Publications

2020

• Bacterial Metabolic Heterogeneity: from Stochastic to Deterministic Models
  
  • Graham Carl
  • Harmand Jérôme
  • Méléard Sylvie
  • Tchouanti Josué
  Mathematical Biosciences and Engineering, AIMS Press, 2020, 17 (5), pp.5120-5133. We revisit the modeling of the diauxic growth of a pure microorganism on two distinct sugars which was first described by Monod. Most available models are deterministic and make the assumption that all cells of the microbial ecosystem behave homogeneously with respect to both sugars, all consuming the first one and then switching to the second when the first is exhausted. We propose here a stochastic model which describes what is called "metabolic heterogeneity". It allows to consider small populations as in microfluidics as well as large populations where billions of individuals coexist in the medium in a batch or chemostat. We highlight the link between the stochastic model and the deterministic behavior in real large cultures using a large population approximation. Then the influence of model parameter values on model dynamics is studied, notably with respect to the lag-phase observed in real systems depending on the sugars on which the microorganism grows. It is shown that both metabolic parameters as well as initial conditions play a crucial role on system dynamics. (10.3934/mbe.2020276)
  DOI : 10.3934/mbe.2020276
• Optimal control of state constrained age-structured problems
  
  • Bonnans Joseph Frédéric
  • Gianatti Justina
  SIAM Journal on Control and Optimization, Society for Industrial and Applied Mathematics, 2020, 58 (4), pp.2206-–2235. The aim of this work is to study an optimal control problem with state constraints where the state is given by an age-structured, abstract parabolic differential equation. We prove the existence and uniqueness of solution for the state equation and provide first and second parabolic estimates. We analyze the differentiability of the cost function and, based on the general theory of Lagrange multipliers, we give a first order optimality condition. We also define and analyze the regularity of the costate.
• Asymptotical estimates for some algorithms for data and image processing : a study of the Sinkhorn algorithm and a numerical analysis of total variation minimization
  
  • Caillaud Corentin
  , 2020. This thesis deals with discrete optimization problems and investigates estimates of their convergence rates. It is divided into two independent parts.The first part addresses the convergence rate of the Sinkhorn algorithm and of some of its variants. This algorithm appears in the context of Optimal Transportation (OT) through entropic regularization. Its iterations, and the ones of the Sinkhorn-like variants, are written as componentwise products of nonnegative vectors and matrices. We propose a new approach to analyze them, based on simple convex inequalities and leading to the linear convergence rate that is observed in practice. We extend this result to a particular type of variants of the algorithm that we call 1D balanced Sinkhorn-like algorithms. In addition, we present some numerical techniques dealing with the convergence towards zero of the regularizing parameter of the OT problems. Lastly, we conduct the complete analysis of the convergence rate in dimension 2. In the second part, we establish error estimates for two discretizations of the total variation (TV) in the Rudin-Osher-Fatemi (ROF) model. This image denoising problem, that is solved by computing the proximal operator of the total variation, enjoys isotropy properties ensuring the preservation of sharp discontinuities in the denoised images in every direction. When the problem is discretized into a square mesh of size h and one uses a standard discrete total variation -- the so-called isotropic TV -- this property is lost. We show that in a particular direction the error in the energy is of order h^{2/3} which is relatively large with respect to what one can expect with better discretizations. Our proof relies on the analysis of an equivalent 1D denoising problem and of the perturbed TV it involves. The second discrete total variation we consider mimics the definition of the continuous total variation replacing the usual dual fields by discrete Raviart-Thomas fields. Doing so, we recover an isotropic behavior of the discrete ROF model. Finally, we prove a O(h) error estimate for this variant under standard hypotheses.
• On Stability of a Class of Filters for Nonlinear Stochastic Systems
  
  • Karvonen Toni
  • Bonnabel Silvère
  • Moulines Eric
  • Särkkä Simo
  SIAM Journal on Control and Optimization, Society for Industrial and Applied Mathematics, 2020, 58, pp.2023 - 2049. This article develops a comprehensive framework for stability analysis of a broad class of commonly used continuous-and discrete-time filters for stochastic dynamic systems with nonlinear state dynamics and linear measurements under certain strong assumptions. The class of filters encompasses the extended and unscented Kalman filters and most other Gaussian assumed density filters and their numerical integration approximations. The stability results are in the form of time-uniform mean square bounds and exponential concentration inequalities for the filtering error. In contrast to existing results, it is not always necessary for the model to be exponentially stable or fully observed. We review three classes of models that can be rigorously shown to satisfy the stringent assumptions of the stability theorems. Numerical experiments using synthetic data validate the derived error bounds. (10.1137/19m1285974)
  DOI : 10.1137/19m1285974
• Imputation and low-rank estimation with Missing Not At Random data
  
  • Sportisse Aude
  • Boyer Claire
  • Josse Julie
  Statistics and Computing, Springer Verlag (Germany), 2020. Missing values challenge data analysis because many supervised and unsupervised learning methods cannot be applied directly to incomplete data. Matrix completion based on low-rank assumptions are very powerful solution for dealing with missing values. However, existing methods do not consider the case of informative missing values which are widely encountered in practice. This paper proposes matrix completion methods to recover Missing Not At Random (MNAR) data. Our first contribution is to suggest a model-based estimation strategy by modelling the missing mechanism distribution. An EM algorithm is then implemented, involving a Fast Iterative Soft-Thresholding Algorithm (FISTA). Our second contribution is to suggest a computationally efficient surrogate estimation by implicitly taking into account the joint distribution of the data and the missing mechanism: the data matrix is concatenated with the mask coding for the missing values; a low-rank structure for exponential family is assumed on this new matrix, in order to encode links between variables and missing mechanisms. The methodology that has the great advantage of handling different missing value mechanisms is robust to model specification errors. The performances of our methods are assessed on the real data collected from a trauma registry (TraumaBase ) containing clinical information about over twenty thousand severely traumatized patients in France. The aim is then to predict if the doctors should administrate tranexomic acid to patients with traumatic brain injury, that would limit excessive bleeding. (10.1007/s11222-020-09963-5)
  DOI : 10.1007/s11222-020-09963-5
• A convex programming approach to solve posynomial systems
  
  • Akian Marianne
  • Allamigeon Xavier
  • Boyet Marin
  • Gaubert Stéphane
  , 2020, 12097. We exhibit a class of classical or tropical posynomial systems which can be solved by reduction to linear or convex programming problems. This relies on a notion of colorful vectors with respect to a collection of Newton polytopes. This extends the convex programming approach of one player stochastic games.
• Efficient Estimation of Equilibria in Large Aggregative Games with Coupling Constraints
  
  • Jacquot Paulin
  • Wan Cheng
  • Beaude Olivier
  • Oudjane Nadia
  IEEE Transactions on Automatic Control, Institute of Electrical and Electronics Engineers, 2020. Aggregative games have many industrial applications, and computing an equilibrium in those games is challenging when the number of players is large. In the framework of atomic aggregative games with coupling constraints, we show that variational Nash equilibria of a large aggregative game can be approximated by a Wardrop equilibrium of an auxiliary population game of smaller dimension. Each population of this auxiliary game corresponds to a group of atomic players of the initial large game. This approach enables an efficient computation of an approximated equilibrium, as the variational inequality characterizing the Wardrop equilibrium is of smaller dimension than the initial one. This is illustrated on an example in the smart grid context. (10.1109/TAC.2020.3008649)
  DOI : 10.1109/TAC.2020.3008649
• On Convergence-Diagnostic based Step Sizes for Stochastic Gradient Descent
  
  • Pesme Scott
  • Dieuleveut Aymeric
  • Flammarion Nicolas
  , 2020, 119, pp.119:7641-7651. Constant step-size Stochastic Gradient Descent exhibits two phases: a transient phase during which iterates make fast progress towards the optimum, followed by a stationary phase during which iterates oscillate around the optimal point. In this paper, we show that efficiently detecting this transition and appropriately decreasing the step size can lead to fast convergence rates. We analyse the classical statistical test proposed by Pflug (1983), based on the inner product between consecutive stochastic gradients. Even in the simple case where the objective function is quadratic we show that this test cannot lead to an adequate convergence diagnostic. We then propose a novel and simple statistical procedure that accurately detects stationarity and we provide experimental results showing state-of-the-art performance on synthetic and real-word datasets.
• An Eco-routing algorithm for HEVs under traffic conditions
  
  • Le Rhun Arthur
  • Bonnans Frédéric
  • de Nunzio Giovanni
  • Leroy Thomas
  • Martinon Pierre
  , 2020. An extension of the bi-level optimization for the energy management of hybrid electric vehicles (HEVs) proposed in Le Rhun et al. (2019a) to the eco-routing problem is presented. Using the knowledge of traffic conditions over the entire road network, we search both the optimal path and state of charge trajectory. This problem results in finding the shortest path on a weighted graph whose nodes are (position, state of charge) pairs for the vehicle, the edge cost being evaluated thanks to the cost maps from optimization at the 'micro' level of a bi-level decomposition. The error due to the discretization of the state of charge is proven to be linear if the cost maps are Lipschitz. The classical A * algorithm is used to solve the problem, with a heuristic based on a lower bound of the energy needed to complete the travel. The eco-routing method is validated by numerical simulations and compared to the fastest path on a synthetic road network.
• Fast and Consistent Learning of Hidden Markov Models by Incorporating Non-Consecutive Correlations
  
  • Mattila Robert
  • Rojas Cristian R
  • Moulines Eric
  • Krishnamurthy Vikram
  • Wahlberg Bo
  , 2020. Can the parameters of a hidden Markov model (HMM) be estimated from a single sweep through the observations-and additionally, without being trapped at a local optimum in the likelihood surface? That is the premise of recent method of moments algorithms devised for HMMs. In these, correlations between consecutive pair-or tripletwise observations are empirically estimated and used to compute estimates of the HMM parameters. Albeit computationally very attractive, the main drawback is that by restricting to only loworder correlations in the data, information is being neglected which results in a loss of accuracy (compared to standard maximum likelihood schemes). In this paper, we propose extending these methods (both pair-and triplet-based) by also including non-consecutive correlations in a way which does not significantly increase the computational cost (which scales linearly with the number of additional lags included). We prove strong consistency of the new methods, and demonstrate an improved performance in numerical experiments on both synthetic and real-world financial timeseries datasets.
• Kernel Regression for Vehicle Trajectory Reconstruction under Speed and Inter-vehicular Distance Constraints
  
  • Aubin-Frankowski Pierre-Cyril
  • Petit Nicolas
  • Szabó Zoltán
  , 2020. This work tackles the problem of reconstructing vehicle trajectories with the side information of physical constraints, such as inter-vehicular distance and speed limits. It is notoriously difficult to perform a regression while enforcing these hard constraints on time intervals. Using reproducing kernel Hilbert spaces, we propose a convex reformulation which can be directly implemented in classical solvers such as CVXGEN. Numerical experiments on a simple dataset illustrate the efficiency of the method, especially with sparse and noisy data.
• Coupled optimization of macroscopic structures and lattice infill
  
  • Geoffroy-Donders Perle
  • Allaire Grégoire
  • Michailidis Georgios
  • Pantz Olivier
  International Journal for Numerical Methods in Engineering, Wiley, 2020, 123 (13), pp.2963-2985. This paper is concerned with the coupled optimization of the external boundary of a structure and its infill made of some graded lattice material. The lattice material is made of a periodic cell, macroscopically modulated and oriented. The external boundary may be coated by a layer of pure material with a fixed prescribed thickness. The infill is optimized by the ho-mogenization method while the macroscopic shape is geometrically optimized by the Hadamard method of shape sensitivity. A first original feature of the proposed approach is that the infill material follows the displacement on the exterior boundary during the geometric optimization step. A second key feature is the dehomogenization or projection step which build a smoothly varying lattice infill from the optimal homogenized properties. Several numerical examples illustrate the effectiveness of our approach in 2-d, which is especially convenient when considering design-dependent loads. (10.1002/nme.6392)
  DOI : 10.1002/nme.6392
• Finite Time Analysis of Linear Two-timescale Stochastic Approximation with Markovian Noise
  
  • Kaledin Maxim
  • Moulines Eric
  • Naumov Alexey
  • Tadic Vladislav
  • Wai Hoi-To
  , 2020. Linear two-timescale stochastic approximation (SA) scheme is an important class of algorithms which has become popular in reinforcement learning (RL), particularly for the policy evaluation problem. Recently, a number of works have been devoted to establishing the finite time analysis of the scheme, especially under the Markovian (non-i.i.d.) noise settings that are ubiquitous in practice. In this paper, we provide a finite-time analysis for linear two timescale SA. Our bounds show that there is no discrepancy in the convergence rate between Markovian and martingale noise, only the constants are affected by the mixing time of the Markov chain. With an appropriate step size schedule, the transient term in the expected error bound is o(1/k c) and the steady-state term is O(1/k), where c > 1 and k is the iteration number. Furthermore, we present an asymptotic expansion of the expected error with a matching lower bound of Ω(1/k). A simple numerical experiment is presented to support our theory.
• On the Root solution to the Skorokhod embedding problem given full marginals
  
  • Richard Alexandre
  • Tan Xiaolu
  • Touzi Nizar
  SIAM Journal on Control and Optimization, Society for Industrial and Applied Mathematics, 2020, 58 (4), pp.1874–1892. This paper examines the Root solution of the Skorohod embedding problem given full marginals on some compact time interval. Our results are obtained by limiting arguments based on finitely-many marginals Root solution of Cox, Oblój, and Touzi. Our main result provides a characterization of the corresponding potential function by means of a convenient parabolic PDE. (10.1137/18M1222594)
  DOI : 10.1137/18M1222594
• A Kinetic derivation of diffuse interface fluid models
  
  • Giovangigli Vincent
  Physical Review E, American Physical Society (APS), 2020, 102 (012110). We present a full derivation of capillary fluid equations from the kinetic theory of dense gases. These equations involve van der Waals' gradient energy, Korteweg's tensor, Dunn and Serrin's heat flux as well as viscous and heat dissipative fluxes. Starting from macroscopic equations obtained from the kinetic theory of dense gases, we use a new second order expansion of the pair distribution function in order to derive the diffuse interface model. The capillary extra terms and the capillarity coefficient are then associated with intermolecular forces and the pair interaction potential.
• Some aspects of the central role of financial market microstructure : Volatility dynamics, optimal trading and market design
  
  • Jusselin Paul
  , 2020. This thesis is made of three parts. In the first one, we study the connections between the dynamics of the market at the microscopic and macroscopic scales, with a focus on the properties of the volatility. In the second part we deal with optimal control for point processes. Finally in the third part we study two questions of market design.We begin this thesis with studying the links between the no-arbitrage principle and the (ir)regularity of volatility. Using a microscopic to macroscopic approach, we show that we can connect those two notions through the market impact of metaorders. We model the market order flow using linear Hawkes processes and show that the no-arbitrage principle together with the existence of a non-trivial market impact imply that the volatility process has to be rough, more precisely a rough Heston model. Then we study a class of microscopic models where order flows are driven by quadratic Hawkes processes. The objective is to extend the rough Heston model building continuous models that reproduce the feedback of price trends on volatility: the so-called Zumbach effect. We show that using appropriate scaling procedures the microscopic models converge towards price dynamics where volatility is rough and that reproduce the Zumbach effect. Finally we use one of those models, the quadratic rough Heston model, to solve the longstanding problem of joint calibration of SPX and VIX options smiles.Motivated by the extensive use of point processes in the first part of our work we focus in the second part on stochastic control for point processes. Our aim is to provide theoretical guarantees for applications in finance. We begin with considering a general stochastic control problem driven by Hawkes processes. We prove the existence of a solution and more importantly provide a method to implement the optimal control in practice. Then we study the scaling limits of solutions to stochastic control problems in the framework of population modeling. More precisely we consider a sequence of models for the dynamics of a discrete population converging to a model with continuous population. For each model we consider a stochastic control problem. We prove that the sequence of optimal controls associated to the discrete models converges towards the optimal control associated to the continuous model. This result relies on the continuity of the solution to a backward stochastic differential equation with respect to the driving martingale and terminal value.In the last part we address two questions of market design. We are first interested in designing a liquid electronic market of derivatives. We focus on options and propose a two steps method that can be easily applied in practice. The first step is to select the listed options. For this we use a quantization algorithm enabling us to pick the options capturing most of market demand. The second step is to design a make-take fees policy for market makers to incentivize them to set attractive quotes. We formalize this issue as a principal agent problem that we explicitly solve. Finally we look for the optimal auction duration that should be used on a market organized in sequential auctions, the case of auctions with 0 second duration corresponding to the continuous double auctions situation. To do so, we use an agent based model where market takers are competing. We consider that the optimal auction duration is the one leading to the best quality of price formation process. After proving existence of a Nash equilibrium for the competition between market takers we apply our results on stocks market data. We find that for most of the stocks, the optimal auction duration lies between 2 and 10 minutes.
• Metropolized flow: from invertible flow to mcmc
  
  • Thin Achille
  • Kotelevskii Nikita
  • Oliviero-Durmus Alain
  • Moulines Éric
  • Panov Maxim
  , 2020.
• Intersection multiplicity of a sparse curve and a low-degree curve
  
  • Koiran Pascal
  • Skomra Mateusz
  Journal of Pure and Applied Algebra, Elsevier, 2020, 224 (7), pp.106279. Let $F(x, y)∈C[x, y]$ be a polynomial of degreed and let $G(x, y)∈C[x, y]$ be a polynomial with t monomials. We want to estimate the maximal multiplicity of a solution of the system $F(x, y) =G(x, y) = 0$. Our main result is that the multiplicity of any isolated solution $(a, b)∈C2$ with nonzero coordinates is no greater than $52d2t2$. We ask whether this intersection multiplicity can be polynomially boundedin the number of monomials of F and G, and we briefly review someconnections between sparse polynomials and algebraic complexity the-ory. (10.1016/j.jpaa.2019.106279)
  DOI : 10.1016/j.jpaa.2019.106279
• AN EXTENSION RESULT FOR GENERALISED SPECIAL FUNCTIONS OF BOUNDED DEFORMATION
  
  • Cagnetti Filippo
  • Chambolle Antonin
  • Perugini Matteo
  • Scardia Lucia
  Journal of Convex Analysis, Heldermann, 2020, 28 (2), pp.457--470. We show an extension result for functions in GSBD^p , for every p > 1 and any dimension n ≥ 2. The proof is based on a recent result in [9], where it is shown that a function u in GSBD^p with a "small" jump set coincides with a W^{1,p} function, up to a small set whose perimeter and volume are controlled by the size of the jump of u.
• Artemis: tight convergence guarantees for bidirectional compression in heterogeneous settings for federated learning
  
  • Philippenko Constantin
  • Dieuleveut Aymeric
  , 2020. We introduce a framework - Artemis - to tackle the problem of learning in a distributed or federated setting with communication constraints. Several workers (randomly sampled) perform the optimization process using a central server to aggregate their computations. To alleviate the communication cost, Artemis allows to compress the information sent in both directions (from the workers to the server and conversely) combined with a memory mechanism. It improves on existing algorithms that only consider unidirectional compression (to the server), or use very strong assumptions on the compression operator. We provide fast rates of convergence (linear up to a threshold) under weak assumptions on the stochastic gradients (noise's variance bounded only at optimal point) in non-i.i.d. setting, highlight the impact of memory for unidirectional and bidirectional compression, and analyze Polyak-Ruppert averaging. We use convergence in distribution to obtain a lower bound of the asymptotic variance that highlights practical limits of compression.
• Topology of tensor ranks
  
  • Comon Pierre
  • Lek-Heng Lim
  • Qi Yang
  • Ye Ke
  Advances in Mathematics, Elsevier, 2020, 367 (June), pp.107128. We study path-connectedness and homotopy groups of sets of tensors defined by tensor rank, border rank, multilinear rank, as well as their symmetric counterparts for symmetric tensors. We show that over C, the set of rank-r tensors and the set of symmetric rank-r symmetric tensors are both path-connected if r is not more than the complex generic rank; these results also extend to border rank and symmetric border rank over C. Over R, the set of rank-r tensors is path-connected if it has the expected dimension but the corresponding result for symmetric rank-r symmetric d- tensors depends on the order d: connected when d is odd but not when d is even. Border rank and symmetric border rank over R have essentially the same path-connectedness properties as rank and symmetric rank over R. When r is greater than the complex generic rank, we are unable to discern any general pattern. Beyond path-connectedness, we determine, over both R and C, the fundamental and higher homotopy groups of the set of tensors of a fixed small rank, and, taking advantage of Bott periodicity, those of the manifold of tensors of a fixed multilinear rank. We also obtain analogues of these results for symmetric tensors of a fixed symmetric rank or a fixed symmetric multilinear rank. (10.1016/j.aim.2020.107128)
  DOI : 10.1016/j.aim.2020.107128
• Formalizing the Face Lattice of Polyhedra
  
  • Allamigeon Xavier
  • Katz Ricardo
  • Strub Pierre-Yves
  , 2020, pp.185-203. (10.1007/978-3-030-51054-1_11)
  DOI : 10.1007/978-3-030-51054-1_11
• A Diffuse Interface Approach For Disperse Two-Phase Flows Involving Dual-Scale Kinematics Of Droplet Deformation Based On Geometrical Variables
  
  • Cordesse Pierre
  • Di Battista Ruben
  • Chevalier Quentin
  • Matuszewski Lionel
  • Ménard Thibaut
  • Kokh Samuel
  • Massot Marc
  ESAIM: Proceedings and Surveys, EDP Sciences, 2020, 69, pp.24-46. The purpose of this contribution is to derive a reduced-order two-phase flow model including interface subscale modeling through geometrical variables based on Stationary Action Principle (SAP) and Second Principle of Thermodynamics in the spirit of [1, 2]. The derivation is conducted in the disperse phase regime for the sake of clarity but the resulting paradigm can be used in a more general framework. One key issue is the definition of the proper potential and kinetic energies in the Lagrangian of the system based on geometrical variables (Interface area density, mean and Gauss curvatures...), which will drive the subscale kinematics and dissipation, and their coupling with large scales of the flow. While [1] relied on bubble pulsation, that is normal deformation of the interface with shape preservation related to pressure changes, we aim here at tackling inclusion deformation at constant volume, thus describing self-sustained oscillations. In order to identify the proper energies, we use Direct Numerical Simulations (DNS) of oscillating droplets using ARCHER code and recently developed library, Mercur(v)e, for mean geometrical variable evaluation and analysis preserving topological invariants. This study is combined with historical analytical studies conducted in the small perturbation regime and shows that the proper potential energy is related to the surface difference compared to the spherical minimal surface. A geometrical quasi-invariant is also identified and a natural definition of subscale momentum is proposed. The set of Partial Differential Equations (PDEs) including the conservation equations as well as dissipation source terms are eventually derived leading to an original two-scale diffuse interface model involving geometrical variables. (10.1051/proc/202069024)
  DOI : 10.1051/proc/202069024
• Contribution to the study of combustion instabilities in cryotechnic rocket engines : coupling diffuse interface models with kinetic-based moment methods for primary atomization simulations
  
  • Cordesse Pierre
  , 2020. Gatekeepers to the open space, launchers are subject to intense and competitive enhancements, through experimental and numerical test campaigns. Predictive numerical simulations have become mandatory to increase our understanding of the physics. Adjustable, they provide early-stage optimization processes, in particular of the combustion chamber, to guaranty safety and maximize efficiency. One of the major physical phenomenon involved in the combustion of the fuel and oxidizer is the jet atomization, which pilotes both the droplet distributions and the potential high-frequency instabilities in subcritical conditions. It encompasses a large sprectrum of two-phase flow topologies, from separated phases to disperse phase, with a mixed region where the small scale physics and topology of the flow are very complex. Reduced-order models are good candidates to perform predictive but low CPU demanding simulations on industrial configurations but have only been able so far to capture large scale dynamics and have to be coupled to disperse phase models through adjustable and weakly reliable parameters in order to predict spray formation. Improving the hierarchy of reduced order models in order to better describe both the mixed region and the disperse region requires a series of building blocks at the heart of the present work and give on to complex problems in the mathematical analysis and physical modelling of these systems of PDE as well as their numerical discretization and implementation in CFD codes for industrial uses. Thanks to the extension of the theory on supplementary conservative equations to system of non-conservation laws and the formalism of the multi-fluid thermodynamics accounting for non-ideal effects, we give some new leads to define a strictly convex mixture entropy consistent with the system of equations and the pressure laws, which would allow to recover the entropic symmetrization of two-phase flow models, prove their hyperbolicity and obtain generalized source terms. Furthermore, we have departed from a geometric approach of the interface and proposed a multi-scale rendering of the interface to describe multi-fluid flow with complex interface dynamics. The Stationary Action Principle has returned a single velocity two-phase flow model coupling large and small scales of the flow. We then have developed a splitting strategy based on a Finite Volume discretization and have implemented the new model in the industrial CFD software CEDRE of ONERA to proceed to a numerical verification. Finally, we have constituted and investigated a first building block of a hierarchy of test-cases designed to be amenable to DNS while close enough to industrial configurations in order to assess the simulation results of the new model but also to any up-coming models.
• Random trees, laminations of the disk and factorizations
  
  • Thevenin Paul
  , 2020. This work is devoted to the study of asymptotic properties of large random combinatorial structures. Three particular structures are the main objects of our interest: trees, factorizations of permutations and configurations of noncrossing chords in the unit disk (or laminations).First, we are specifically interested in the number of vertices with fixed degree in Galton-Watson trees that are conditioned in different ways, for example by their number of vertices with even degree, or by their number of leaves. When the offspring distribution of the tree is critical and in the domain of attraction of a stable law, we notably prove the asymptotic normality of these quantities. We are also interested in the spread of these vertices with fixed degree in the tree, when one explores it from left to right.Then, we consider configurations of chords that do not cross in the unit disk. Such configurations notably code trees in a natural way. We define in particular a nondecreasing sequence of laminations coding a fragmentation of a given tree, that is, a way of cutting this tree at points chosen randomly. This geometric point of view then allows us to study some properties of a factorization of the cycle (1, 2,⋯, n) as a product of n-1 transpositions, chosen uniformly at random, by coding it in the disk by a random lamination and by remarking a connection between this model and a Galton-Watson tree conditioned by its total number of vertices. Finally, we present a generalization of these results to random factorizations of the same cycle, that are not necessarily as a product of transpositions anymore, but may involve cycles of larger lengths. We highlight this way a connection between conditioned Galton-Watson trees, factorizations of large permutations and the theory of fragmentations.