Publications

2018

• Minimization of the eigenvalues of the Dirichlet-LapIacian with a diameter constraint
  
  • Bogosel B
  • Henrot A
  • Lucardesi I
  SIAM Journal on Mathematical Analysis, Society for Industrial and Applied Mathematics, 2018, 50 (5), pp.5337-5361. In this paper we look for the domains minimizing the h-th eigenvalue of the Dirichlet-Laplacian λ h with a constraint on the diameter. Existence of an optimal domain is easily obtained, and is attained at a constant width body. In the case of a simple eigenvalue, we provide non standard (i.e., non local) optimality conditions. Then we address the question whether or not the disk is an optimal domain in the plane, and we give the precise list of the 17 eigenvalues for which the disk is a local minimum. We conclude by some numerical simulations showing the 20 first optimal domains in the plane. (10.1137/17M1162147)
  DOI : 10.1137/17M1162147
• Global acoustic daylight imaging in a stratified Earth-like model
  
  • Garnier Josselin
  • de Hoop Maarten V.
  • Sølna Knut
  Inverse Problems, IOP Publishing, 2018, 34 (1). (10.1088/1361-6420/aa9ad7)
  DOI : 10.1088/1361-6420/aa9ad7
• A NON-INTRUSIVE STRATIFIED RESAMPLER FOR REGRESSION MONTE CARLO: APPLICATION TO SOLVING NON-LINEAR EQUATIONS
  
  • Gobet Emmanuel
  • Liu Gang
  • Zubelli Jorge
  SIAM Journal on Numerical Analysis, Society for Industrial and Applied Mathematics, 2018, 56 (1), pp.50-77. Our goal is to solve certain dynamic programming equations associated to a given Markov chain X, using a regression-based Monte Carlo algorithm. More specifically, we assume that the model for X is not known in full detail and only a root sample X1, . . . , XM of such process is available. By a stratification of the space and a suitable choice of a probability measure ν, we design a new resampling scheme that allows to compute local regressions (on basis functions) in each stratum. The combination of the stratification and the resampling allows to compute the solution to the dynamic programming equation (possibly in large dimensions) using only a relatively small set of root paths. To assess the accuracy of the algorithm, we establish non-asymptotic error estimates in L2(ν). Our numerical experiments illustrate the good performance, even with M = 20 − 40 root paths. (10.1137/16M1066865)
  DOI : 10.1137/16M1066865
• Optimization of classification and regression analysis of four monoclonal antibodies from Raman spectra using collaborative machine learning approach
  
  • Le Laetitia Minh Maï
  • Kégl Balázs
  • Gramfort Alexandre
  • Marini Camille
  • Nguyen David
  • Cherti Mehdi
  • Tfaili Sana
  • Tfayli Ali
  • Baillet-Guffroy Arlette
  • Prognon Patrice
  • Chaminade Pierre
  • Caudron Eric
  Talanta, Elsevier, 2018, 184, pp.260-265. The use of monoclonal antibodies (mAbs) constitutes one of the most important strategies to treat patients suffering from cancers such as hematological malignancies and solid tumors. These antibodies are prescribed by the physician and prepared by hospital pharmacists. An analytical control enables the quality of the preparations to be ensured. The aim of this study was to explore the development of a rapid analytical method for quality control. The method used four mAbs (Infliximab, Bevacizumab, Rituximab and Ramucirumab) at various concentrations and was based on recording Raman data and coupling them to a traditional chemometric and machine learning approach for data analysis. Compared to conventional linear approach, prediction errors are reduced with a data-driven approach using statistical machine learning methods. In the latter, preprocessing and predictive models are jointly optimized. An additional original aspect of the work involved on submitting the problem to a collaborative data challenge platform called Rapid Analytics and Model Prototyping (RAMP). This allowed using solutions from about 300 data scientists in collaborative work. Using machine learning, the prediction of the four mAbs samples was considerably improved. The best predictive model showed a combined error of 2.4% versus 14.6% using linear approach. The concentration and classification errors were 5.8% and 0.7%, only three spectra were misclassified over the 429 spectra of the test set. This large improvement obtained with machine learning techniques was uniform for all molecules but maximal for Bevacizumab with an 88.3% reduction on combined errors (2.1% versus 17.9%). (10.1016/j.talanta.2018.02.109)
  DOI : 10.1016/j.talanta.2018.02.109
• Parking 3-sphere swimmer I. Energy minimizing strokes
  
  • Alouges François
  • Di Fratta Giovanni
  Discrete and Continuous Dynamical Systems - Series B, American Institute of Mathematical Sciences, 2018, 23 (4), pp.1797-1817. The paper is about the parking 3-sphere swimmer (sPr$_3$), a low-Reynolds number model swimmer composed of three balls of equal radii. The three balls can move along three horizontal axes (supported in the same plane) that mutually meet at the center of sPr$_3$ with angles of 120°. The governing dynamical system is introduced and the implications of its geometric symmetries revealed. It is then shown that, in the first order range of small strokes, optimal periodic strokes are ellipses embedded in 3d space, i.e., closed curves of the form t ∈ [0,2π ] → (cos t)${u}$ +(sin t)${v}$ for suitable vectors ${u}$ and ${v}$ of $\mathbb{R}^3$. A simple analytic expression for the vectors ${u}$ and ${v}$ is derived (10.3934/dcdsb.2018085)
  DOI : 10.3934/dcdsb.2018085
• Adiabatic ensemble control of a continuum of quantum systems
  
  • Augier Nicolas
  • Boscain Ugo
  • Sigalotti Mario
  SIAM Journal on Control and Optimization, Society for Industrial and Applied Mathematics, 2018, 56, pp.4045-4068. In this article we discuss how to control a parameter-dependent family of quantum systems. Our technique is based on adiabatic approximation theory and on the presence of curves of conical eigenvalue intersections of the controlled Hamiltonian. As particular cases, we recover chirped pulses for two-level quantum systems and counter-intuitive solutions for three-level stimulated Raman adiabatic passage (STIRAP). The proposed technique works for systems evolving both in finite-dimensional and infinite-dimensional Hilbert spaces. We show that the assumptions guaranteeing ensemble controllability are structurally stable with respect to perturbations of the parametrized family of systems.
• Adaptive estimation in the nonparametric random coefficients binary choice model by needlet thresholding
  
  • Gautier Eric
  • Le Pennec Erwan
  Electronic Journal of Statistics, Shaker Heights, OH : Institute of Mathematical Statistics, 2018, 12 (1), pp.277-320. (10.1214/17-EJS1383)
  DOI : 10.1214/17-EJS1383
• High order moment model for polydisperse evaporating sprays towards interfacial geometry
  
  • Essadki Mohamed
  • de Chaisemartin Stephane
  • Laurent Frédérique
  • Massot Marc
  SIAM Journal on Applied Mathematics, Society for Industrial and Applied Mathematics, 2018, 78 (4), pp.2003-2027. In this paper we propose a new Eulerian modeling and related accurate and robust numerical methods, describing polydisperse evaporating sprays, based on high order moment methods in size. The main novelty of this model is its capacity to describe some geometrical variables of the droplet-gas interface, by analogy with the liquid-gas interface in interfacial flows. For this purpose, we use fractional size-moments, where the size variable is taken as the droplet surface. In order to evaluate the evaporation of the polydisperse spray, we use a smooth reconstruction which maximizes the Shannon entropy. However, the use of fractional moments introduces some theoretical and numerical difficulties, which need to be tackled. First, relying on a study of the moment space, we extend the Maximum Entropy (ME) reconstruction of the size distribution to the case of fractional moments. Then, we propose a new accurate and realizable algorithm to solve the moment evolution due to evaporation, which preserves the structure of the moment space. This algorithm is based on a mathematical analysis of the kinetic evolution due to evaporation, where it shown that the evolution of some negative order fractional moments have to be properly predicted, a peculiarity related to the use of fractional moments. The present model and numerical schemes yield an accurate and stable evaluation of the moment dynamics with minimal number of variables, as well as a minimal computational cost as with the EMSM model, but with the very interesting additional capacity of coupling with diffuse interface model and transport equation of averaged geometrical interface variables, which are essential in oder to describe atomization. (10.1137/16M1108364)
  DOI : 10.1137/16M1108364
• Transport et diffusion
  
  • Allaire Grégoire
  • Blanc Xavier
  • Després Bruno
  • Golse François
  , 2018, pp.328. Ce livre est issu d'un cours enseigné par les auteurs en troisième année de l'Ecole Polytechnique, ce qui correspond à un niveau de première année de Master. Le sujet en est l'étude mathématique et numérique de modèles d'équations aux dérivées partielles, dits de transport et diffusion. Ces équations modélisent l'évolution d'une densité de particules ou d'individus en interaction avec leur milieu. L'origine de ces modèles est très variée. Ils proviennent classiquement de la physique et servent à décrire des particules neutres comme les neutrons et les photons. Dans le premier cas on parle aussi de neutronique alors que le dernier cas est appelé transfert radiatif. La biologie fait aussi appel aux équations de transport pour modéliser la dynamique de populations structurées. Citons aussi pour mémoire la dynamique des gaz raréfiés, le transport d'électrons dans les semi-conducteurs ou encore la physique des plasmas qui sont des phénomènes modélisés par des équations où l'opérateur de transport est une brique de base essentielle.
• The derivation of homogenized diffusion kurtosis models for diffusion MRI
  
  • Haddar Houssem
  • Kchaou Marwa
  • Moakher Maher
  Journal of Magnetic Resonance, Elsevier, 2018, 298, pp.48-57.
• Quadratic backward stochastic differential equations driven by $G$-Brownian motion: discrete solutions and approximation
  
  • Hu Ying
  • Lin Yiqing
  • Soumana-Hima Abdoulaye
  Stochastic Processes and their Applications, Elsevier, 2018, 128 (11), pp.3724-3750. In this paper, we consider backward stochastic differential equations driven by $G$-Brownian motion (GBSDEs) under quadratic assumptions on coefficients. We prove the existence and uniqueness of solution for such equations. On the one hand, a priori estimates are obtained by applying the Girsanov type theorem in the $G$-framework, from which we deduce the uniqueness. On the other hand, to prove the existence of solutions, we first construct solutions for discrete GBSDEs by solving corresponding fully nonlinear PDEs, and then approximate solutions for general quadratic GBSDEs in Banach spaces. (10.1016/j.spa.2017.12.004)
  DOI : 10.1016/j.spa.2017.12.004
• Shadow prices, fractional Brownian motion, and portfolio optimisation under transaction costs
  
  • Czichowsky Christoph
  • Peyre Rémi
  • Schachermayer Walter
  • Yang Junjian
  Finance and Stochastics, Springer Verlag (Germany), 2018, 22 (1), pp.161-180. The present paper accomplishes a major step towards a reconciliation of two conflicting approaches in mathematical finance: on the one hand, the mainstream approach based on the notion of no arbitrage (Black, Merton & Scholes); and on the other hand, the consideration of non-semimartingale price processes, the archetype of which being fractional Brownian motion (Mandelbrot). Imposing (arbitrarily small) proportional transaction costs and considering logarithmic utility optimisers, we are able to show the existence of a semimartingale, frictionless shadow price process for an exponential fractional Brownian financial market. (10.1007/s00780-017-0351-5)
  DOI : 10.1007/s00780-017-0351-5
• Drift Theory in Continuous Search Spaces: Expected Hitting Time of the (1+1)-ES with 1/5 Success Rule
  
  • Akimoto Youhei
  • Auger Anne
  • Glasmachers Tobias
  , 2018. This paper explores the use of the standard approach for proving runtime bounds in discrete domains---often referred to as drift analysis---in the context of optimization on a continuous domain. Using this framework we analyze the (1+1) Evolution Strategy with one-fifth success rule on the sphere function. To deal with potential functions that are not lower-bounded, we formulate novel drift theorems. We then use the theorems to prove bounds on the expected hitting time to reach a certain target fitness in finite dimension $d$. The bounds are akin to linear convergence. We then study the dependency of the different terms on $d$ proving a convergence rate dependency of $\Theta(1/d)$. Our results constitute the first non-asymptotic analysis for the algorithm considered as well as the first explicit application of drift analysis to a randomized search heuristic with continuous domain.
• Maximal lower bounds in the Löwner order
  
  • Stott Nikolas
  Proceedings of the American Mathematical Society, American Mathematical Society, 2018. We show that the set of maximal lower bounds of two symmetric matrices with respect to the L\"owner order can be identified to the quotient set O(p,q)/(O(p)×O(q)). Here, (p,q) denotes the inertia of the difference of the two matrices, O(p) is the p-th orthogonal group, and O(p,q) is the indefinite orthogonal group arising from a quadratic form with inertia (p,q). We also show that a similar result holds for positive semidefinite maximal lower bounds with maximal rank of two positive semidefinite matrices. We exhibit a correspondence between the maximal lower bounds C of two matrices A,B and certain pairs of subspaces, describing the directions on which the quadratic form associated with C is tangent to the one associated with A or B. The present results refines a theorem from Kadison that characterizes the existence of the infimum of two symmetric matrices and a theorem from Moreland, Gudder and Ando on the existence of the positive semidefinite infimum of two positive semidefinite matrices. (10.1090/proc/13785)
  DOI : 10.1090/proc/13785
• Efficient semiparametric estimation and model selection for multidimensional mixtures
  
  • Gassiat Elisabeth
  • Rousseau Judith
  • Vernet Elodie
  Electronic Journal of Statistics, Shaker Heights, OH : Institute of Mathematical Statistics, 2018, 12 (1), pp.703-740. In this paper, we consider nonparametric multidimensional finite mixture models and we are interested in the semiparametric estimation of the population weights. Here, the i.i.d. observations are assumed to have at least three components which are independent given the population. We approximate the semiparametric model by projecting the conditional distributions on step functions associated to some partition. Our first main result is that if we refine the partition slowly enough, the associated sequence of maximum likelihood estimators of the weights is asymptotically efficient, and the posterior distribution of the weights, when using a Bayesian procedure, satisfies a semiparametric Bernstein von Mises theorem. We then propose a cross-validation like procedure to select the partition in a finite horizon. Our second main result is that the proposed procedure satisfies an oracle inequality. Numerical experiments on simulated data illustrate our theoretical results. (10.1214/17-ejs1387)
  DOI : 10.1214/17-ejs1387
• Efficient Bayesian Computation by Proximal Markov Chain Monte Carlo: When Langevin Meets Moreau.
  
  • Durmus Alain
  • Moulines Éric
  • Pereyra Marcelo
  SIAM Journal on Imaging Sciences, Society for Industrial and Applied Mathematics, 2018, 11 (1). In this paper, two new algorithms to sample from possibly non-smooth log-concave probability measures are introduced. These algorithms use Moreau-Yosida envelope combined with the Euler-Maruyama discretization of Langevin diffusions. They are applied to a de-convolution problem in image processing, which shows that they can be practically used in a high dimensional setting. Finally, non-asymptotic bounds for one of the proposed methods are derived. These bounds follow from non-asymptotic results for ULA applied to probability measures with a convex continuously differentiable log-density with respect to the Lebesgue measure. (10.1137/16M110834)
  DOI : 10.1137/16M110834
• Non reflection and perfect reflection via Fano resonance in waveguides
  
  • Chesnel Lucas
  • Nazarov Sergei A
  Communications in Mathematical Sciences, International Press, 2018. We investigate a time-harmonic wave problem in a waveguide. By means of asymptotic analysis techniques, we justify the so-called Fano resonance phenomenon. More precisely, we show that the scattering matrix considered as a function of a geometrical parameter ε and of the frequency λ is in general not continuous at a point $(ε, λ) = (0,λ_0$) where trapped modes exist. In particular, we prove that for a given $ε = 0$ small, the scattering matrix exhibits a rapid change for frequencies varying in a neighbourhood of $λ_0$. We use this property to construct examples of waveguides such that the energy of an incident wave propagating through the structure is perfectly transmitted (non reflection) or perfectly reflected in monomode regime. We provide numerical results to illustrate our theorems.
• Monochromatic identities for the Green function and uniqueness results for passive imaging
  
  • Agaltsov Alexey
  • Hohage Thorsten
  • Novikov Roman
  SIAM Journal on Applied Mathematics, Society for Industrial and Applied Mathematics, 2018, 78 (5), pp.2865–2890. For many wave propagation problems with random sources it has been demonstrated that cross correlations of wave fields are proportional to the imaginary part of the Green function of the underlying wave equation. This leads to the inverse problem to recover coefficients of a wave equation from the imaginary part of the Green function on some measurement manifold. In this paper we prove, in particular, local uniqueness results for the Schrödinger equation with one frequency and for the acoustic wave equation with unknown density and sound speed and two frequencies. As the main tool of our analysis, we establish new algebraic identities between the real and the imaginary part of Green's function, which in contrast to the well-known Kramers-Kronig relations involve only one frequency. (10.1137/18M1182218)
  DOI : 10.1137/18M1182218
• Volume Viscosity and Internal Energy Relaxation: Error Estimates
  
  • Giovangigli Vincent
  • Yong Wen An
  Nonlinear Analysis: Real World Applications, Elsevier, 2018. We investigate the fast relaxation of internal energy in nonequilibrium gas models derived from the kinetic theory of gases. We establish uniform a priori estimates and existence theorems for symmetric hyperbolic-parabolic systems of partial differential equations with small second order terms and stiff sources. We prove local in time error estimates between the out of equilibrium solution and the one-temperature equilibrium fluid solution for well prepared data and justify the apparition of volume viscosity terms.
• The Asymptotic of Transmission Eigenvalues for a Domain with a Thin Coating
  
  • Boujlida Hanen
  • Haddar Houssem
  • Khenissi Moez
  SIAM Journal on Applied Mathematics, Society for Industrial and Applied Mathematics, 2018, 78 (5), pp.2348-2369. We consider the transmission eigenvalue problem for a medium surrounded by a thin layer of inhomogeneous material with different refractive index. We derive explicit asymptotic expansion for the transmission eigenvalues with respect to the thickness of the thin layer. We prove error estimate for the asymptotic expansion up to order 1 for simple eigenvalues. This expansion can be used to obtain explicit expressions for constant index of refraction.
• The geometry of random minimal factorizations of a long cycle via biconditioned bitype random trees
  
  • Féray Valentin
  • Kortchemski Igor
  Annales Henri Lebesgue, UFR de Mathématiques - IRMAR, 2018, 1, pp.149-226. (10.5802/ahl.5)
  DOI : 10.5802/ahl.5
• Epidemics and the Eden model : a detailed study of robustness
  
  • Gerin Lucas
  , 2018, Emergence, Complexity and Computation, vol 27, pp.165-178. We present some well-known and less well-known properties of the Probabilistic Cellular Automaton \emph{Epidemics} on a finite grid and its analogous on the infinite square lattice: the Eden model. (10.1007/978-3-319-65558-1_12)
  DOI : 10.1007/978-3-319-65558-1_12
• On the Whitney extension property for continuously differentiable horizontal curves in sub-Riemannian manifolds
  
  • Sacchelli Ludovic
  • Sigalotti Mario
  Calculus of Variations and Partial Differential Equations, Springer Verlag, 2018. In this article we study the validity of the Whitney $C^1$ extension property for horizontal curves in sub-Riemannian manifolds endowed with 1-jets that satisfy a first-order Taylor expansion compatibility condition. We first consider the equiregular case, where we show that the extension property holds true whenever a suitable non-singularity property holds for the input-output maps on the Carnot groups obtained by nilpotent approximation. We then discuss the case of sub-Riemannian manifolds with singular points and we show that all step-2 manifolds satisfy the $C^1$ extension property. We conclude by showing that the $C^1$ extension property implies a Lusin-like approximation theorem for horizontal curves on sub-Riemannian manifolds.
• Sufficient optimality conditions for bilinear optimal control of the linear damped wave equation
  
  • Bethke Franz
  • Kröner Axel
  , 2018. In this paper we discuss sufficient optimality conditions for an optimal control problem for the linear damped wave equation with the damping parameter as the control. We address the case that the control enters quadratic in the cost function as well as the singular case that the control enters affine. For the non-singular case we consider strong and weak local minima , in the singular case we derive sufficient optimality conditions for weak local minima. Thereby, we take advantage of the Goh transformation applying techniques recently established in Aronna, Bonnans, and Kröner [Math. Program. 168(1):717–757, 2018] and [INRIA research report, 2017]. Moreover, a numerical example for the singular case is presented.
• Optimal Control of Infinite Dimensional Bilinear Systems: Application to the Heat and Wave Equations
  
  • Aronna Maria Soledad
  • Bonnans Joseph Fréderic
  • Kröner Axel
  Mathematical Programming, Springer Verlag, 2018, 168 (1-2), pp.717-757. In this paper we consider second order optimality conditions for a bilinear optimal control problem governed by a strongly continuous semigroup operator, the control entering linearly in the cost function. We derive first and second order optimality conditions, taking advantage of the Goh transform. We then apply the results to the heat and wave equations.