Publications

2021

• Handling Hard Affine SDP Shape Constraints in RKHSs
  
  • Aubin-Frankowski Pierre-Cyril
  • Szabó Zoltán
  , 2021. Shape constraints, such as non-negativity, monotonicity, convexity or supermodularity, play a key role in various applications of machine learning and statistics. However, incorporating this side information into predictive models in a hard way (for example at all points of an interval) for rich function classes is a notoriously challenging problem. We propose a unified and modular convex optimization framework, relying on second-order cone (SOC) tightening, to encode hard affine SDP constraints on function derivatives, for models belonging to vector-valued reproducing kernel Hilbert spaces (vRKHSs). The modular nature of the proposed approach allows to simultaneously handle multiple shape constraints, and to tighten an infinite number of constraints into finitely many. We prove the consistency of the proposed scheme and that of its adaptive variant, leveraging geometric properties of vRKHSs. The efficiency of the approach is illustrated in the context of shape optimization, safety-critical control and econometrics.
• Long-Time Correlations For A Hard-Sphere Gas At Equilibrium
  
  • Bodineau Thierry
  • Gallagher Isabelle
  • Saint-Raymond Laure
  • Simonella Sergio
  Communications on Pure and Applied Mathematics, Wiley, 2021. It has been known since Lanford [22] that the dynamics of a hard sphere gas is described in the low density limit by the Boltzmann equation, at least for short times. The classical strategy of proof fails for longer times, even close to equilibrium. In this paper, we introduce a weak convergence method coupled with a sampling argument to prove that the covariance of the fluctuation field around equilibrium is governed by the linearized Boltzmann equation globally in time (including in diffusive regimes). This method is much more robust and simple than the one devised in [4] which was specific to the 2D case. (10.48550/arXiv.2012.03813)
  DOI : 10.48550/arXiv.2012.03813
• Finite state N-agent and mean field control problems
  
  • Cecchin Alekos
  ESAIM: Control, Optimisation and Calculus of Variations, EDP Sciences, 2021, 27, pp.31. We examine mean field control problems on a finite state space, in continuous time and over a finite time horizon. We characterize the value function of the mean field control problem as the unique viscosity solution of a Hamilton-Jacobi-Bellman equation in the simplex. In absence of any convexity assumption, we exploit this characterization to prove convergence, as N grows, of the value functions of the centralized N-agent optimal control problem to the limit mean field control problem value function, with a convergence rate of order . Then, assuming convexity, we show that the limit value function is smooth and establish propagation of chaos, i.e. convergence of the N-agent optimal trajectories to the unique limiting optimal trajectory, with an explicit rate. (10.1051/cocv/2021032)
  DOI : 10.1051/cocv/2021032
• Kinetic Theory of Chemical Reactions on Crystal Surfaces
  
  • Aoki Kazuo
  • Giovangigli Vincent
  Physica A: Statistical Mechanics and its Applications, Elsevier, 2021, 565, pp.125573. A kinetic theory describing chemical reactions on crystal surfaces is introduced. Kinetic equations are used to model physisorbed-gas particles and chemisorbed particles interacting with fixed potentials and colliding with phonons. The phonons are assumed to be at equilibrium and the physisorbed-gas and chemisorbed species equations are coupled to similar kinetic equations describing crystal atoms on the surface. An arbitrary number of surface species and heterogeneous chemical reactions are considered, covering Langmuir-Hinshelwood as well as Eley-Rideal mechanisms and the species may be polyatomic. A kinetic entropy is introduced for the coupled system and the H theorem is established. Using a fluid scaling and a Chapman-Enskog method, fluid boundary conditions are derived from the kinetic model and involve complex surface chemistry as well as surface tangential multicomponent diffusion. (10.1016/j.physa.2020.125573)
  DOI : 10.1016/j.physa.2020.125573
• Stability estimates for reconstruction from the Fourier transform on the ball
  
  • Isaev Mikhail
  • Novikov Roman
  Journal of Inverse and Ill-posed Problems, De Gruyter, 2021, 29 (3), pp.421–433. Abstract We prove Hölder-logarithmic stability estimates for the problem of finding an integrable function v on ℝ d {{\mathbb{R}}^{d}} with a super-exponential decay at infinity from its Fourier transform ℱ ⁢ v {\mathcal{F}v} given on the ball B r {B_{r}} . These estimates arise from a Hölder-stable extrapolation of ℱ ⁢ v {\mathcal{F}v} from B r {B_{r}} to a larger ball. We also present instability examples showing an optimality of our results. (10.1515/jiip-2020-0106)
  DOI : 10.1515/jiip-2020-0106
• COCO: A Platform for Comparing Continuous Optimizers in a Black-Box Setting
  
  • Hansen Nikolaus
  • Auger Anne
  • Ros Raymond
  • Mersmann Olaf
  • Tušar Tea
  • Brockhoff Dimo
  Optimization Methods and Software, Taylor & Francis, 2021, 36 (1), pp.114-144. We introduce COCO, an open source platform for Comparing Continuous Optimizers in a black-box setting. COCO aims at automatizing the tedious and repetitive task of benchmarking numerical optimization algorithms to the greatest possible extent. The platform and the underlying methodology allow to benchmark in the same framework deterministic and stochastic solvers for both single and multiobjective optimization. We present the rationales behind the (decade-long) development of the platform as a general proposition for guidelines towards better benchmarking. We detail underlying fundamental concepts of COCO such as the definition of a problem as a function instance, the underlying idea of instances, the use of target values, and runtime defined by the number of function calls as the central performance measure. Finally, we give a quick overview of the basic code structure and the currently available test suites. (10.1080/10556788.2020.1808977)
  DOI : 10.1080/10556788.2020.1808977
• Wave Propagation in Periodic and Random Time-Dependent Media
  
  • Garnier Josselin
  Multiscale Modeling and Simulation: A SIAM Interdisciplinary Journal, Society for Industrial and Applied Mathematics, 2021, 19 (3), pp.1190-1211. (10.1137/20M1377734)
  DOI : 10.1137/20M1377734
• Passive Communication with Ambient Noise
  
  • Garnier Josselin
  SIAM Journal on Applied Mathematics, Society for Industrial and Applied Mathematics, 2021, 81 (3), pp.814-833. (10.1137/20M1366848)
  DOI : 10.1137/20M1366848
• Enhanced Backscattering of a partially coherent field from an anisotropic random lossy medium
  
  • Garnier Josselin
  • Sølna Knut
  Discrete and Continuous Dynamical Systems - Series B, American Institute of Mathematical Sciences, 2021, 26 (2), pp.1171-1195. (10.3934/dcdsb.2020158)
  DOI : 10.3934/dcdsb.2020158