Publications

2018

• Efficient Bayesian Computation by Proximal Markov Chain Monte Carlo: When Langevin Meets Moreau.
  
  • Durmus Alain
  • Moulines Éric
  • Pereyra Marcelo
  SIAM Journal on Imaging Sciences, Society for Industrial and Applied Mathematics, 2018, 11 (1). In this paper, two new algorithms to sample from possibly non-smooth log-concave probability measures are introduced. These algorithms use Moreau-Yosida envelope combined with the Euler-Maruyama discretization of Langevin diffusions. They are applied to a de-convolution problem in image processing, which shows that they can be practically used in a high dimensional setting. Finally, non-asymptotic bounds for one of the proposed methods are derived. These bounds follow from non-asymptotic results for ULA applied to probability measures with a convex continuously differentiable log-density with respect to the Lebesgue measure. (10.1137/16M110834)
  DOI : 10.1137/16M110834
• Volume Viscosity and Internal Energy Relaxation: Error Estimates
  
  • Giovangigli Vincent
  • Yong Wen An
  Nonlinear Analysis: Real World Applications, Elsevier, 2018. We investigate the fast relaxation of internal energy in nonequilibrium gas models derived from the kinetic theory of gases. We establish uniform a priori estimates and existence theorems for symmetric hyperbolic-parabolic systems of partial differential equations with small second order terms and stiff sources. We prove local in time error estimates between the out of equilibrium solution and the one-temperature equilibrium fluid solution for well prepared data and justify the apparition of volume viscosity terms.
• 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.
• 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
• 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.