Publications

2015

• Analysis of Some Qualitative Methods for Inverse Electromagnetic Scattering Problems
  
  • Haddar Houssem
  , 2015, pp.51. This chapter provides a comprehensive presentation of some qualitative methods associated with inverse 3D electromagnetic scattering problem from inhomogeneous and anisotropic media. We first discuss the problem in the framework of so-called Born approximation, that leads to a linearisation of the inverse problem. We second present and analyze the application of the Linear Sampling Method to the full non linear problem using multistatic data at a given frequency. We especially focus on a generalization of this method based on an exact characterization of the inclusion shape in terms of the available data. We then discuss the closely related interior transmission problem and associated transmission eigenvalues. We complement each chapter with some open challenging questions as well as references for further readings. (10.1007/978-3-319-19306-9)
  DOI : 10.1007/978-3-319-19306-9
• Sparse and spurious: dictionary learning with noise and outliers
  
  • Gribonval Rémi
  • Jenatton Rodolphe
  • Bach Francis
  IEEE Transactions on Information Theory, Institute of Electrical and Electronics Engineers, 2015, 61 (11), pp.6298-6319. A popular approach within the signal processing and machine learning communities consists in modelling signals as sparse linear combinations of atoms selected from a learned dictionary. While this paradigm has led to numerous empirical successes in various fields ranging from image to audio processing, there have only been a few theoretical arguments supporting these evidences. In particular, sparse coding, or sparse dictionary learning, relies on a non-convex procedure whose local minima have not been fully analyzed yet. In this paper, we consider a probabilistic model of sparse signals, and show that, with high probability, sparse coding admits a local minimum around the reference dictionary generating the signals. Our study takes into account the case of over-complete dictionaries, noisy signals, and possible outliers, thus extending previous work limited to noiseless settings and/or under-complete dictionaries. The analysis we conduct is non-asymptotic and makes it possible to understand how the key quantities of the problem, such as the coherence or the level of noise, can scale with respect to the dimension of the signals, the number of atoms, the sparsity and the number of observations. (10.1109/TIT.2015.2472522)
  DOI : 10.1109/TIT.2015.2472522
• Finite element approximation of level set motion by powers of the mean curvature
  
  • Kröner Axel
  • Kröner Eva
  • Kröner Heiko
  , 2015. In this paper we study the level set formulations of certain geometric evolution equations from a numerical point of view. Specifically, we consider the flow by powers greater than one of the mean curvature and the inverse mean curvature flow. Since the corresponding equations in level set form are quasilinear, degenerate and especially possibly singular a regularization method is used in the literature to approximate these equations to overcome the singularities of the equations. Motivated by the paper [29] which studies the finite element approximation of inverse mean curvature flow we prove error estimates for the finite element approximation of the regularized equations for the flow by powers of the mean curvature. We validate the rates with numerical examples. Additionally, the regularization error in the rotational symmetric case for both flows is analyzed numerically. All calculations are performed in the 2D case.
• Approximate Controllability of the Two Trapped Ions System
  
  • Paduro Esteban
  • Sigalotti Mario
  Quantum Information Processing, Springer Verlag, 2015, 14, pp.2397-2418. We prove the approximate controllability of a bilinear Schr\"odinger equation modelling a two trapped ions system. A new spectral decoupling technique is introduced, which allows to analyze the controllability of the infinite-dimensional system through finite-dimensional considerations.
• Second order BSDEs with jumps: formulation and uniqueness
  
  • Kazi-Tani Mohamed Nabil
  • Possamaï Dylan
  • Zhou Chao
  The Annals of Applied Probability, Institute of Mathematical Statistics (IMS), 2015, 25 (5), pp.2867-2908. (10.1214/14-AAP1063)
  DOI : 10.1214/14-AAP1063
• New high order sufficient conditions for configuration tracking
  
  • Barbero-Liñán M.
  • Sigalotti M.
  Automatica, Elsevier, 2015, 62, pp.222-226. In this paper, we propose new conditions guaranteeing that the trajectories of a mechanical control system can track any curve on the configuration manifold. We focus on systems that can be represented as forced affine connection control systems and we generalize the sufficient conditions for tracking known in the literature. The new results are proved by a combination of averaging procedures by highly oscillating controls with the notion of kinematic reduction. (10.1016/j.automatica.2015.09.032)
  DOI : 10.1016/j.automatica.2015.09.032
• Remarks on the internal exponential stabilization to a nonstationary solution for 1D Burgers equations
  
  • Kröner Axel
  • Rodrigues Sergio S.
  SIAM Journal on Control and Optimization, Society for Industrial and Applied Mathematics, 2015, 53 (2), pp.1020–1055. The feedback stabilization of the Burgers system to a nonstationary solution using finite-dimensional internal controls is considered. Estimates for the dimension of the controller are derived. In the particular case of no constraint on the support of the controla better estimate is derived and the possibility of getting an analogous estimate for the general case is discussed; some numerical examplesare presented illustrating the stabilizing effect of the feedback control, and suggesting that the existence of an estimatein the general case analogous to that in the particular one is plausible.
• Mathematical justification of macroscopic models for diffusion MRI through the periodic unfolding method
  
  • Coatléven Julien
  Asymptotic Analysis, IOS Press, 2015, 93 (3), pp.219-258. Diffusion Magnetic Resonance Imaging (dMRI) is a promising tool to obtain useful information on cellular structure when applied to biological tissues. A coupled macroscopic model has been introduced recently through formal homogenization to model dMRI's signal attenuation. This model was based on a particular scaling of the permeability condition modeling cellular membranes. In this article, we explore all the possible scalings and mathematically justify the corresponding limit models, using the periodic unfolding method. We also illustrate through numerical simulations the respective behavior of the limit models when compared to dMRI measurements. (10.3233/ASY-151294)
  DOI : 10.3233/ASY-151294
• Control of a Quantum Model for Two Trapped Ions
  
  • Paduro Esteban
  • Sigalotti Mario
  , 2015.
• Origin and diversification of living cycads: a cautionary tale on the impact of the branching process prior in Bayesian molecular dating
  
  • Condamine Fabien L.
  • Nagalingum Nathalie S.
  • Marshall Charles R
  • Morlon Hélène
  BMC Evolutionary Biology, BioMed Central, 2015, 15 (1). Background: Bayesian relaxed-clock dating has significantly influenced our understanding of the timeline of biotic evolution. This approach requires the use of priors on the branching process, yet little is known about their impact on divergence time estimates. We investigated the effect of branching priors using the iconic cycads. We conducted phylogenetic estimations for 237 cycad species using three genes and two calibration strategies incorporating up to six fossil constraints to (i) test the impact of two different branching process priors on age estimates, (ii) assess which branching prior better fits the data, (iii) investigate branching prior impacts on diversification analyses, and (iv) provide insights into the diversification history of cycads. Results: Using Bayes factors, we compared divergence time estimates and the inferred dynamics of diversification when using Yule versus birth-death priors. Bayes factors were calculated with marginal likelihood estimated with stepping-stone sampling. We found striking differences in age estimates and diversification dynamics depending on prior choice. Dating with the Yule prior suggested that extant cycad genera diversified in the Paleogene and with two diversification rate shifts. In contrast, dating with the birth-death prior yielded Neogene diversifications, and four rate shifts, one for each of the four richest genera. Nonetheless, dating with the two priors provided similar age estimates for the divergence of cycads from Ginkgo (Carboniferous) and their crown age (Permian). Of these, Bayes factors clearly supported the birth-death prior. Conclusions: These results suggest the choice of the branching process prior can have a drastic influence on our understanding of evolutionary radiations. Therefore, all dating analyses must involve a model selection process using Bayes factors to select between a Yule or birth-death prior, in particular on ancient clades with a potential pattern of high extinction. We also provide new insights into the history of cycad diversification because we found (i) periods of extinction along the long branches of the genera consistent with fossil data, and (ii) high diversification rates within the Miocene genus radiations. (10.1186/s12862-015-0347-8)
  DOI : 10.1186/s12862-015-0347-8
• A highly anisotropic nonlinear elasticity model for vesicles I. Eulerian formulation, rigidity estimates and vanishing energy limit
  
  • Merlet Benoit
  Archive for Rational Mechanics and Analysis, Springer Verlag, 2015, 217 (2), pp.651--680. We propose a nonlinear elasticity model for vesicle membranes which is an Eulerian version of a model introduced by Pantz and Trabelsi.We describe the limit behavior of sequences of configurations whose energy goes to 0 in a fixed domain. The material is highly anisotropic and the analysis is based on some rigidity estimates adapted to this anisotropy. The main part of the paper is devoted to these estimates and to some of their consequences. The strongest form of these estimates are used in a second article to derive the thin-shell limit bending theory of the model. (10.1007/s00205-014-0839-5)
  DOI : 10.1007/s00205-014-0839-5
• Dobrushin ergodicity coefficient for Markov operators on cones
  
  • Gaubert Stéphane
  • Qu Zheng
  Integral Equations and Operator Theory, Springer Verlag, 2015, 1 (81), pp.127-150. Doeblin and Dobrushin characterized the contraction rate of Markov operators with respect the total variation norm. We generalize their results by giving an explicit formula for the contraction rate of a Markov operator over a cone in terms of pairs of extreme points with disjoint support in a set of abstract probability measures. By duality, we derive a characterization of the contraction rate of consensus dynamics over a cone with respect to Hopf’s oscillation seminorm (the infinitesimal seminorm associated with Hilbert’s projective metric). We apply these results to Kraus maps (noncommutative Markov chains, representing quantum channels), and characterize the ultimate contraction of the map in terms of the existence of a rank one matrix in a certain subspace. (10.1007/s00020-014-2193-2)
  DOI : 10.1007/s00020-014-2193-2
• Phase retrieval for the Cauchy wavelet transform
  
  • Waldspurger Irène
  • Mallat Stéphane
  Journal of Fourier Analysis and Applications, Springer Verlag, 2015. We consider the phase retrieval problem in which one tries to reconstruct a function from the modulus of its wavelet transform. We study the unicity and stability of the reconstruction. In the case where the wavelets are Cauchy wavelets, we prove that the modulus of the wavelet transform uniquely determines the function up to a global phase. We show that the reconstruction operator is continuous but not uniformly continuous. We describe how to construct pairs of functions which are far away in L 2-norm but whose wavelet transforms are very close, in modulus. The principle is to modulate the wavelet transform of a fixed initial function by a phase which varies slowly in both time and frequency. This construction seems to cover all the instabilities that we observe in practice; we give a partial formal justification to this fact. Finally, we describe an exact reconstruction algorithm and use it to numerically confirm our analysis of the stability question.
• Analytical approximations of BSDEs with non-smooth driver
  
  • Gobet Emmanuel
  • Pagliarani Stefano
  SIAM Journal on Financial Mathematics, Society for Industrial and Applied Mathematics, 2015, 6 (1), pp.919-958. We provide and analyse analytical approximations of BSDEs in the limit of small non-linearity {and short time}, in the case of non-smooth drivers. We identify the first and the second order approximations within this asymptotics and consider two topical financial applications: the two interest rates problem and the Funding Value Adjustment. In high dimensional diffusion setting, we show how to compute explicitly the first order formula by taking advantage of recent proxy techniques. Numerical tests up to dimension 10 illustrate the efficiency of the numerical schemes. (10.1137/14100021X)
  DOI : 10.1137/14100021X
• An iterative approach to non-overdetermined inverse scattering at fixed energy
  
  • Novikov Roman
  Sbornik: Mathematics, Turpion, 2015, 206 (1), pp.120-134. We propose an iterative approximate reconstruction algorithm for non-overdetermined inverse scattering at fixed energy E with incomplete data in dimension d >= 2. In particular, we obtain rapidly converging approximate reconstructions for this inverse scattering for E --> +infinity.
• Zubov's equation for state-constrained perturbed nonlinear systems
  
  • Grüne Lars
  • Zidani Hasnaa
  Mathematical Control and Related Fields, AIMS, 2015, 5 (1), pp.55-71. The paper gives a characterization of the uniform robust domain of attraction for a nite non-linear controlled system subject to perturbations and state constraints. We extend the Zubov approach to characterize this domain by means of the value function of a suitable in nite horizon state-constrained control problem which at the same time is a Lyapunov function for the system. We provide associated Hamilton-Jacobi-Bellman equations and prove existence and uniqueness of the solutions of these generalized Zubov equations. (10.3934/mcrf.2015.5.55)
  DOI : 10.3934/mcrf.2015.5.55
• Analysis and simulation of rare events for SPDEs
  
  • Bréhier Charles-Edouard
  • Gazeau Maxime
  • Goudenège Ludovic
  • Rousset Mathias
  ESAIM: Proceedings and Surveys, EDP Sciences, 2015, 48, pp.364-384. In this work, we consider the numerical estimation of the probability for a stochastic process to hit a set B before reaching another set A. This event is assumed to be rare. We consider reactive trajectories of the stochastic Allen-Cahn partial differential evolution equation (with double well potential) in dimension 1. Reactive trajectories are defined as the probability distribution of the trajectories of a stochastic process, conditioned by the event of hitting B before A. We investigate the use of the so-called Adaptive Multilevel Splitting algorithm in order to estimate the rare event and simulate reactive trajectories. This algorithm uses a reaction coordinate (a real valued function of state space defining level sets), and is based on (i) the selection, among several replicas of the system having hit A before B, of those with maximal reaction coordinate; (ii) iteration of the latter step. We choose for the reaction coordinate the average magnetization, and for B the minimum of the well opposite to the initial condition. We discuss the context, prove that the algorithm has a sense in the usual functional setting, and numerically test the method (estimation of rare event, and transition state sampling). (10.1051/proc/201448017)
  DOI : 10.1051/proc/201448017
• Sharp asymptotics of metastable transition times for one dimensional SPDEs
  
  • Barret Florent
  Annales de l'Institut Henri Poincaré (B) Probabilités et Statistiques, Institut Henri Poincaré (IHP), 2015, 51 (1), pp.129-166. We consider a class of parabolic semi-linear stochastic partial differential equations driven by space-time white noise on a compact space interval. Our aim is to obtain precise asymptotics of the transition times between metastable states. A version of the so-called Eyring-Kramers Formula is proven in an infinite dimensional setting. The proof is based on a spatial finite difference discretization of the stochastic partial differential equation. The expected transition time is computed for the finite dimensional approximation and controlled uniformly in the dimension.
• Sample Complexity of Dictionary Learning and other Matrix Factorizations
  
  • Gribonval Rémi
  • Jenatton Rodolphe
  • Bach Francis
  • Kleinsteuber Martin
  • Seibert Matthias
  IEEE Transactions on Information Theory, Institute of Electrical and Electronics Engineers, 2015, 61 (6), pp.3469-3486. Many modern tools in machine learning and signal processing, such as sparse dictionary learning, principal component analysis (PCA), non-negative matrix factorization (NMF), $K$-means clustering, etc., rely on the factorization of a matrix obtained by concatenating high-dimensional vectors from a training collection. While the idealized task would be to optimize the expected quality of the factors over the underlying distribution of training vectors, it is achieved in practice by minimizing an empirical average over the considered collection. The focus of this paper is to provide sample complexity estimates to uniformly control how much the empirical average deviates from the expected cost function. Standard arguments imply that the performance of the empirical predictor also exhibit such guarantees. The level of genericity of the approach encompasses several possible constraints on the factors (tensor product structure, shift-invariance, sparsity \ldots), thus providing a unified perspective on the sample complexity of several widely used matrix factorization schemes. The derived generalization bounds behave proportional to $\sqrt{\log(n)/n}$ w.r.t.\ the number of samples $n$ for the considered matrix factorization techniques. (10.1109/TIT.2015.2424238)
  DOI : 10.1109/TIT.2015.2424238
• Derivation of nonlinear shell models combining shear and flexure: application to biological membranes
  
  • Pantz Olivier
  • Trabelsi Karim
  Mathematics and Mechanics of Complex Systems, International Research Center for Mathematics & Mechanics of Complex Systems (M&MoCS),University of L’Aquila in Italy, 2015, 3 (2), pp.101--138. Biological membranes are often idealized as incompressible elastic surfaces whose strain energy only depends on their mean curvature and pos-sibly on their shear. We show that this type of model can be derived using a formal asymptotic method by considering biological membranes to be thin, strongly anisotropic, elastic, locally homogeneous bodies.
• Quadratic BSDEs with jumps: a fixed-point approach
  
  • Kazi-Tani Mohamed Nabil
  • Possamaï Dylan
  • Zhou Chao
  Electronic Journal of Probability, Institute of Mathematical Statistics (IMS), 2015, 20 (66), pp.1-28. (10.1214/EJP.v20-3363)
  DOI : 10.1214/EJP.v20-3363
• Time-Optimal Synthesis for Three Relevant Problems: The Brockett Integrator, the Grushin Plane and the Martinet Distribution
  
  • Barilari Davide
  • Boscain Ugo
  • Le Donne Enrico
  • Sigalotti Mario
  , 2015.
• Accelerated Share Repurchase: pricing and execution strategy
  
  • Guéant Olivier
  • Pu Jiang
  • Guillaume Royer
  International Journal of Theoretical and Applied Finance, World Scientific Publishing, 2015, 18 (3). In this article, we consider a specific optimal execution problem associated to accelerated share repurchase contracts. When firms want to repurchase their own shares, they often enter such a contract with a bank. The bank buys the shares for the firm and is paid the average market price over the execution period, the length of the period being decided upon by the bank during the buying process. Mathematically, the problem is new and related to both option pricing (Asian and Bermudan options) and optimal execution. We provide a model, along with associated numerical methods, to determine the optimal stopping time and the optimal buying strategy of the bank. (10.1142/S0219024915500193)
  DOI : 10.1142/S0219024915500193
• Robust utility maximization in nondominated models with 2BSDE: the uncertain volatility model
  
  • Matoussi Anis
  • Possamaï Dylan
  • Zhou Chao
  Mathematical Finance, Wiley, 2015, 25 (2), pp.258-287. The problem of robust utility maximization in an incomplete market with volatility uncertainty is considered, in the sense that the volatility of the market is only assumed to lie between two given bounds. The set of all possible models (probability measures) considered here is non-dominated. We propose to study this problem in the framework of second-order backward stochastic differential equations (2BSDEs for short) with quadratic growth generators. We show for exponential, power and logarithmic utilities that the value function of the problem can be written as the initial value of a particular 2BSDE and prove existence of an optimal strategy. Finally several examples which shed more light on the problem and its links with the classical utility maximization one are provided. In particular, we show that in some cases, the upper bound of the volatility interval plays a central role, exactly as in the option pricing problem with uncertain volatility models of [2]. (10.1111/mafi.12031)
  DOI : 10.1111/mafi.12031
• Convergent stochastic Expectation Maximization algorithm with efficient sampling in high dimension. Application to deformable template model estimation
  
  • Allassonnière Stéphanie
  • Kuhn Estelle
  Computational Statistics and Data Analysis, Elsevier, 2015, 91, pp.4-19. Estimation in the deformable template model is a big challenge in image analysis. The issue is to estimate an atlas of a population. This atlas contains a template and the corresponding geometrical variability of the observed shapes. The goal is to propose an accurate estimation algorithm with low computational cost and with theoretical guaranties of relevance. This becomes very demanding when dealing with high dimensional data, which is particularly the case of medical images. The use of an optimized Monte Carlo Markov Chain method for a stochastic Expectation Maximization algorithm, is proposed to estimate the model parameters by maximizing the likelihood. A new Anisotropic Metropolis Adjusted Langevin Algorithm is used as transition in the MCMC method. First it is proven that this new sampler leads to a geometrically uniformly ergodic Markov chain. Furthermore, it is proven also that under mild conditions, the estimated parameters converge almost surely and are asymptotically Gaussian distributed. The methodology developed is then tested on handwritten digits and some 2D and 3D medical images for the deformable model estimation. More widely, the proposed algorithm can be used for a large range of models in many fields of applications such as pharmacology or genetic. The technical proofs are detailed in an appendix. (10.1016/j.csda.2015.04.011)
  DOI : 10.1016/j.csda.2015.04.011