Publications

2018

• Maximal lower bounds in the Löwner order
  
  • Stott Nikolas
  Proceedings of the American Mathematical Society, American Mathematical Society, 2018. We show that the set of maximal lower bounds of two symmetric matrices with respect to the L\"owner order can be identified to the quotient set O(p,q)/(O(p)×O(q)). Here, (p,q) denotes the inertia of the difference of the two matrices, O(p) is the p-th orthogonal group, and O(p,q) is the indefinite orthogonal group arising from a quadratic form with inertia (p,q). We also show that a similar result holds for positive semidefinite maximal lower bounds with maximal rank of two positive semidefinite matrices. We exhibit a correspondence between the maximal lower bounds C of two matrices A,B and certain pairs of subspaces, describing the directions on which the quadratic form associated with C is tangent to the one associated with A or B. The present results refines a theorem from Kadison that characterizes the existence of the infimum of two symmetric matrices and a theorem from Moreland, Gudder and Ando on the existence of the positive semidefinite infimum of two positive semidefinite matrices. (10.1090/proc/13785)
  DOI : 10.1090/proc/13785
• Monochromatic identities for the Green function and uniqueness results for passive imaging
  
  • Agaltsov Alexey
  • Hohage Thorsten
  • Novikov Roman
  SIAM Journal on Applied Mathematics, Society for Industrial and Applied Mathematics, 2018, 78 (5), pp.2865–2890. For many wave propagation problems with random sources it has been demonstrated that cross correlations of wave fields are proportional to the imaginary part of the Green function of the underlying wave equation. This leads to the inverse problem to recover coefficients of a wave equation from the imaginary part of the Green function on some measurement manifold. In this paper we prove, in particular, local uniqueness results for the Schrödinger equation with one frequency and for the acoustic wave equation with unknown density and sound speed and two frequencies. As the main tool of our analysis, we establish new algebraic identities between the real and the imaginary part of Green's function, which in contrast to the well-known Kramers-Kronig relations involve only one frequency. (10.1137/18M1182218)
  DOI : 10.1137/18M1182218
• The Asymptotic of Transmission Eigenvalues for a Domain with a Thin Coating
  
  • Boujlida Hanen
  • Haddar Houssem
  • Khenissi Moez
  SIAM Journal on Applied Mathematics, Society for Industrial and Applied Mathematics, 2018, 78 (5), pp.2348-2369. We consider the transmission eigenvalue problem for a medium surrounded by a thin layer of inhomogeneous material with different refractive index. We derive explicit asymptotic expansion for the transmission eigenvalues with respect to the thickness of the thin layer. We prove error estimate for the asymptotic expansion up to order 1 for simple eigenvalues. This expansion can be used to obtain explicit expressions for constant index of refraction.
• Drift Theory in Continuous Search Spaces: Expected Hitting Time of the (1+1)-ES with 1/5 Success Rule
  
  • Akimoto Youhei
  • Auger Anne
  • Glasmachers Tobias
  , 2018. This paper explores the use of the standard approach for proving runtime bounds in discrete domains---often referred to as drift analysis---in the context of optimization on a continuous domain. Using this framework we analyze the (1+1) Evolution Strategy with one-fifth success rule on the sphere function. To deal with potential functions that are not lower-bounded, we formulate novel drift theorems. We then use the theorems to prove bounds on the expected hitting time to reach a certain target fitness in finite dimension $d$. The bounds are akin to linear convergence. We then study the dependency of the different terms on $d$ proving a convergence rate dependency of $\Theta(1/d)$. Our results constitute the first non-asymptotic analysis for the algorithm considered as well as the first explicit application of drift analysis to a randomized search heuristic with continuous domain.
• Efficient semiparametric estimation and model selection for multidimensional mixtures
  
  • Gassiat Elisabeth
  • Rousseau Judith
  • Vernet Elodie
  Electronic Journal of Statistics, Shaker Heights, OH : Institute of Mathematical Statistics, 2018, 12 (1), pp.703-740. In this paper, we consider nonparametric multidimensional finite mixture models and we are interested in the semiparametric estimation of the population weights. Here, the i.i.d. observations are assumed to have at least three components which are independent given the population. We approximate the semiparametric model by projecting the conditional distributions on step functions associated to some partition. Our first main result is that if we refine the partition slowly enough, the associated sequence of maximum likelihood estimators of the weights is asymptotically efficient, and the posterior distribution of the weights, when using a Bayesian procedure, satisfies a semiparametric Bernstein von Mises theorem. We then propose a cross-validation like procedure to select the partition in a finite horizon. Our second main result is that the proposed procedure satisfies an oracle inequality. Numerical experiments on simulated data illustrate our theoretical results.
• Non reflection and perfect reflection via Fano resonance in waveguides
  
  • Chesnel Lucas
  • Nazarov Sergei A
  Communications in Mathematical Sciences, International Press, 2018. We investigate a time-harmonic wave problem in a waveguide. By means of asymptotic analysis techniques, we justify the so-called Fano resonance phenomenon. More precisely, we show that the scattering matrix considered as a function of a geometrical parameter ε and of the frequency λ is in general not continuous at a point $(ε, λ) = (0,λ_0$) where trapped modes exist. In particular, we prove that for a given $ε = 0$ small, the scattering matrix exhibits a rapid change for frequencies varying in a neighbourhood of $λ_0$. We use this property to construct examples of waveguides such that the energy of an incident wave propagating through the structure is perfectly transmitted (non reflection) or perfectly reflected in monomode regime. We provide numerical results to illustrate our theorems.