Publications

2017

• Regularity for the optimal compliance problem with length penalization
  
  • Chambolle Antonin
  • Lamboley Jimmy
  • Lemenant Antoine
  • Stepanov Eugene
  SIAM Journal on Mathematical Analysis, Society for Industrial and Applied Mathematics, 2017. We prove some regularity results for a connected set S in the planar domain O, which minimizes the compliance of its complement O\S, plus its length. This problem, interpreted as to find the best location for attaching a membrane subject to a given external force f so as to minimize the compliance, can be seen as an elliptic PDE version of the average distance problem/irrigation problem (in a penalized version rather than a constrained one), which has been extensively studied in the literature. We prove that minimizers consist of a finite number of smooth curves meeting only by three at 120 degree angles, containing no loop, and possibly touching the boundary of the domain only tangentially. Several new technical tools together with the classical ones are developed for this purpose.
• 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
• 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
• High dimensional matrix estimation with unknown variance of the noise
  
  • Klopp Olga
  • Gaiffas Stéphane
  Statistica Sinica, Taipei : Institute of Statistical Science, Academia Sinica, 2017. We propose a new pivotal method for estimating high-dimensional matrices. Assume that we observe a small set of entries or linear combinations of entries of an unknown matrix $A_0$ corrupted by noise. We propose a new method for estimating $A_0$ which does not rely on the knowledge or an estimation of the standard deviation of the noise $\sigma$. Our estimator achieves, up to a logarithmic factor, optimal rates of convergence under the Frobenius risk and, thus, has the same prediction performance as previously proposed estimators which rely on the knowledge of $\sigma$. Our method is based on the solution of a convex optimization problem which makes it computationally attractive.
• 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.
• Optimal scaling of the Random Walk Metropolis algorithm under Lp mean differentiability
  
  • Durmus Alain
  • Le Corff Sylvain
  • Moulines Éric
  • Roberts Gareth O. O.
  Journal of Applied Probability, Cambridge University press, 2017, 54 (4), pp.1233 -1260. This paper considers the optimal scaling problem for high-dimensional random walk Metropolis algorithms for densities which are differentiable in Lp mean but which may be irregular at some points (like the Laplace density for example) and/or are supported on an interval. Our main result is the weak convergence of the Markov chain (appropriately rescaled in time and space) to a Langevin diffusion process as the dimension d goes to infinity. Because the log-density might be non-differentiable, the limiting diffusion could be singular. The scaling limit is established under assumptions which are much weaker than the one used in the original derivation of [6]. This result has important practical implications for the use of random walk Metropolis algorithms in Bayesian frameworks based on sparsity inducing priors. (10.1017/jpr.2017.61)
  DOI : 10.1017/jpr.2017.61
• 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
• On Jacobi fields and canonical connection in sub-Riemannian geometry
  
  • Barilari Davide
  • Rizzi Luca
  Archivum Mathematicum, Masarykova Universita, 2017, 53 (2), pp.77-92. In sub-Riemannian geometry the coefficients of the Jacobi equation define curvature-like invariants. We show that these coefficients can be interpreted as the curvature of a canonical Ehresmann connection associated to the metric, first introduced in [Zelenko-Li]. We show why this connection is naturally nonlinear, and we discuss some of its properties. (10.5817/AM2017-2-77)
  DOI : 10.5817/AM2017-2-77
• Shape optimisation with the level set method for contact problems in linearised elasticity
  
  • Maury Aymeric
  • Allaire Grégoire
  • Jouve François
  SMAI Journal of Computational Mathematics, Société de Mathématiques Appliquées et Industrielles (SMAI), 2017, 3, pp.249-292. This article is devoted to shape optimisation of contact problems in linearised elasticity, thanks to the level set method. We circumvent the shape non-differentiability, due to the contact boundary conditions, by using penalised and regularised versions of the mechanical problem. This approach is applied to five different contact models: the frictionless model, the Tresca model, the Coulomb model, the normal compliance model and the Norton-Hoff model. We consider two types of optimisation problems in our applications: first, we minimise volume under a compliance constraint, second, we optimise the normal force, with a volume constraint, which is useful to design compliant mechanisms. To illustrate the validity of the method, 2D and 3D examples are performed, the 3D examples being computed with an industrial software. (10.5802/smai-jcm.27)
  DOI : 10.5802/smai-jcm.27
• Sampling from a log-concave distribution with compact support with proximal Langevin Monte Carlo
  
  • Brosse Nicolas
  • Durmus Alain
  • Moulines Éric
  • Pereyra Marcelo
  Proceedings of Machine Learning Research, PMLR, 2017, 65, pp.319-342. This paper presents a detailed theoretical analysis of the Langevin Monte Carlo sampling algorithm recently introduced in [DMP16] when applied to log-concave probability distributions that are restricted to a convex body K. This method relies on a regularisation procedure involving the Moreau-Yosida envelope of the indicator function associated with K. Explicit convergence bounds in total variation norm and in Wasserstein distance of order 1 are established. In particular, we show that the complexity of this algorithm given a first order oracle is polynomial in the dimension of the state space. Finally, some numerical experiments are presented to compare our method with competing MCMC approaches from the literature.
• Finite volume approximation of a degenerate immiscible two-phase flow model of {C}ahn-{H}illiard type
  
  • Cancès Clément
  • Nabet Flore
  , 2017, 199, pp.431-438. We propose a two-point flux approximation Finite Volume scheme for a model of incompressible and immiscible two-phase flow of Cahn-Hilliard type with degenerate mobility. This model was derived from a variational principle and can be interpreted as the Wasserstein gradient flow of the free energy. The fundamental properties of the continuous model, namely the positivity of the concentrations, the decay of the free energy, and the boundedness of the Boltzmann entropy, are preserved by the numerical scheme. Numerical simulations are provided to illustrate the behavior of the model and of the numerical scheme.
• Avis sur les nouvelles techniques d’obtention de plantes (New plant breeding techniques-NPBT)
  
  • Angevin Frédérique
  • Bagnis Claude
  • Bar-Hen Avner
  • Barny Marie-Anne
  • Boireau Pascal
  • Brévault Thierry
  • Chauvel Bruno B.
  • Collonnier Cécile
  • Couvet Denis
  • Dassa Elie
  • de Verneuil Hubert
  • Demeneix Barbara
  • Franche Claudine
  • Guerche Philippe
  • Guillemain Joël
  • Hernandez Raquet Guillermina
  • Khalife Jamal
  • Klonjkowski Bernard
  • Lavielle Marc
  • Le Corre Valérie
  • Lefèvre François
  • Lemaire Olivier
  • Lereclus Didier D.
  • Maximilien Rémy
  • Meurs Eliane
  • Naffakh Nadia
  • Négre Didier
  • Noyer Jean-Louis
  • Ochatt Sergio
  • Pages Jean-Christophe
  • Raynaud Xavier
  • Regnault-Roger Catherine
  • Renard Michel M.
  • Renault Tristan
  • Saindrenan Patrick
  • Simonet Pascal
  • Troadec Marie-Bérengère
  • Vaissière Bernard
  • Vilotte Jean-Luc
  , 2017. Le sigle NPBT désigne un ensemble hétérogène de techniques utilisables dans le cadre de l’obtention de variétés végétales. une question centrale, aujourd’hui débattue dans l’union européenne, concerne le cadre réglementaire d’utilisation de ces techniques. dans ce contexte se posent de multiples questions relatives à l’évaluation sanitaire et environnementale des plantes issues de NPBT , mais aussi à leur détection, leur traçabilité, et leur éventuel étiquetage. le comité scientifique (CS) du HCB a rédigé le présent avis en s’appuyant sur le rapport d’un groupe de travail et sur les discussions menées lors de quatre séances plénières. Le cs avait pour mandat en réponse à une saisine des ministres en charges de l’environnement et de l’agriculture de se prononcer sur : • les méthodes d'analyse et de traçabilité des plantes et des produits issus des NPBT ; • en lien avec le point précédent, les enjeux pour la coexistence des filières ; • les risques directs pour la santé et l'environnement liés aux caractéristiques nouvelles des plantes et des produits obtenus ; • les mesures de gestion à mettre en place pour prévenir et limiter les risques pour la santé et l'environnement liés à l'utilisation des plantes et des produits issus de ces nouvelles techniques, si de tels risques étaient mis en évidence ; • des propositions de pistes intermédiaires entre les dispositions du catalogue européen et celles de la directive 2001/18/CE, qui paraîtraient utiles pour encadrer l'usage de ces nouvelles techniques sur le territoire européen.
• Fast and privacy preserving distributed low-rank regression
  
  • Wai Hoi-To
  • Lafond Jean
  • Scaglione Anna
  • Moulines Éric
  , 2017.
• Probabilistic and Piecewise Deterministic models in Biology
  
  • Cloez Bertrand
  • Dessalles Renaud
  • Genadot Alexandre
  • Malrieu Florent
  • Marguet Aline
  • Yvinec Romain
  ESAIM: Proceedings and Surveys, EDP Sciences, 2017, 60, pp.225-245. We present recent results on Piecewise Deterministic Markov Processes (PDMPs), involved in biological modeling. PDMPs, first introduced in the probabilistic literature by Davis (1984), are a very general class of Markov processes and are being increasingly popular in biological applications. They also give new interesting challenges from the theoretical point of view. We give here different examples on the long time behavior of switching Markov models applied to population dynamics, on uniform sampling in general branching models applied to structured population dynamic, on time scale separation in integrate-and-fire models used in neuroscience, and, finally, on moment calculus in stochastic models of gene expression. (10.1051/proc/201760225)
  DOI : 10.1051/proc/201760225
• Galerkin approximations of nonlinear optimal control problems in Hilbert spaces
  
  • Chekroun Mickaël D.
  • Kröner Axel
  • Liu Honghu
  Electronic Journal of Differential Equations, Texas State University, Department of Mathematics, 2017, 2017 (189), pp.1-40. Nonlinear optimal control problems in Hilbert spaces are considered for which we derive approximation theorems for Galerkin approximations. Approximation theorems are available in the literature. The originality of our approach relies on the identification of a set of natural assumptions that allows us to deal with a broad class of nonlinear evolution equations and cost functionals for which we derive convergence of the value functions associated with the optimal control problem of the Galerkin approximations. This convergence result holds for a broad class of nonlinear control strategies as well. In particular, we show that the framework applies to the optimal control of semilinear heat equations posed on a general compact manifold without boundary. The framework is then shown to apply to geoengineering and mitigation of greenhouse gas emissions formulated for the first time in terms of optimal control of energy balance climate models posed on the sphere $\mathbb{S}^2$.
• Numerical analysis of the Gross-Pitaevskii Equation with a randomly varying potential in time
  
  • Poncet Romain
  , 2017. The Gross-Pitaevskii equation with white noise in time perturbations of the harmonic potential is considered. In this article we define a Crank-Nicolson scheme based on a spectral discretization and we show the convergence of this scheme in the case of cubic non-linearity and when the exact solution is uniquely defined and global in time. We prove that the strong order of convergence in probability is at least one.
• MCMC design-based non-parametric regression for rare event. Application to nested risk computation.
  
  • Fort Gersende
  • Gobet Emmanuel
  • Moulines Éric
  Monte Carlo Methods and Applications, De Gruyter, 2017.