Publications

2017

• On the heat diffusion for generic Riemannian and sub-Riemannian structures
  
  • Barilari Davide
  • Boscain Ugo
  • Charlot Grégoire
  • Neel Robert W.
  International Mathematics Research Notices, Oxford University Press (OUP), 2017, 15, pp.4639-4672. In this paper we provide the small-time heat kernel asymptotics at the cut locus in three relevant cases: generic low-dimensional Riemannian manifolds, generic 3D contact sub-Riemannian manifolds (close to the starting point) and generic 4D quasi-contact sub-Riemannian manifolds (close to a generic starting point). As a byproduct, we show that, for generic low-dimensional Riemannian manifolds, the only singularities of the exponential map, as a Lagragian map, that can arise along a minimizing geodesic are $A_3$ and $A_5$ (in the classification of Arnol'd's school). We show that in the non-generic case, a cornucopia of asymptotics can occur, even for Riemannian surfaces. (10.1093/imrn/rnw141)
  DOI : 10.1093/imrn/rnw141
• Some discussions on the Read Paper "Beyond subjective and objective in statistics" by A. Gelman and C. Hennig
  
  • Robert Christian
  • Celeux Gilles
  • Jewson Jack
  • Josse Julie
  • Marin Jean-Michel
  • Robert Christian P.
  , 2017. This note is a collection of several discussions of the paper "Beyond subjective and objective in statistics", read by A. Gelman and C. Hennig to the Royal Statistical Society on April 12, 2017, and to appear in the Journal of the Royal Statistical Society, Series A.
• Transport of power in random waveguides with turning points
  
  • Garnier Josselin
  • Borcea Liliana
  • Wood Derek
  Communications in Mathematical Sciences, International Press, 2017, 15 (8), pp.2327 - 2371. (10.4310/CMS.2017.v15.n8.a9)
  DOI : 10.4310/CMS.2017.v15.n8.a9
• Multi-Wave Medical Imaging: Mathematical Modelling and Imaging Reconstruction
  
  • Ammari Habib
  • Garnier Josselin
  • Kang Hyeonbae
  • Nguyen Loc Hoang
  • Seppecher Laurent
  , 2017.
• The Generalized Linear Sampling Method for limited aperture measurements
  
  • Audibert Lorenzo
  • Haddar Houssem
  SIAM Journal on Imaging Sciences, Society for Industrial and Applied Mathematics, 2017, 10 (2), pp.845–870. We extend the so-called qeneralized Linear Sampling Method (GLSM) to the case of limited aperture data at a fixed frequency.in this case thefactorization of the samplig operatort does not obey the symmetry required in the justification of the GLSM introduced in Audibert-Haddar [Inverse Problems, 2014]. We propose a new formulation by adding an extra penalty term that asymptotically correct the non symmetry of the GLSM original penalty term. The analysis of the new formulation is first presented in an abstact frmeworkF e then show how to pply our setting to the slr W prolem with fr eld mesurements or ner eld mesurements on limited pertureF e nlly IH vlidte the method through some numeril tests in two dimensions nd for fr eld mesurementsF II Key words. snverse sttering prolemsD viner mpling wethodD qenerlized viner mpling wethodD pE IP toriztion wethodD ulittive methods IQ (10.1137/16M110112X)
  DOI : 10.1137/16M110112X
• Online Learning and Blackwell Approachability with Partial Monitoring: Optimal Convergence Rates
  
  • Kwon Joon
  • Perchet Vianney
  JMLR Papers, 2017, 54, pp.604-613. Blackwell approachability is an online learning setup generalizing the classical problem of regret minimization by allowing for instance multi-criteria optimization, global (online) optimization of a convex loss, or online linear optimization under some cumulative constraint. We consider partial monitoring where the decision maker does not necessarily observe the outcomes of his decision (unlike the traditional regret/bandit literature). Instead, he receives a random signal correlated to the decision-outcome pair, or only to the outcome. We construct, for the first time, approachability algorithms with convergence rate of order O(T −1/2) when the signal is independent of the decision and of order O(T −1/3) in the case of general signals. Those rates are optimal in the sense that they cannot be improved without further assumption on the structure of the objectives and/or the signals.
• Équations de Navier-Stokes incompressibles et multirésolution spatiale adaptative: sur la question des modes parasites en maillage collocalisé.
  
  • N'Guessan Marc-Arthur
  • Massot Marc
  • Tenaud Christian
  • Series Laurent
  , 2017. La simulation numérique directe (DNS) de la combustion avec chimie détaillée et transport multi-espèces représente l'un des défis les plus importants en matière de calcul scientifique dans de nombreuses applications industrielles. Un des enjeux est de coupler un solveur hydrodynamique pour la résolution des équations de Navier-Stokes, pour un mélange réactif dans la limite des faibles nombres de Mach, à une stratégie de résolution de systèmes de convection-réaction-diffusion, tout en maintenant l'efficacité algorithmique, l'adaptation temps-espace et le contrôle d'erreur. La présente communication vise à proposer une stratégie optimale pour l'élimination des modes parasites dans un contexte de maillages colocalisés en volume fini, dans un cadre de multirésolution spatiale.
• Generalized and hybrid Metropolis-Hastings overdamped Langevin algorithms
  
  • Poncet Romain
  , 2017. It has been shown that the nonreversible overdamped Langevin dynamics enjoy better convergence properties in terms of spectral gap and asymptotic variance than the reversible one. In this article we propose a variance reduction method for the Metropolis-Hastings Adjusted Langevin Algorithm (MALA) that makes use of the good behaviour of the these nonreversible dynamics. It consists in constructing a nonreversible Markov chain (with respect to the target invariant measure) by using a Generalized Metropolis-Hastings adjustment on a lifted state space. We present two variations of this method and we discuss the importance of a well-chosen proposal distribution in terms of average rejection probability. We conclude with numerical experimentations to compare our algorithms with the MALA, and show variance reduction of several orders of magnitude in some favourable toy cases.
• Exploring the complexity of the integer image problem in the max-algebra
  
  • Maccaig Marie
  Discrete Applied Mathematics, Elsevier, 2017, 217 (2), pp.261--275. We investigate the complexity of the problem of finding an integer vector in the max-algebraic column span of a matrix, which we call the integer image problem. We show some cases where we can determine in strongly polynomial time whether such an integer vector exists, and find such an integer vector if it does exist. On the other hand we also describe a group of related problems each of which we prove to be NP-hard. Our main results demonstrate that the integer image problem is equivalent to finding a special type of integer image of a matrix satisfying a property we call column typical. For a subclass of matrices this problem is polynomially solvable but if we remove the column typical assumption then it becomes NP-hard. (10.1016/j.dam.2016.09.016)
  DOI : 10.1016/j.dam.2016.09.016
• Quantitative DLA-based compressed sensing for T1-weighted acquisitions.
  
  • Svehla Pavel
  • Nguyen Khieu-Van
  • Li Jing-Rebecca
  • Ciobanu Luisa
  Journal of Magnetic Resonance, Elsevier, 2017. (10.1016/j.jmr.2017.05.002)
  DOI : 10.1016/j.jmr.2017.05.002
• Information-Geometric Optimization Algorithms: A Unifying Picture via Invariance Principles
  
  • Ollivier Yann
  • Arnold Ludovic
  • Auger Anne
  • Hansen Nikolaus
  Journal of Machine Learning Research, Microtome Publishing, 2017, 18 (18), pp.1-65. We present a canonical way to turn any smooth parametric family of probability distributions on an arbitrary search space X into a continuous-time black-box optimization method on X, the information-geometric optimization (IGO) method. Invariance as a major design principle keeps the number of arbitrary choices to a minimum. The resulting IGO flow is the flow of an ordinary differential equation conducting the natural gradient ascent of an adaptive, time-dependent transformation of the objective function. It makes no particular assumptions on the objective function to be optimized. The IGO method produces explicit IGO algorithms through time discretization. It naturally recovers versions of known algorithms and offers a systematic way to derive new ones. In continuous search spaces, IGO algorithms take a form related to natural evolution strategies (NES). The cross-entropy method is recovered in a particular case with a large time step, and can be extended into a smoothed, parametrization-independent maximum likelihood update (IGO-ML). When applied to the family of Gaussian distributions on R^d, the IGO framework recovers a version of the well-known CMA-ES algorithm and of xNES. For the family of Bernoulli distributions on {0, 1}^d, we recover the seminal PBIL algorithm and cGA. For the distributions of restricted Boltzmann machines, we naturally obtain a novel algorithm for discrete optimization on {0, 1}^d. All these algorithms are natural instances of, and unified under, the single information-geometric optimization framework. The IGO method achieves, thanks to its intrinsic formulation, maximal invariance properties: invariance under reparametrization of the search space X, under a change of parameters of the probability distribution, and under increasing transformation of the function to be optimized. The latter is achieved through an adaptive, quantile-based formulation of the objective. Theoretical considerations strongly suggest that IGO algorithms are essentially characterized by a minimal change of the distribution over time. Therefore they have minimal loss in diversity through the course of optimization, provided the initial diversity is high. First experiments using restricted Boltzmann machines confirm this insight. As a simple consequence, IGO seems to provide, from information theory, an elegant way to simultaneously explore several valleys of a fitness landscape in a single run.
• Resolution Analysis of Passive Synthetic Aperture Imaging of Fast Moving Objects
  
  • Garnier Josselin
  • Borcea L.
  • Papanicolaou G.
  • Solna K.
  • Tsogka C.
  SIAM Journal on Imaging Sciences, Society for Industrial and Applied Mathematics, 2017, 10 (2), pp.665 - 710. (10.1137/16M109716X)
  DOI : 10.1137/16M109716X
• Functional Characterization of Intrinsic and Extrinsic Geometry
  
  • Corman Etienne
  • Solomon Justin
  • Ben-Chen Mirela
  • Guibas Leonidas
  • Ovsjanikov Maks
  ACM Transactions on Graphics, Association for Computing Machinery, 2017, 17. We propose a novel way to capture and characterize distortion between pairs of shapes by extending the recently proposed framework of shape differences built on functional maps. We modify the original definition of shape differences slightly and prove that, after this change, the discrete metric is fully encoded in two shape difference operators and can be recovered by solving two linear systems of equations. Then, we introduce an extension of the shape difference operators using offset surfaces to capture extrinsic or embedding-dependent distortion, complementing the purely intrinsic nature of the original shape differences. Finally, we demonstrate that a set of four operators is complete, capturing intrinsic and extrinsic structure and fully encoding a shape up to rigid motion in both discrete and continuous settings. We highlight the usefulness of our constructions by showing the complementary nature of our extrinsic shape differences in capturing distortion ignored by previous approaches. We additionally provide examples where we recover local shape structure from the shape difference operators, suggesting shape editing and analysis tools based on manipulating shape differences.
• An equilibrated fluxes approach to the certified descent algorithm for shape optimization using conforming finite element and discontinuous Galerkin discretizations
  
  • Giacomini Matteo
  Journal of Scientific Computing, Springer Verlag, 2017. The certified descent algorithm (CDA) is a gradient-based method for shape optimization which certifies that the direction computed using the shape gradient is a genuine descent direction for the objective functional under analysis. It relies on the computation of an upper bound of the error introduced by the finite element approximation of the shape gradient. In this paper, we present a goal-oriented error estimator which depends solely on local quantities and is fully-computable. By means of the equilibrated fluxes approach, we construct a unified strategy valid for both conforming finite element approximations and discontinuous Galerkin discretizations. The new variant of the CDA is tested on the inverse identification problem of electrical impedance tomography: both its ability to identify a genuine descent direction at each iteration and its reliable stopping criterion are confirmed. (10.1007/s10915-017-0545-1)
  DOI : 10.1007/s10915-017-0545-1
• Dependence of tropical eigenspaces
  
  • Niv Adi
  • Rowen Louis
  Communications in Algebra, Taylor & Francis, 2017, 45 (3), pp.924-942. We study the pathology that causes tropical eigenspaces of distinct su-pertropical eigenvalues of a non-singular matrix A, to be dependent. We show that in lower dimensions the eigenvectors of distinct eigenvalues are independent, as desired. The index set that differentiates between subsequent essential monomials of the characteristic polynomial, yields an eigenvalue λ, and corresponds to the columns of adj(A + λI) from which the eigenvectors are taken. We ascertain the cause for failure in higher dimensions, and prove that independence of the eigenvectors is recovered in case the " difference criterion " holds, defined in terms of disjoint differences between index sets of subsequent coefficients. We conclude by considering the eigenvectors of the matrix A ∇ := 1 det(A) adj(A) and the connection of the independence question to generalized eigenvectors. (10.1080/00927872.2016.1172603)
  DOI : 10.1080/00927872.2016.1172603
• A characterization of switched linear control systems with finite L 2 -gain
  
  • Chitour Yacine
  • Mason Paolo
  • Sigalotti Mario
  IEEE Transactions on Automatic Control, Institute of Electrical and Electronics Engineers, 2017, 62, pp.1825-1837. Motivated by an open problem posed by J.P. Hespanha, we extend the notion of Barabanov norm and extremal trajectory to classes of switching signals that are not closed under concatenation. We use these tools to prove that the finiteness of the L2-gain is equivalent, for a large set of switched linear control systems, to the condition that the generalized spectral radius associated with any minimal realization of the original switched system is smaller than one. (10.1109/tac.2016.2593678)
  DOI : 10.1109/tac.2016.2593678
• Log-majorization of the moduli of the eigenvalues of a matrix polynomial by tropical roots
  
  • Akian Marianne
  • Gaubert Stéphane
  • Sharify Meisam
  Linear Algebra and its Applications, Elsevier, 2017, 528, pp.394--435. We show that the sequence of moduli of the eigenvalues of a matrix polynomial is log-majorized, up to universal constants, by a sequence of "tropical roots" depending only on the norms of the matrix coefficients. These tropical roots are the non-differentiability points of an auxiliary tropical polynomial, or equivalently, the opposites of the slopes of its Newton polygon. This extends to the case of matrix polynomials some bounds obtained by Hadamard, Ostrowski and Pólya for the roots of scalar polynomials. We also obtain new bounds in the scalar case, which are accurate for "fewnomials" or when the tropical roots are well separated. (10.1016/j.laa.2016.11.004)
  DOI : 10.1016/j.laa.2016.11.004
• Matched-Filter and Correlation-Based Imaging for Fast Moving Objects Using a Sparse Network of Receivers
  
  • Garnier Josselin
  • Fournier J.
  • Papanicolaou G.
  • Tsogka C.
  SIAM Journal on Imaging Sciences, Society for Industrial and Applied Mathematics, 2017, 10 (4), pp.2165 - 2216. (10.1137/17M112364X)
  DOI : 10.1137/17M112364X
• Certified Descent Algorithm for shape optimization driven by fully-computable a posteriori error estimators
  
  • Giacomini Matteo
  • Pantz Olivier
  • Trabelsi Karim
  ESAIM: Control, Optimisation and Calculus of Variations, EDP Sciences, 2017, 23 (3), pp.977-1001. In this paper we introduce a novel certified shape optimization strategy-named Certified Descent Algorithm (CDA)-to account for the numerical error introduced by the Finite Element approximation of the shape gradient. We present a goal-oriented procedure to derive a certified upper bound of the error in the shape gradient and we construct a fully-computable, constant-free a posteriori error estimator inspired by the complementary energy principle. The resulting CDA is able to identify a genuine descent direction at each iteration and features a reliable stopping criterion. After validating the error estimator, some numerical simulations of the resulting certified shape optimization strategy are presented for the well-known inverse identification problem of Electrical Impedance Tomography. (10.1051/cocv/2016021)
  DOI : 10.1051/cocv/2016021
• The Purcell Three-link swimmer: some geometric and numerical aspects related to periodic optimal controls
  
  • Bettiol Piernicola
  • Bonnard Bernard
  • Giraldi Laetitia
  • Martinon Pierre
  • Rouot Jérémy
  Radon Series on Computational and Applied Mathematics, De Gruyter, 2017, 18, pp.314–343. The maximum principle combined with numerical methods is a powerful tool to compute solutions for optimal control problems. This approach turns out to be extremely useful in applications, including solving problems which require establishing periodic trajectories for Hamiltonian systems, optimizing the production of photobioreactors over a one-day period, finding the best periodic controls for locomotion models (e.g. walking, flying and swimming). In this article we investigate some geometric and numerical aspects related to optimal control problems for the so-called Purcell Three-link swimmer [20], in which the cost to minimize represents the energy consumed by the swimmer. More precisely, employing the maximum principle and shooting methods we derive optimal trajectories and controls, which have particular periodic features. Moreover, invoking a linearization procedure of the control system along a reference extremal, we estimate the conjugate points, which play a crucial role for the second order optimality conditions. We also show how, making use of techniques imported by the sub-Riemannian geometry, the nilpotent approximation of the system provides a model which is integrable, obtaining explicit expressions in terms of elliptic functions. This approximation allows to compute optimal periodic controls for small deformations of the body, allowing the swimmer to move minimizing its energy. Numerical simulations are presented using Hampath and Bocop codes. (10.1515/9783110430394-010)
  DOI : 10.1515/9783110430394-010
• Pulse Reflection in a Random Waveguide with a Turning Point
  
  • Garnier Josselin
  • Borcea Liliana
  Multiscale Modeling and Simulation: A SIAM Interdisciplinary Journal, Society for Industrial and Applied Mathematics, 2017, 15 (4), pp.1472 - 1501. (10.1137/16M1094154)
  DOI : 10.1137/16M1094154
• Introducing a level-set based shape and topology optimization method for the wear of composite materials with geometric constraints
  
  • Feppon Florian
  • Michailidis G
  • Sidebottom M.A.
  • Allaire Grégoire
  • Krick B.A.
  • Vermaak N
  Structural and Multidisciplinary Optimization, Springer Verlag, 2017, 55 (2), pp.547-568. The wear of materials continues to be a limiting factor in the lifetime and performance of mechanical systems with sliding surfaces. As the demand for low wear materials grows so does the need for models and methods to systematically optimize tribological systems. Elastic foundation models offer a simplified framework to study the wear of multimaterial composites subject to abrasive sliding. Previously, the evolving wear profile has been shown to converge to a steady-state that is characterized by a time-independent elliptic equation. In this article, the steady-state formulation is generalized and integrated with shape optimization to improve the wear performance of bi-material composites. Both macroscopic structures and periodic material microstructures are considered. Several common tribological objectives for systems undergoing wear are identified and mathematically formalized with shape derivatives. These include (i) achieving a planar wear surface from multimaterial composites and (ii) minimizing the run-in volume of material lost before steady-state wear is achieved. A level-set based topology optimization algorithm that incorporates a novel constraint on the level-set function is presented. In particular, a new scheme is developed to update material interfaces ; the scheme (i) conveniently enforces volume constraints at each iteration, (ii) controls the complexity of design features using perimeter penalization, and (iii) nucleates holes or inclusions with the topological gradient. The broad applicability of the proposed formulation for problems beyond wear is discussed, especially for problems where convenient control of the complexity of geometric features is desired. (10.1007/s00158-016-1512-4)
  DOI : 10.1007/s00158-016-1512-4
• Application of the Sparse Cardinal Sine Decomposition to 3D Stokes Flows
  
  • Alouges F.
  • Aussal M.
  • Lefebvre-Lepot A.
  • Pigeonneau Franck
  • Sellier Antoine
  International Journal of Computational Methods and Experimental Measurements, WIT Press, 2017, 5 (3), pp.387 - 394. In boundary element method (BEM), one encounters linear system with a dense and non-symmetric square matrix which might be so large that inverting the linear system is too prohibitive in terms of cpu time and/or memory. Each usual powerful treatment (Fast Multipole Method, H-matrices) developed to deal with this issue is optimized to efficiently perform matrix vector products. This work presents a new technique to adequately and quickly handle such products: the Sparse Cardinal Sine Decomposition. This approach, recently pioneered for the Laplace and Helmholtz equations, rests on the decomposition of each encountered kernel as series of radial Cardinal Sine functions. Here, we achieve this decomposition for the Stokes problem and implement it in MyBEM, a new fast solver for multi-physical BEM. The reported computational examples permit us to compare the advocated method against a usual BEM in terms of both accuracy and convergence. (10.2495/CMEM-V5-N3-387-394)
  DOI : 10.2495/CMEM-V5-N3-387-394
• Multipoint scatterers with zero-energy bound states
  
  • Grinevich Piotr
  • Novikov Roman
  Theoretical and Mathematical Physics, Consultants bureau, 2017, 193 (2), pp.1675-1679. We study multipoint scatterers with zero-energy bound states in three dimensions. We present examples of such scatterers with multiple zero eigenvalue or with strong multipole localization of zero-energy bound states.
• Dimensional Reduction of a Multiscale Model Based on Long Time Asymptotics
  
  • Clément Frédérique
  • Coquel Frédéric
  • Postel Marie
  • Tran Kim Long
  Multiscale Modeling and Simulation: A SIAM Interdisciplinary Journal, Society for Industrial and Applied Mathematics, 2017, 15 (3), pp.1198 - 1241. We consider a class of kinetic models for which a moment equation has a natural interpretation. We show that, depending on their velocity field, some models lead to moment equations that enable one to compute monokinetic solutions economically. We detail the example of a multiscale structured cell population model, consisting of a system of 2D transport equations. The reduced model, a system of 1D transport equations, is obtained by computing the moments of the 2D model with respect to one variable. The 1D solution is defined from the solution of the 2D model starting from an initial condition that is a Dirac mass in the direction removed by reduction. Long time properties of the 1D model solution are obtained in connection with properties of the support of the 2D solution for general case initial conditions. Finite volume numerical approximations of the 1D reduced model can be used to compute the moments of the 2D solution with proper accuracy. The numerical robustness is studied in the scalar case, and a full scale vector case is presented. (10.1137/16M1062545)
  DOI : 10.1137/16M1062545