Publications

2015

• Ensuring robustness of domain decomposition methods by block strategies
  
  • Gosselet Pierre
  • Rixen Daniel
  • Spillane Nicole
  • Roux François-Xavier
  , 2015. no abstract
• Lyapunov and Minimum-Time Path Planning for Drones
  
  • Maillot Thibault
  • Boscain Ugo
  • Gauthier Jean-Paul
  • Serres Ulysse
  Journal of Dynamical and Control Systems, Springer Verlag, 2015, 21 (1), pp.1-34. (10.1007/s10883-014-9222-y)
  DOI : 10.1007/s10883-014-9222-y
• Coupling techniques for nonlinear hyperbolic equations. IV. Well-balanced schemes for scalar multi-dimensional and multi-component laws
  
  • Boutin Benjamin
  • Coquel Frédéric
  • LeFloch Philippe G.
  Mathematics of Computation, American Mathematical Society, 2015, 84 (294), pp.1663-1702. This series of papers is devoted to the formulation and the approximation of coupling problems for nonlinear hyperbolic equations. The coupling across an interface in the physical space is formulated in term of an augmented system of partial differential equations. In an earlier work, this strategy allowed us to develop a regularization method based on a thick interface model in one space variable for coupling scalar equations. In the present paper, we significantly extend this framework and, in addition, encompass equations in several space variables. This new formulation includes the coupling of several distinct scalar conservation laws and allows for a possible covering in space. Our main contributions are, on one hand, the design and analysis of a well–balanced finite volume method on general triangulations and, on the other hand, a proof of convergence of this method toward entropy solutions, extending Coquel, Cockburn, and LeFloch's theory (restricted to a single conservation law without coupling). The core of our analysis is, first, the derivation of entropy inequalities as well as a discrete entropy dissipation estimate and, second, a proof of convergence toward the entropy solution of the coupling problem. (10.1090/S0025-5718-2015-02933-0)
  DOI : 10.1090/S0025-5718-2015-02933-0
• Developmental Partial Differential Equations
  
  • Pouradier Duteil Nastassia
  • Rossi Francesco
  • Boscain Ugo
  • Piccoli Benedetto
  , 2015. In this paper, we introduce the concept of Developmental Partial Differential Equation (DPDE), which consists of a Partial Differential Equation (PDE) on a time-varying manifold with complete coupling between the PDE and the manifold’s evolution. In other words, the manifold’s evolution depends on the solution to the PDE, and vice versa the differential operator of the PDE depends on the manifold’s geometry. DPDE is used to study a diffusion equation with source on a growing surface whose growth depends on the intensity of the diffused quantity. The surface may, for instance, represent the membrane of an egg chamber and the diffused quantity a protein activating a signaling pathway leading to growth. Our main objective is to show controllability of the surface shape using a fixed source with variable intensity for the diffusion. More specifically, we look for a control driving a symmetric manifold shape to any other symmetric shape in a given time interval. For the diffusion we take directly the Laplace-Beltrami operator of the surface, while the surface growth is assumed to be equal to the value of the diffused quantity. We introduce a theoretical framework, provide approximate controllability and show numerical results. Future applications include a specific model for the oogenesis of Drosophila melanogaster.
• Phaseless inverse scattering in the one-dimensional case
  
  • Novikov Roman
  Eurasian Journal of Mathematical and Computer Applications, Eurasian National University, Kazakhstan (Nur-Sultan), 2015, 3 (1), pp.64-70. We consider the one-dimensional Schrödinger equation with a potential satisfying the standard assumptions of the inverse scattering theory and supported on the half-line x ≥ 0. For this equation at fixed positive energy we give explicit formulas for finding the full complex valued reflection coefficient to the left from appropriate phaseless scattering data measured on the left, i.e. for x < 0. Using these formulas and known inverse scattering results we obtain global uniqueness and reconstruction results for phaseless inverse scattering in dimension d = 1.
• Energy release rate for non smooth cracks in planar elasticity
  
  • Babadjian Jean-François
  • Chambolle Antonin
  • Lemenant Antoine
  Journal de l'École polytechnique — Mathématiques, École polytechnique, 2015, 2, pp.117-152. This paper is devoted to the characterization of the energy release rate of a crack which is merely closed, connected, and with density $1/2$ at the tip. First, the blow-up limit of the displacement is analyzed, and the convergence to the corresponding positively $1/2$-homogenous function in the cracked plane is established. Then, the energy release rate is obtained as the derivative of the elastic energy with respect to an infinitesimal additional crack increment.
• Training Schr\"odinger's cat: quantum optimal control
  
  • Glaser Stefffen J.
  • Boscain Ugo
  • Calarco Tommaso
  • Koch Christiane P.
  • Köckenberger Walter
  • Kosloff Ronnie
  • Kuprov Ilya
  • Luy Burkard
  • Schirmer Sophie
  • Schulte-Herbrüggen Thomas
  • Sugny Dominique
  • Wilhelm Frank K.
  , 2015. It is control that turns scientific knowledge into useful technology: in physics and engineering it provides a systematic way for driving a system from a given initial state into a desired target state with minimized expenditure of energy and resources -- as famously applied in the Apollo programme. As one of the cornerstones for enabling quantum technologies, optimal quantum control keeps evolving and expanding into areas as diverse as quantum-enhanced sensing, manipulation of single spins, photons, or atoms, optical spectroscopy, photochemistry, magnetic resonance (spectroscopy as well as medical imaging), quantum information processing and quantum simulation. --- Here state-of-the-art quantum control techniques are reviewed and put into perspective by a consortium uniting expertise in optimal control theory and applications to spectroscopy, imaging, quantum dynamics of closed and open systems. We address key challenges and sketch a roadmap to future developments.
• Artificial boundary conditions for axisymmetric eddy current probe problems
  
  • Haddar Houssem
  • Jiang Zixian
  • Lechleiter Armin
  Computers & Mathematics with Applications, Elsevier, 2015, 68 (12, Part A,), pp.1844–1870. We study different strategies for the truncation of computational domains in the simulation of eddy current probes of elongated axisymmetric tubes. For axial fictitious boundaries, an exact Dirichlet-to-Neumann map is proposed and mathematically analyzed via a non-selfadjoint spectral problem: under general assumptions we show convergence of the solution to an eddy current problem involving a truncated Dirichlet-to-Neumann map to the solution on the entire, unbounded axisymmetric domain as the truncation parameter tends to infinity. Under stronger assumptions on the physical parameters of the eddy current problem, convergence rates are shown. We further validate our theoretical results through numerical experiments for a realistic physical setting inspired by eddy current probes of nuclear reactor core tubes. (10.1016/j.camwa.2014.10.008)
  DOI : 10.1016/j.camwa.2014.10.008
• 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.
• Control of a Quantum Model for Two Trapped Ions
  
  • Paduro Esteban
  • Sigalotti Mario
  , 2015.
• 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.
• 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
• 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
• 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
• 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
• 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
• 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
• 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
• 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.
• 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.
• 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.
• 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
• 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