Publications

2022

• Fixed-distance multipoint formulas for the scattering amplitude from phaseless measurements
  
  • Novikov Roman
  • Sivkin Vladimir
  Inverse Problems, IOP Publishing, 2022, 38 (2), pp.025012. We give new formulas for finding the complex (phased) scattering amplitude at fixed frequency and angles from absolute values of the scattering wave function at several points $x_1,..., x_m$. In dimension $d\geq 2$, for $m>2$, we significantly improve previous results in the following two respects. First, geometrical constraints on the points needed in previous results are significantly simplified. Essentially, the measurement points $x_j$ are assumed to be on a ray from the origin with fixed distance $\tau=|x_{j+1}- x_j|$, and high order convergence (linearly related to $m$) is achieved as the points move to infinity with fixed $\tau$. Second, our new asymptotic reconstruction formulas are significantly simpler than previous ones. In particular, we continue studies going back to [Novikov, Bull. Sci. Math. 139(8), 923-936, 2015]. (10.1088/1361-6420/ac44db)
  DOI : 10.1088/1361-6420/ac44db
• Points and lines configurations for perpendicular bisectors of convex cyclic polygons
  
  • Melotti Paul
  • Ramassamy Sanjay
  • Thévenin Paul
  The Electronic Journal of Combinatorics, Open Journal Systems, 2022, 29 (1), pp.P1.59. We characterize the topological configurations of points and lines that may arise when placing n points on a circle and drawing the n perpendicular bisectors of the sides of the corresponding convex cyclic n-gon. We also provide a functional central limit theorem describing the shape of a large realizable configuration of points and lines taken uniformly at random among realizable configurations.
• Multiresolution-based mesh adaptation and error control for lattice Boltzmann methods with applications to hyperbolic conservation laws
  
  • Bellotti Thomas
  • Gouarin Loïc
  • Graille Benjamin
  • Massot Marc
  SIAM Journal on Scientific Computing, Society for Industrial and Applied Mathematics, 2022, 44 (4), pp.A2599-A2627. Lattice Boltzmann Methods (LBM) stand out for their simplicity and computational efficiency while offering the possibility of simulating complex phenomena. While they are optimal for Cartesian meshes, adapted meshes have traditionally been a stumbling block since it is difficult to predict the right physics through various levels of meshes. In this work, we design a class of fully adaptive LBM methods with dynamic mesh adaptation and error control relying on multiresolution analysis. This wavelet-based approach allows to adapt the mesh based on the regularity of the solution and leads to a very efficient compression of the solution without loosing its quality and with the preservation of the properties of the original LBM method on the finest grid. This yields a general approach for a large spectrum of schemes and allows precise error bounds, without the need for deep modifications on the reference scheme. An error analysis is proposed. For the purpose of assessing the approach, we conduct a series of test-cases for various schemes and scalar and systems of conservation laws, where solutions with shocks are to be found and local mesh adaptation is especially relevant. Theoretical estimates are retrieved while a reduced memory footprint is observed. It paves the way to an implementation in a multi-dimensional framework and high computational efficiency of the method for both parabolic and hyperbolic equations, which is the subject of a companion paper. (10.1137/21M140256X)
  DOI : 10.1137/21M140256X
• Asymptotic Analysis of a Matrix Latent Decomposition Model
  
  • Mantoux Clément
  • Durrleman​ Stanley
  • Allassonnière Stéphanie
  ESAIM: Probability and Statistics, EDP Sciences, 2022, 26, pp.208-242. Matrix data sets arise in network analysis for medical applications, where each network belongs to a subject and represents a measurable phenotype. These large dimensional data are often modeled using lower-dimensional latent variables, which explain most of the observed variability and can be used for predictive purposes. In this paper, we provide asymptotic convergence guarantees for the estimation of a hierarchical statistical model for matrix data sets. It captures the variability of matrices by modeling a truncation of their eigendecomposition. We show that this model is identifiable, and that consistent Maximum A Posteriori (MAP) estimation can be performed to estimate the distribution of eigenvalues and eigenvectors. The MAP estimator is shown to be asymptotically normal for a restricted version of the model. (10.1051/ps/2022004)
  DOI : 10.1051/ps/2022004
• Firm non-expansive mappings in weak metric spaces
  
  • Gutiérrez Armando W.
  • Walsh Cormac
  Archiv der Mathematik, Springer Verlag, 2022,  119, pp.389-400. We introduce the notion of firm non-expansive mapping in weak metric spaces, extending previous work for Banach spaces and certain geodesic spaces. We prove that, for firm non-expansive mappings, the minimal displacement, the linear rate of escape, and the asymptotic step size are all equal. This generalises a theorem by Reich and Shafrir.
• Weak Langmuir turbulence in disordered multimode optical fibers
  
  • Baudin Kilian
  • Garnier Josselin
  • Fusaro Adrien
  • Berti Nicolas
  • Millot Guy
  • Picozzi Antonio
  Physical Review A, American Physical Society, 2022, 105 (1), pp.013528. We consider the propagation of temporally incoherent waves in multimode optical fibers (MMFs) in the framework of the multimode nonlinear Schrödinger (NLS) equation accounting for the impact of the natural structural disorder that affects light propagation in standard MMFs (random mode coupling and polarization fluctuations). By averaging the dynamics over the fast disordered fluctuations, we derive a Manakov equation from the multimode NLS equation, which reveals that the Raman effect introduces a previously unrecognized nonlinear coupling among the modes. Applying the wave turbulence theory on the Manakov equation, we derive a very simple scalar kinetic equation describing the evolution of the multimode incoherent waves. The structure of the kinetic equation is analogous to that developed in plasma physics to describe weak Langmuir turbulence. The extreme simplicity of the derived kinetic equation provides physical insight into the multimode incoherent wave dynamics. It reveals the existence of different collective behaviors where all modes self-consistently form a multimode spectral incoherent soliton state. Such an incoherent soliton can exhibit a discrete behavior characterized by collective synchronized spectral oscillations in frequency space. The theory is validated by accurate numerical simulations: The simulations of the generalized multimode NLS equation are found in quantitative agreement with those of the derived scalar kinetic equation without using adjustable parameters. (10.1103/PhysRevA.105.013528)
  DOI : 10.1103/PhysRevA.105.013528
• EV-GAN: Simulation of extreme events with ReLU neural networks
  
  • Allouche Michaël
  • Girard Stéphane
  • Gobet Emmanuel
  Journal of Machine Learning Research, Microtome Publishing, 2022, 23 (150), pp.1--39. Feedforward neural networks based on Rectified linear units (ReLU) cannot efficiently approximate quantile functions which are not bounded, especially in the case of heavy-tailed distributions. We thus propose a new parametrization for the generator of a Generative adversarial network (GAN) adapted to this framework, basing on extreme-value theory. An analysis of the uniform error between the extreme quantile and its GAN approximation is provided: We establish that the rate of convergence of the error is mainly driven by the second-order parameter of the data distribution. The above results are illustrated on simulated data and real financial data. It appears that our approach outperforms the classical GAN in a wide range of situations including high-dimensional and dependent data.
• Leveraging Deep Learning for Efficient Explicit MPC of High-Dimensional and Non-linear Chemical Processes
  
  • Shokry Ahmed
  • El Qassime Mehdi Abou
  • Moulines Eric
  , 2022, 51, pp.1171-1176. (10.1016/B978-0-323-95879-0.50196-X)
  DOI : 10.1016/B978-0-323-95879-0.50196-X
• Numerical reconstruction from the Fourier transform on the ball using prolate spheroidal wave functions
  
  • Isaev Mikhail
  • Novikov Roman
  • Sabinin Grigory
  Inverse Problems, IOP Publishing, 2022. We implement numerically formulas of [Isaev, Novikov, arXiv:2107.07882, hal-03289374] for finding a compactly supported function v on R^d , d ≥ 1, from its Fourier transform F[v] given within the ball B_r. For the one-dimensional case, these formulas are based on the theory of prolate spheroidal wave functions, which arise, in particular, in the singular value decomposition of the aforementioned band-limited Fourier transform for d = 1. In multidimensions, these formulas also include inversion of the Radon transform. In particular, we give numerical examples of super-resolution, that is, recovering details beyond the diffraction limit. (10.1088/1361-6420/ac87cb)
  DOI : 10.1088/1361-6420/ac87cb