tool is an encoding of the factorization by an edge and vertex-labelled tree, which is shown to converge to Kesten’s infinite Bienaymé-Galton-Watson tree with Poisson offspring distribution, uniform i.i.d. edge labels and vertex labels obtained by a local exploration algorithm. (10.30757/ALEA.v16-27)
  DOI : 10.30757/ALEA.v16-27
• Topology optimization of modulated and oriented periodic microstructures by the homogenization method
  
  • Allaire Grégoire
  • Geoffroy-Donders Perle
  • Pantz Olivier
  Computers & Mathematics with Applications, Elsevier, 2019, 78, pp.2197-2229. This paper is concerned with the topology optimization of structures made of periodically perforated material, where the microscopic periodic cell can be macroscopically modulated and oriented. The main idea is to optimize the homogenized formulation of this problem, which is an easy task of parametric optimization, then to project the optimal microstruc-ture at a desired lengthscale, which is a delicate issue, albeit computa-tionally cheap. The main novelty of our work is, in a plane setting, the conformal treatment of the optimal orientation of the microstructure. In other words, although the periodicity cell has varying parameters and orientation throughout the computational domain, the angles between its members or bars are conserved. The main application of our work is the optimization of so-called lattice materials which are becoming increasingly popular in the context of additive manufacturing. Several numerical examples are presented for compliance minimization in 2-d.
• Log-sum-exp neural networks and posynomial models for convex and log-log-convex data
  
  • Calafiore Giuseppe C.
  • Gaubert Stéphane
  • Possieri Corrado
  IEEE Transactions on Neural Networks and Learning Systems, IEEE, 2019. We show that a one-layer feedforward neural network with exponential activation functions in the inner layer and logarithmic activation in the output neuron is an universal approximator of convex functions. Such a network represents a family of scaled log-sum exponential functions, here named LSET. Under a suitable exponential transformation, the class of LSET functions maps to a family of generalized posynomials GPOST, which we similarly show to be universal approximators for log-log-convex functions. A key feature of an LSET network is that, once it is trained on data, the resulting model is convex in the variables, which makes it readily amenable to efficient design based on convex optimization. Similarly, once a GPOST model is trained on data, it yields a posynomial model that can be efficiently optimized with respect to its variables by using geometric programming (GP). The proposed methodology is illustrated by two numerical examples, in which, first, models are constructed from simulation data of the two physical processes (namely, the level of vibration in a vehicle suspension system, and the peak power generated by the combustion of propane), and then optimization-based design is performed on these models. (10.1109/TNNLS.2019.2910417)
  DOI : 10.1109/TNNLS.2019.2910417
• Stability Properties of Systems of Linear Stochastic Differential Equations with Random Coefficients
  
  • Bishop Adrian N
  • del Moral Pierre
  SIAM Journal on Control and Optimization, Society for Industrial and Applied Mathematics, 2019, 57 (2), pp.1023-1042. This work is concerned with the stability properties of linear stochastic differential equationswith random (drift and diffusion) coefficient matrices, and the stability of a corresponding ran-dom transition matrix (or exponential semigroup). We consider a class of random matrix driftcoefficients that involves random perturbations of an exponentially stable flow of deterministic(time-varying) drift matrices. In contrast with more conventional studies, our analysis is notbased on the existence of Lyapunov functions, and it does notrely on any ergodic properties.These approaches are often difficult to apply in practice whenthe drift/diffusion coefficients arerandom. We present rather weak and easily checked perturbation-type conditions for the asymp-totic stability of time-varying and random linear stochastic differential equations. We providenew log-Lyapunov estimates and exponential contraction inequalities on any time horizon assoon as the fluctuation parameter is sufficiently small. Theseseem to be the first results of thistype for this class of linear stochastic differential equations with random coefficient matrices. (10.1137/18M1182759)
  DOI : 10.1137/18M1182759
• Sound Propagation in a Weakly Turbulent Flow in a Waveguide
  
  • Borcea Liliana
  • Garnier Josselin
  • Sølna Knut
  SIAM Journal on Applied Mathematics, Society for Industrial and Applied Mathematics, 2019, 79 (6), pp.2663-2687. (10.1137/19M1252144)
  DOI : 10.1137/19M1252144
• On the role of bulk viscosity in compressible reactive shear layer developments
  
  • Boukharfane Radouan
  • Martínez Ferrer Pedro José
  • Mura Arnaud
  • Giovangigli Vincent
  European Journal of Mechanics - B/Fluids, Elsevier, 2019, 77, pp.32-47. Despite 150 years of research after the reference work of Stokes, it should be acknowledged that some confusion still remains in the literature regarding the importance of bulk viscosity effects in flows of both academic and practical interests. On the one hand, it can be readily shown that the neglection of bulk viscosity is strictly exact for mono-atomic gases. The corresponding bulk viscosity effects are also unlikely to alter the flowfield dynamics provided that the ratio of the shear viscosity μ to the bulk viscosity κ remains sufficiently large. On the other hand, for polyatomic gases, the scattered available experimental and numerical data show that it is certainly not zero and actually often far from negligible. Therefore, since the ratio κ/μ can display significant variations and may reach very large values (it can exceed thirty for dihydrogen), it remains unclear to what extent the neglection of κ holds. The purpose of the present study is thus to analyze the mechanisms through which bulk viscosity and associated processes may alter a canonical turbulent flow. In this context, we perform direct numerical simulations (DNS) of spatially-developing compressible non-reactive and reactive hydrogen-air shear layers interacting with an oblique shock wave. The corresponding flowfield is of special interest for var- ious reactive high-speed flow applications, e.g., Scramjets. The corresponding computations either neglect the influence of bulk viscosity (κ = 0) or take it into consideration by evaluating its value using the EGlib library. The qualitative inspection of the results obtained for two-dimensional cases in either the presence or the absence of bulk viscosity effects shows that the local and instantaneous structure of the mixing layer may be deeply altered when taking bulk viscosity into account. This contrasts with some mean statistical quantities, e.g., the vorticity thickness growth rate, which do not exhibit any significant sensitivity to the bulk viscosity. Enstrophy, Reynolds stress components, and turbulent kinetic energy (TKE) budgets are then evaluated from three-dimensional re- active simulations. Slight modifications are put into evidence on the energy transfer and dissipation contributions. From the obtained results, one may expect that refined large-eddy simulations (LES) may be rather sensitive to the consideration of bulk viscosity, while Reynolds-averaged Navier-Stokes (RANS) simulations, which are based on statistical averages, are not. (10.1016/j.euromechflu.2019.02.005)
  DOI : 10.1016/j.euromechflu.2019.02.005
• Avis en réponse à la saisine HCB – habilitation agents juillet 2019. Paris, le 19 septembre 2019
  
  • Comité Scientifique Du Haut Conseil Des Biotechnologies .
  • Angevin Frédérique
  • Bagnis Claude
  • Bar-Hen Avner
  • Barny Marie-Anne
  • Boireau Pascal
  • Brévault Thierry
  • Chauvel Bruno B.
  • Collonnier Cécile
  • Couvet Denis
  • Dassa Elie
  • de Verneuil Hubert
  • Demeneix Barbara
  • Franche Claudine
  • Guerche Philippe
  • Guillemain Joël
  • Hernandez Raquet Guillermina
  • Khalife Jamal
  • Klonjkowski Bernard
  • Lavielle Marc
  • Le Corre Valérie
  • Lefèvre François
  • Lemaire Olivier
  • Lereclus Didier D.
  • Maximilien Rémy
  • Meurs Eliane
  • Naffakh Nadia
  • Négre Didier
  • Noyer Jean-Louis
  • Ochatt Sergio
  • Pages Jean-Christophe
  • Raynaud Xavier
  • Regnault-Roger Catherine
  • Renard Michel M.
  • Renault Tristan
  • Saindrenan Patrick
  • Simonet Pascal
  • Troadec Marie-Bérengère
  • Vaissière Bernard
  • Vilotte Jean-Luc
  , 2019, pp.2 p..
• Avis en réponse à la saisine HCB - dossier EFSA-GMO-NL-2018-153. Paris, le 24 mai 2019
  
  • Comité Scientifique Du Haut Conseil Des Biotechnologies .
  • Angevin Frédérique
  • Bagnis Claude
  • Bar-Hen Avner
  • Barny Marie-Anne
  • Boireau Pascal
  • Brévault Thierry
  • Chauvel Bruno B.
  • Collonnier Cécile
  • Couvet Denis
  • Dassa Elie
  • de Verneuil Hubert
  • Demeneix Barbara
  • Franche Claudine
  • Guerche Philippe
  • Guillemain Joël
  • Hernandez Raquet Guillermina
  • Khalife Jamal
  • Klonjkowski Bernard
  • Lavielle Marc
  • Le Corre Valérie
  • Lefèvre François
  • Lemaire Olivier
  • Lereclus Didier D.
  • Maximilien Rémy
  • Meurs Eliane
  • Naffakh Nadia
  • Négre Didier
  • Noyer Jean-Louis
  • Ochatt Sergio
  • Pages Jean-Christophe
  • Raynaud Xavier
  • Regnault-Roger Catherine
  • Renard Michel M.
  • Renault Tristan
  • Saindrenan Patrick
  • Simonet Pascal
  • Troadec Marie-Bérengère
  • Vaissière Bernard
  • Vilotte Jean-Luc
  , 2019.
• Computing Relative Binding Affinity of Ligands to Receptor: An Effective Hybrid Single-Dual-Topology Free-Energy Perturbation Approach in NAMD
  
  • Jiang Wei
  • Chipot Christophe
  • Roux Benoit
  Journal of Chemical Information and Modeling, American Chemical Society, 2019, 59 (9), pp.3794--3802. (10.1021/acs.jcim.9b00362)
  DOI : 10.1021/acs.jcim.9b00362
• Matrix versions of the Hellinger distance
  
  • Bhatia Rajendra
  • Gaubert Stéphane
  • Jain Tanvi
  Letters in Mathematical Physics, Springer Verlag, 2019, 109, pp.1777-1804. On the space of positive definite matrices we consider distance functions of the form $d(A,B)=\left[\operatorname{tr}\mathcal{A}(A,B)-\operatorname{tr}\mathcal{G}(A,B)\right]^{1/2},$ where $\mathcal{A}(A,B)$ is the arithmetic mean and $\mathcal{G}(A,B)$ is one of the different versions of the geometric mean. When $\mathcal{G}(A,B)=A^{1/2}B^{1/2}$ this distance is $\|A^{1/2}-B^{1/2}\|_2,$ and when $\mathcal{G}(A,B)=(A^{1/2}BA^{1/2})^{1/2}$ it is the Bures-Wasserstein metric. We study two other cases: $\mathcal{G}(A,B)=A^{1/2}(A^{-1/2}BA^{-1/2})^{1/2}A^{1/2},$ the Pusz-Woronowicz geometric mean, and $\mathcal{G}(A,B)=\exp\big(\frac{\log A+\log B}{2}\big),$ the log Euclidean mean. With these choices $d(A,B)$ is no longer a metric, but it turns out that $d^2(A,B)$ is a divergence. We establish some (strict) convexity properties of these divergences. We obtain characterisations of barycentres of $m$ positive definite matrices with respect to these distance measures. One of these leads to a new interpretation of a power mean introduced by Lim and Palfia, as a barycentre. The other uncovers interesting relations between the log Euclidean mean and relative entropy. (10.1007/s11005-019-01156-0)
  DOI : 10.1007/s11005-019-01156-0
• Time-Optimal Trajectories of Generic Control-Affine Systems Have at Worst Iterated Fuller Singularities
  
  • Boarotto Francesco
  • Sigalotti Mario
  Annales de l'Institut Henri Poincaré (C), Analyse non linéaire (Nonlinear Analysis), EMS, 2019, 36 (2), pp.327-346. We consider in this paper the regularity problem for time-optimal trajectories of a single-input control-affine system on a n-dimensional manifold. We prove that, under generic conditions on the drift and the controlled vector field, any control u associated with an optimal trajectory is smooth out of a countable set of times. More precisely, there exists an integer K, only depending on the dimension n, such that the non-smoothness set of u is made of isolated points, accumulations of isolated points, and so on up to K-th order iterated accumulations.
• Energy-optimal strokes for multi-link microswimmers: Purcell's loops and Taylor's waves reconciled
  
  • Wiezel Oren
  • Giraldi Laetitia
  • Desimone Antonio
  • Or Yizhar
  • Alouges François
  , 2019, pp.043050. Abstract Micron-scale swimmers move in the realm of negligible inertia, dominated by viscous drag forces. In this paper, we formulate the leading-order dynamics of a slender multi-link ( N -link) microswimmer assuming small-amplitude undulations about its straight configuration. The energy-optimal stroke to achieve a given prescribed displacement in a given time period is obtained as the largest eigenvalue solution of a constrained optimal control problem. Remarkably, the optimal stroke is an ellipse lying within a two-dimensional plane in the ( N – 1)-dimensional space of joint angles, where N can be arbitrarily large. For large N , the optimal stroke is a traveling wave of bending, modulo edge effects. If the number of shape variables is small, we can consider the same problem when the prescribed displacement in one time period is large, and not attainable with small variations of the joint angles. The fully nonlinear optimal control problem is solved numerically for the cases N = 3 (Purcell’s three-link swimmer) and N = 5 showing that, as the prescribed displacement becomes small, the optimal solutions obtained using the small-amplitude assumption are recovered. We also show that, when the prescribed displacements become large, the picture is different. For N = 3 we recover the non-convex planar loops already known from previous studies. For N = 5 we obtain non-planar loops, raising the question of characterizing the geometry of complex high-dimensional loops. (10.1088/1367-2630/ab1142)
  DOI : 10.1088/1367-2630/ab1142
• A tale of a principal and many many agents
  
  • Elie Romuald
  • Mastrolia Thibaut
  • Possamaï Dylan
  Mathematics of Operations Research, INFORMS, 2019, 44 (2), pp.440-467. In this paper, we investigate a moral hazard problem in finite time with lump–sum and continuous payments, involving infinitely many Agents with mean field type interactions, hired by one Principal. By reinterpreting the mean-field game faced by each Agent in terms of a mean field forward backward stochastic differential equation (FBSDE for short), we are able to rewrite the Principal's problem as a control problem of McKean–Vlasov SDEs. We review two general approaches to tackle it: the first one introduced recently in [2, 66, 67, 68, 69] using dynamic programming and Hamilton–Jacobi– Bellman (HJB for short) equations, the second based on the stochastic Pontryagin maximum principle, which follows [16]. We solve completely and explicitly the problem in special cases, going beyond the usual linear–quadratic framework. We finally show in our examples that the optimal contract in the N −players' model converges to the mean–field optimal contract when the number of agents goes to +∞, this illustrating in our specific setting the general results of [12]. (10.1287/moor.2018.0931)
  DOI : 10.1287/moor.2018.0931