Publications

2019

• Existence of strong solutions to the Dirichlet problem for the Griffith energy
  
  • Chambolle Antonin
  • Crismale Vito
  Calculus of Variations and Partial Differential Equations, Springer Verlag, 2019, 58 (136). In this paper we continue the study of the Griffith brittle fracture energy minimisation under Dirichlet boundary conditions, suggested by Francfort and Marigo in 1998. In a recent paper, we proved the existence of weak minimisers of the problem. Now we show that these minimisers are indeed strong solutions, namely their jump set is closed and they are smooth away from the jump set and continuous up to the Dirichlet boundary. This is obtained by extending up to the boundary the recent regularity results of Conti, Focardi, Iurlano, and of Chambolle, Conti, Iurlano. (10.1007/s00526-019-1571-7)
  DOI : 10.1007/s00526-019-1571-7
• The asymptotic geometry of the Teichmüller metric
  
  • Walsh Cormac
  Geometriae Dedicata, Springer Verlag, 2019, 200 (1), pp.115-152. We determine the asymptotic behaviour of extremal length along arbitrary Teichmüller rays. This allows us to calculate the endpoint in the Gardiner-Masur boundary of any Teichmüller ray. We give a proof that this compactification is the same as the horofunction compactification. An important subset of the latter is the set of Busemann points. We show that the Busemann points are exactly the limits of the Teichmüller rays, and we give a necessary and sufficient condition for a sequence of Busemann points to converge to a Busemann point. Finally, we determine the detour metric on the boundary. (10.1007/s10711-018-0364-z)
  DOI : 10.1007/s10711-018-0364-z
• Imputation of mixed data with multilevel singular value decomposition
  
  • Husson François
  • Josse Julie
  • Narasimhan Balasubramanian
  • Robin Geneviève
  Journal of Computational and Graphical Statistics, Taylor & Francis, 2019, 28 (3), pp.552-566. Statistical analysis of large data sets offers new opportunities to better understand many processes. Yet, data accumulation often implies relaxing acquisition procedures or compounding diverse sources. As a consequence, such data sets often contain mixed data, i.e. both quantitative and qualitative and many missing values. Furthermore, aggregated data present a natural \textit{multilevel} structure, where individuals or samples are nested within different sites, such as countries or hospitals. Imputation of multilevel data has therefore drawn some attention recently, but current solutions are not designed to handle mixed data, and suffer from important drawbacks such as their computational cost. In this article, we propose a single imputation method for multilevel data, which can be used to complete either quantitative, categorical or mixed data. The method is based on multilevel singular value decomposition (SVD), which consists in decomposing the variability of the data into two components, the between and within groups variability, and performing SVD on both parts. We show on a simulation study that in comparison to competitors, the method has the great advantages of handling data sets of various size, and being computationally faster. Furthermore, it is the first so far to handle mixed data. We apply the method to impute a medical data set resulting from the aggregation of several data sets coming from different hospitals. This application falls in the framework of a larger project on Trauma patients. To overcome obstacles associated to the aggregation of medical data, we turn to distributed computation. The method is implemented in an R package. (10.1080/10618600.2019.1585261)
  DOI : 10.1080/10618600.2019.1585261
• Mean field model for collective motion bistability
  
  • Garnier Josselin
  • Papanicolaou George
  • Yang Tzu-Wei
  Discrete and Continuous Dynamical Systems - Series B, American Institute of Mathematical Sciences, 2019, 24 (2), pp.851-879. (10.3934/dcdsb.2018210)
  DOI : 10.3934/dcdsb.2018210
• On the Essential Self-Adjointness of Singular Sub-Laplacians
  
  • Franceschi Valentina
  • Prandi Dario
  • Rizzi Luca
  Potential Analysis, Springer Verlag, 2019, 53, pp.89-112. (10.1007/s11118-018-09760-w)
  DOI : 10.1007/s11118-018-09760-w
• Quantifying uncertainties in signal position in non-resolved object images: application to space object observation
  
  • Sanson Francois
  • Frueh Carolin
  Advances in Space Research, Elsevier, 2019. Charged Coupled Devices (CCDs) and subsequently Complementary metal-oxide-semiconductor (CMOS) detectors revolutionized scientific imaging. On both the CCD and CMOS detector, the generated images are degraded by inevitable noise. In many applications, such as in astronomy or for satellite tracking , only unresolved object images are available. Strategies to estimate the center of the non-resolved image their results are affected by the detector noise. The uncertainty in the center is classically estimated by running prohibitively costly Monte Carlo simulations, but in this paper, we propose analytic uncertainty estimates of the center position. The expressions that depend on the pixel size, the signal to noise ratio and the extension of the object signal relative to the pixel size are validated against rigorous Monte Carlo simulations with very satisfying results. Numerical tests show that our analytic expression is an efficient substitute to the Monte Carlo simulation thereby reducing computational cost. (10.1016/j.asr.2018.12.040)
  DOI : 10.1016/j.asr.2018.12.040
• 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
• 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
• 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
• 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
• 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
• 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.
• 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
• 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
• 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
• 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
• 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
• 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
• 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
• Efficient sequential experimental design for surrogate modeling of nested codes
  
  • Marque-Pucheu Sophie
  • Perrin Guillaume
  • Garnier Josselin
  ESAIM: Probability and Statistics, EDP Sciences, 2019, 23, pp.245-270. Thanks to computing power increase, the certification and the conception of complex systems relies more and more on simulation. To this end, predictive codes are needed, which have generally to be evaluated in a huge number of input points. When the computational cost of these codes is high, surrogate models are introduced to emulate the response of these codes. In this paper, we consider the situation when the system response can be modeled by two nested computer codes. By two nested computer codes, we mean that some inputs of the second code are outputs of the first code. More precisely, the idea is to propose sequential designs to improve the accuracy of the nested code's predictor by exploiting the nested structure of the codes. In particular, a selection criterion is proposed to allow the modeler to choose the code to call, depending on the expected learning rate and the computational cost of each code. The sequential designs are based on the minimization of the prediction variance, so adaptations of the Gaussian process formalism are proposed for this particular configuration in order to quickly evaluate the mean and the variance of the predictor. The proposed methods are then applied to examples. (10.1051/ps/2018011)
  DOI : 10.1051/ps/2018011
• 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, 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.
• 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
• 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
• 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..