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Séminaire Calcul Scientifique et Modélisation

Responsables : Christele Etchegaray et Martin Parisot

  • Le 4 avril 2024 à 14:00
  • Séminaire de Calcul Scientifique et Modélisation
    Salle 2
    Lorenzo Audibert (EDF)
    [Séminaire CSM] An unexpected role of transmission eigenvalues in imaging algorithms
    Transmission eigenvalues are frequencies. Appearing naturally in the study of inverse scattering problems for inhomogeneous media, the associated spectral problem has a deceptively simple formulation but presents a puzzling mathematical structure, in particular it is a non self-adjoint-eigenvalue problem. It triggered a rich literature with a variety of theoretical results on the structure of the spectrum and also on applications for uniqueness results.
    For inverse shape problems, these special frequencies were first considered as bad values, for some imaging algorithms, e.g., sampling methods, as they are associated with non injectivity of the measurement operator. It later turned out that transmission eigenvalues can be used in the design of an imaging algorithm capable of revealing density of cracks in highly fractured domains, thus exceeding the capabilities of traditional approaches to address this problem. This new imaging concept has been further developed to produce average properties of highly heterogeneous scattering media at a fixed frequency, not necessarily a transmission eigenvalue, by encoding a special spectral parameter in the background that acts as transmission eigenvalues.
    While targeting this unexpected additional value of transmission eigenvalues in imaging algorithms, the talk will also provide an opportunity to highlight some key results and open problems related to this active research area.
    This is a joint work with Houssem Haddar, Fioralba Cakoni, Lucas Chesnel, Kevish Napal and Fabien Pourre.
  • Le 11 avril 2024 à 14:00
  • Séminaire de Calcul Scientifique et Modélisation
    Salle 2
    Christian Klingenberg Université de Wurzburg
    .

  • Le 2 mai 2024 à 14:00
  • Séminaire de Calcul Scientifique et Modélisation
    Salle 2
    Victor Michel-Dansac Inria Strasbourg
    .

  • Le 13 juin 2024 à 14:00
  • Séminaire de Calcul Scientifique et Modélisation
    Salle 2
    Firas Dhaouadi University of Trento
    .

  • Le 27 juin 2024 à 14:00
  • Séminaire de Calcul Scientifique et Modélisation
    Salle 2
    Davide Torlo SISSA Trieste
    Structure preserving methods via Global Flux quadrature: divergence-free preservation with continuous Finite Element
    In many problems, the emergence of physical structures and equilibrium solutions, such as divergence-free solutions in contexts like shallow water and magneto-hydrodynamics, poses a significant challenge. A simple linear approximation of such systems that already show these behavior is the linear acoustic system of equations. We focus on Cartesian grid discretizations of such system in 2 dimensions and in the preservation of stationary solutions that arise due to a truly multidimensional balance of terms, which corresponds to the divergence-free solutions for acoustic systems.
    Conventional methods, like the continuous Finite Element SUPG, face limitations in maintaining these structures due to the stabilization techniques employed, which do not effectively vanish when the discrete divergence is zero.
    What we propose is to use the Global Flux procedure, which has proven to be successful in preserving 1-dimensional equilibria [1,2], to define some auxiliary variables guiding a suitable discretization of both the divergence and stabilization operators [3]. This approach enables the natural preservation of divergence-free solutions and more intricate equilibria involving various sources. Moreover, this strategy facilitates the identification of discrete equilibria of the scheme that verify boundary or initial conditions. We use the Deferred Correction time discretization, obtaining explicit arbitrarily high order methods.
    Numerous numerical tests validate the accuracy of our proposed scheme compared to classical approaches. Our method not only excels in preserving (discretely) divergence-free solutions and their perturbations but also maintains the original order of accuracy on smooth solutions.

    [1] Y. Cheng, A. Chertock, M. Herty, A. Kurganov and T. Wu. A new approach for designing moving-water equilibria preserving schemes for the shallow water equations. J. Sci. Comput. 80(1): 538–554, 2019.
    [2] M. Ciallella, D. Torlo and M. Ricchiuto. Arbitrary high order WENO finite volume scheme with flux globalization for moving equilibria preservation. Journal of Scientific Computing, 96(2):53, 2023.
    [3] W. Barsukow, M. Ricchiuto and D. Torlo. Structure preserving methods via Global Flux quadrature: divergence-free preservation with continuous Finite Element. In preparation, 2024.

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