Publications

2018

• Normalizing constants of log-concave densities
  
  • Brosse Nicolas
  • Durmus Alain
  • Moulines Éric
  Electronic Journal of Statistics, Shaker Heights, OH : Institute of Mathematical Statistics, 2018, 12 (1), pp.851-889. We derive explicit bounds for the computation of normalizing constants Z for log-concave densities $\pi= \mathrm{e}^{−U} /Z$ w.r.t. the Lebesgue measure on $\mathbb{R}^d$. Our approach relies on a Gaussian annealing combined with recent and precise bounds on the Unadjusted Langevin Algorithm (Durmus, A. and Moulines, E. (2016). High-dimensional Bayesian inference via the Unadjusted Langevin Algorithm). Polynomial bounds in the dimension $d$ are obtained with an exponent that depends on the assumptions made on $U$. The algorithm also provides a theoretically grounded choice of the annealing sequence of variances. A numerical experiment supports our findings. Results of independent interest on the mean squared error of the empirical average of locally Lipschitz functions are established. (10.1214/18-EJS1411)
  DOI : 10.1214/18-EJS1411
• Une nouvelle représentation de la polysomnographie par technique de machine learning non supervisée
  
  • Solelhac G.
  • Brigham M.
  • Marini C.
  • Bouchequet P.
  • Chennaoui M.
  • Le Pennec E.
  • Leger D.
  Médecine du sommeil, Elsevier Masson, 2018, 15 (1), pp.49. (10.1016/j.msom.2018.01.132)
  DOI : 10.1016/j.msom.2018.01.132
• Density analysis of non-Markovian BSDEs and applications to biology and finance
  
  • Mastrolia Thibaut
  Stochastic Processes and their Applications, Elsevier, 2018, 128 (3), pp.897-938. (10.1016/j.spa.2017.06.009)
  DOI : 10.1016/j.spa.2017.06.009
• Perfect transmission invisibility for waveguides with sound hard walls
  
  • Bonnet-Ben Dhia Anne-Sophie
  • Chesnel Lucas
  • Nazarov Sergei
  Journal de Mathématiques Pures et Appliquées, Elsevier, 2018. We are interested in a time harmonic acoustic problem in a waveguide with locally perturbed sound hard walls. We consider a setting where an observer generates incident plane waves at −∞ and probes the resulting scattered field at −∞ and +∞. Practically, this is equivalent to measure the reflection and transmission coefficients respectively denoted R and T. In [9], a technique has been proposed to construct waveguides with smooth walls such that R = 0 and |T | = 1 (non reflection). However the approach fails to ensure T = 1 (perfect transmission without phase shift). In this work, first we establish a result explaining this observation. More precisely, we prove that for wavenumbers smaller than a given bound k depending on the geometry, we cannot have T = 1 so that the observer can detect the presence of the defect if he/she is able to measure the phase at +∞. In particular, if the perturbation is smooth and small (in amplitude and in width), k is very close to the threshold wavenumber. Then, in a second step, we change the point of view and, for a given wavenumber, working with singular perturbations of the domain, we show how to obtain T = 1. In this case, the scattered field is exponentially decaying both at −∞ and +∞. We implement numerically the method to provide examples of such undetectable defects.
• Cache Miss Estimation for Non-Stationary Request Processes
  
  • Olmos Felipe
  • Graham Carl
  • Simonian Alain
  Stochastic Systems, INFORMS Applied Probability Society, 2018, 8 (1), pp.75-90. The goal of the paper is to evaluate the miss probability of a Least Recently Used (LRU) cache, when it is offered a non-stationary request process given by a Poisson cluster point process. First, we construct a probability space using Palm theory, describing how to consider a tagged document with respect to the rest of the request process. This framework allows us to derive a fundamental integral formula for the expected number of misses of the tagged document. Then, we consider the limit when the cache size and the arrival rate go to infinity in proportion, and use the integral formula to derive an asymptotic expansion of the miss probability in powers of the inverse of the cache size. This enables us to quantify and improve the accuracy of the so-called Che approximation. (10.1287/stsy.2017.0009)
  DOI : 10.1287/stsy.2017.0009
• Kinetic theory of two-temperature polyatomic plasmas
  
  • Orlac’h Jean-Maxime
  • Giovangigli Vincent
  • Novikova Tatiana
  • Roca I Cabarrocas Pere
  Physica A: Statistical Mechanics and its Applications, Elsevier, 2018, 494, pp.503-546. (10.1016/j.physa.2017.11.151)
  DOI : 10.1016/j.physa.2017.11.151
• Ordonnancement sous contraintes d’énergie avec stockage et couts linéaires par morceaux
  
  • Absi Nabil
  • Artigues Christian
  • Kedad-Sidhoum Safia
  • Ngueveu Sandra Ulrich
  • Rannou Janik
  • Saadi Omar
  , 2018.
• Market impact in a latent order book
  
  • Lemhadri Ismael
  , 2018. We revisit the classical problem of market impact through the lens of a new agent-based model. Drawing from the mean-field approach in Statistical Mechanics and Physics, we assume a large number of 'agents' interacting in the order book. By taking the 'continuum' limit we obtain a set of nonlinear differential equations, the core of our dynamical theory of price formation. And we explicitly solve them using Fourier analysis. One could talk as well of a "micro-macro" approach of equilibrium, where the market price is the consequence of each ("microscopic") agent behaving with respect to his preferences and to global ("macroscopic") information. When a large market order (or "metaorder") perturbs the market, our model recovers the square-root law of impact, providing new insights on the price formation process. In addition, we give various limiting cases, examples and possible extensions.
• Surrogate-Assisted Bounding-Box Approach for Optimization Problems with Tunable Objectives Fidelity
  
  • Rivier Mickael
  • Congedo Pietro Marco
  , 2018, pp.1-32. In this work, we present a novel framework to perform multi-objective optimization when considering expensive objective functions computed with tunable fidelity. This case is typical in many engineering optimization problems, for example with simulators relying on Monte Carlo or on iterative solvers. The objectives can only be estimated, with an accuracy depending on the computational resources allocated by the user. We propose here a heuristic for allocating the resources efficiently to recover an accurate Pareto front at low computational cost. The approach is independent from the choice of the optimizer and overall very flexible for the user. The framework is based on the concept of Bounding-Box, where the estimation error can be regarded with the abstraction of an interval (in one-dimensional problems) or a product of intervals (in multi-dimensional problems) around the estimated value, naturally allowing the computation of an approximated Pareto front. This approach is then supplemented by the construction of a surrogate model on the estimated objective values. We first study the convergence of the approximated Pareto Front toward the true continuous one under some hypotheses. Secondly, a numerical algorithm is proposed and tested on several numerical test-cases.
• An example of non-uniqueness for Radon transforms with continuous positive rotation invariant weights
  
  • Goncharov Fedor O
  • Novikov Roman G
  The Journal of Geometric Analysis, Springer, 2018, 28 (4), pp.3807-3828. We consider weighted Radon transforms $R_W$ along hyperplanes in $R^3$ with strictly positive weights $W$. We construct an example of such a transform with non-trivial kernel $\mathrm{Ker}R_W$ in the space of infinitely smooth compactly supported functions and with continuous weight. Moreover, in this example the weight $W$ is rotation invariant. In particular, by this result we continue studies of Quinto (1983), Markoe, Quinto (1985), Boman (1993) and Goncharov, Novikov (2017). We also extend our example to the case of weighted Radon transforms along two-dimensional planes in $R^d$ , $d \geq 3$. (10.1007/s12220-018-0001-y)
  DOI : 10.1007/s12220-018-0001-y
• The boundary of random planar maps via looptrees
  
  • Kortchemski Igor
  • Richier Loïc
  , 2018. We study the scaling limits of the boundary of Boltzmann planar maps conditioned on having a large perimeter. We first deal with the non-generic critical regime, where the degree of a typical face falls within the domain of attraction of a stable law with parameter α∈(1,2). In the so-called dense phase α∈(1,3/2), it was established by Richier that the scaling limit of the boundary is a stable looptree. In this work, we complete the picture by proving that in the dilute phase α∈(3/2,2) (as well as in the generic critical regime), the scaling limit is a multiple of the unit circle. This establishes the first evidence of a phase transition for the topology of the boundary: in the dense phase, large faces are self-intersecting while in the dilute phase, they are self-avoiding. The subcritical regime is also investigated. In this case, we show that the scaling limit of the boundary is a multiple of the Brownian CRT instead. The strategy consists in studying scaling limits of looptrees associated with specific Bienaymé--Galton--Watson trees. In the first case, it relies on an invariance principle for random walks with negative drift, which is of independent interest. In the second case, we obtain the more general result that the Brownian CRT is the scaling limit of looptrees associated with BGW trees whose offspring distribution is critical and in the domain of attraction of a Gaussian distribution, confirming thereby a prediction of Curien and Kortchemski.
• Dispersion analysis of triangle-based Whitney element methods for electromagnetic wave propagation
  
  • Bonazzoli Marcella
  • Rapetti Francesca
  • Venturini Chiara
  Applied Mathematics and Computation, Elsevier, 2018, 319, pp.274-286. We study the numerical dispersion/dissipation of a triangle-based edge Finite Element Method (edgeFEM) of degree r ≥ 1 when coupled with the Leap-Frog (LF) finite difference scheme to simulate the electromagnetic wave propagation over a structured triangulation of the 2D physical domain. The analysis addresses the discrete eigenvalue problem resulting from the approximation of the dispersion relation. First, we present semi-discrete dispersion graphs by varying the approximation degree r and the number of discrete points per wavelength. The fully-discrete ones are then obtained by varying also the time step. Numerical results for the edgeFEM, resp. edgeFEM-LF, are compared with those for the node Finite Element Method (nodeFEM), resp. nodeFEM-LF, applied to the considered problem. (10.1016/j.amc.2017.03.026)
  DOI : 10.1016/j.amc.2017.03.026
• Relaxation Limit and Initial-Layer for a Class of Hyperbolic-Parabolic Systems
  
  • Giovangigli Vincent
  • Yang Zaibao
  • Yong Wen-An
  , 2018. We consider a class of hyperbolic-parabolic systems with small diffusion terms and stiff sources. Existence of solutions to the Cauchy problem with ill prepared initial data is established by using composite expansions including initial-layer correctors and a convergence-stability lemma. New multitime expansions are introduced and lead to second-order error estimates between the composite expansions and the solution. Reduced equilibrium systems of second-order accuracy are also investigated as well as initial-layers of Chapman-Enskog expansions.
• Contribution à la modélisation eulérienne unifiée de l’injection : de la zone dense au spray polydispersé
  
  • Essadki Mohamed
  , 2018. L’injection directe à haute pression du carburant dans les moteurs à combustion interne permet une atomisation compacte et efficace. Dans ce contexte, la simulation numérique de l’injection est devenue un outil fondamental pour la conception industrielle. Cependant,l’écoulement du carburant liquide dans une chambre occupée initialement par l’air est un écoulement diphasique très complexe ; elle implique une très large gamme d’échelles. L’objectif de cette thèse est d’apporter de nouveaux éléments de modélisation et de simulation afin d’envisager une simulation prédictive de ce type d’écoulement avec un coût de calcul abordable dans un contexte industriel. En effet, au vu du coût de calcul prohibitif de la simulation directe de l’ensemble des échelles spatiales et temporelles, nous devons concevoir une gamme de modèles d’ordre réduit prédictifs. En outre, des méthodes numériques robustes, précises et adaptées au calcul de haute performance sont primordiales pour des simulations complexes.Cette thèse est dédiée au développement d’un modèle d’ordre réduit Eulérien capable de capter tant la polydispersiond’un brouillard de goutte dans la zone dispersée,que la dynamique de l’interface dans le régime de phases séparées. En s’appuyant sur une extension des méthodes de moments d’ordre élevé à des moments fractionnaires qui représentent des quantités géométriques de l’interface, et sur l’utilisation de variables géométrique sen sous-échelle dans la zone où l’interface gaz-liquide ne peut plus être complètement résolue, nous proposons une approche unifiée où un ensemble de variables géométriques sont transportées et valides dans les deux régimes d’écoulement [...].
• Learning in nonatomic anonymous games with applications to first-order mean field games
  
  • Hadikhanloo Saeed
  , 2018. We introduce a model of anonymous games with the player dependent action sets. We propose several learning procedures based on the well-known Fictitious Play and Online Mirror Descent and prove their convergence to equilibrium under the classical monotonicity condition. Typical examples are first-order mean field games.
• The Root solution to the multi-marginal embedding problem: an optimal stopping and time-reversal approach
  
  • Cox Alexander
  • Obłój Jan
  • Touzi Nizar
  Probability Theory and Related Fields, Springer Verlag, 2018, 173 (1-2), pp.211-259. (10.1007/s00440-018-0833-1)
  DOI : 10.1007/s00440-018-0833-1
• Moral Hazard Under Ambiguity
  
  • Mastrolia Thibaut
  • Possamaï Dylan
  Journal of Optimization Theory and Applications, Springer Verlag, 2018, 179 (2), pp.452-500. (10.1007/s10957-018-1230-8)
  DOI : 10.1007/s10957-018-1230-8
• Duality of random planar maps via percolation
  
  • Curien Nicolas
  • Richier Loïc
  , 2018. We discuss duality properties of critical Boltzmann planar maps such that the degree of a typical face is in the domain of attraction of a stable distribution with parameter $\alpha\in(1,2]$. We consider the critical Bernoulli bond percolation model on a Boltzmann map in the dilute and generic regimes $\alpha \in (3/2,2]$, and show that the open percolation cluster of the origin is itself a Boltzmann map in the dense regime $\alpha \in (1,3/2)$, with parameter \[\alpha':= \frac{2\alpha+3}{4\alpha-2}.\] This is the counterpart in random planar maps of the duality property $\kappa \leftrightarrow 16/\kappa$ of Schramm--Loewner Evolutions and Conformal Loop Ensembles, recently established by Miller, Sheffield and Werner. As a byproduct, we identify the scaling limit of the boundary of the percolation cluster conditioned to have a large perimeter. The cases of subcritical and supercritical percolation are also discussed. In particular, we establish the sharpness of the phase transition through the tail distribution of the size of the percolation cluster.
• Minimax representation of nonexpansive functions and application to zero-sum recursive games
  
  • Akian Marianne
  • Gaubert Stéphane
  • Hochart Antoine
  Journal of Convex Analysis, Heldermann, 2018, 25 (1). We show that a real-valued function on a topological vector space is positively homogeneous of degree one and nonexpansive with respect to a weak Minkowski norm if and only if it can be written as a minimax of linear forms that are nonexpansive with respect to the same norm. We derive a representation of monotone, additively and positively homogeneous functions on L∞ spaces and on Rn, which extend results of Kolokoltsov, Rubinov, Singer, and others. We apply this representation to nonconvex risk measures and to zero-sum games. We derive in particular results of representation and polyhedral approximation for the class of Shapley operators arising from games without instantaneous payments (Everett's recursive games).
• Optimal control of branching diffusion processes: A finite horizon problem
  
  • Claisse Julien
  The Annals of Applied Probability, Institute of Mathematical Statistics (IMS), 2018, 28 (1). (10.1214/17-AAP1290)
  DOI : 10.1214/17-AAP1290
• Nonequilibrium Precondensation of Classical Waves in Two Dimensions Propagating through Atomic Vapors
  
  • Šantić Neven
  • Fusaro Adrien
  • Salem Sabeur
  • Garnier Josselin
  • Picozzi Antonio
  • Kaiser Robin
  Physical Review Letters, American Physical Society, 2018, 120 (5), pp.055301. The nonlinear Schrödinger equation, used to describe the dynamics of quantum fluids, is known to be valid not only for massive particles but also for the propagation of light in a nonlinear medium, predicting condensation of classical waves. Here we report on the initial evolution of random waves with Gaussian statistics using atomic vapors as an efficient two dimensional nonlinear medium. Experimental and theoretical analysis of near field images reveal a phenomenon of nonequilibrium precondensation, characterized by a fast relaxation towards a precondensate fraction of up to 75%. Such precondensation is in contrast to complete thermalization to the Rayleigh-Jeans equilibrium distribution, requiring prohibitive long interaction lengths. (10.1103/PhysRevLett.120.055301)
  DOI : 10.1103/PhysRevLett.120.055301
• PAC-Bayesian aggregation of affine estimators
  
  • Montuelle Lucie
  • Le Pennec Erwan
  , 2018. Aggregating estimators using exponential weights depending on their risk appears optimal in expectation but not in probability. We use here a slight overpenalization to obtain oracle inequality in probability for such an explicit aggregation procedure. We focus on the fixed design regression framework and the aggregation of affine estimators and obtain results for a large family of affine estimators under a non necessarily independent sub-Gaussian noise assumptions.
• On the controllability of the quantum dynamics of closed and open systems
  
  • Pinna Lorenzo
  , 2018. We investigate the controllability of quantum systems in two differentsettings: the standard 'closed' setting, in which a quantum system is seen as isolated, the control problem is formulated on the Schroedinger equation; the open setting that describes a quantum system in interaction with a larger one, of which just qualitative parameters are known, by means of the Lindblad equation on states.In the context of closed systems we focus our attention to an interesting class ofmodels, namely the spin-boson models. The latter describe the interaction between a 2-level quantum system and finitely many distinguished modes of a bosonic field. We discuss two prototypical examples, the Rabi model and the Jaynes-Cummings model, which despite their age are still very popular in several fields of quantum physics. Notably, in the context of cavity Quantum Electro Dynamics (C-QED) they provide an approximate yet accurate description of the dynamics of a 2-level atom in a resonant microwave cavity, as in recent experiments of S. Haroche. We investigate the controllability properties of these models, analyzing two different types of control operators acting on the bosonic part, corresponding -in the application to cavity QED- to an external electric and magnetic field, respectively. We review some recent results and prove the approximate controllability of the Jaynes-Cummings model with these controls. This result is based on a spectral analysis exploiting the non-resonances of the spectrum. As far as the relation between the Rabi andthe Jaynes-Cummings Hamiltonians concerns, we treat the so called rotating waveapproximation in a rigorous framework. We formulate the problem as an adiabaticlimit in which the detuning frequency and the interaction strength parameter goes to zero, known as the weak-coupling regime. We prove that, under certain hypothesis on the ratio between the detuning and the coupling, the Jaynes-Cumming and the Rabi dynamics exhibit the same behaviour, more precisely the evolution operators they generate are close in norm.In the framework of open quantum systems we investigate the controllability ofthe Lindblad equation. We consider a control acting adiabatically on the internal part of the system, which we see as a degree of freedom that can be used to contrast the action of the environment. The adiabatic action of the control is chosen to produce a robust transition. We prove, in the prototype case of a two-level system, that the system approach a set of equilibrium points determined by the environment, i.e. the parameters that specify the Lindblad operator. On that set the system can be adiabatically steered choosing a suitable control. The analysis is based on the application of geometrical singular perturbation methods.
• Branching diffusion representation for nonlinear Cauchy problems and Monte Carlo approximation
  
  • Henry-Labordere Pierre
  • Touzi Nizar
  , 2018. We provide a probabilistic representations of the solution of some semilinear hyperbolic and high-order PDEs based on branching diffusions. These representations pave the way for a Monte-Carlo approximation of the solution, thus bypassing the curse of dimensionality. We illustrate the numerical implications in the context of some popular PDEs in physics such as nonlinear Klein-Gordon equation, a simplied scalar version of the Yang-Mills equation, a fourth-order nonlinear beam equation and the Gross-Pitaevskii PDE as an example of nonlinear Schrodinger equations.
• Simulation of (nested/extreme) risks in finance: regression Monte-Carlo, MCMC, stochastic algorithms
  
  • Gobet Emmanuel
  , 2018.