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Houssem Haddar - One of the best experts on this subject based on the ideXlab platform.
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new interior Transmission Problem applied to a single floquet bloch mode imaging of local perturbations in periodic media
Inverse Problems, 2019Co-Authors: Fioralba Cakoni, Houssem Haddar, Thiphong NguyenAbstract:This paper considers the imaging of local perturbations of an infinite penetrable periodic layer. A cell of this periodic layer consists of several bounded inhomogeneities situated in a known homogeneous media. We use a differential linear sampling method to reconstruct the support of perturbations without using the Green's function of the periodic layer nor reconstruct the periodic background inhomogeneities. The justification of this imaging method relies on the well-posedeness of a nonstandard interior Transmission Problem, which until now was an open Problem except for the special case when the local perturbation did not intersect the background inhomogeneities. The analysis of this new interior Transmission Problem is the main focus of this paper. We then complete the justification of our inversion method and present some numerical examples that confirm the theoretical behavior of the differential indicator function determining the reconstructable regions in the periodic layer.
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new interior Transmission Problem applied to a single floquet bloch mode imaging of local perturbations in periodic media
arXiv: Mathematical Physics, 2018Co-Authors: Fioralba Cakoni, Houssem Haddar, Thiphong NguyenAbstract:This paper considers the imaging of local perturbations of an infinite penetrable periodic layer. A cell of this periodic layer consists of several bounded inhomogeneities situated in a known homogeneous media. We use \mfied{a differential linear sampling method} to reconstruct the support of perturbations without using the Green's function of the periodic layer nor reconstruct the periodic background inhomogeneities. The justification of this imaging method relies on the well-posedeness of a nonstandard interior Transmission Problem, which until now was an open Problem except for the special case when the local perturbation didn't intersect the background inhomogeneities. The analysis of this new interior Transmission Problem is the main focus of this paper. We then complete the justification of our inversion method and present some numerical examples that confirm the theoretical behavior of the differential indicator function determining the reconstructable regions in the periodic layer.
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Surface integral formulation of the interior Transmission Problem
Journal of Integral Equations and Applications, 2013Co-Authors: Anne Cossonnière, Houssem HaddarAbstract:We consider a surface integral formulation of the so-called interior Transmission Problem that appears in the study of inverse scattering Problems from dielectric inclusions. In the case where the magnetic permeability contrast is zero, the main originality of our approach consists in still using classical potentials for the Helmholtz equation but in weaker trace space solutions. One major outcome of this study is to establish Fredholm properties of the Problem for relaxed assumptions on the material coefficients. For instance we allow the contrast to change sign inside the medium. We also show how one can retrieve discreteness results for Transmission eigenvalues in some particular situations.
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The Electromagnetic Interior Transmission Problem for Regions with Cavities
SIAM Journal on Mathematical Analysis, 2011Co-Authors: Anne Cossonnière, Houssem HaddarAbstract:We consider the electromagnetic interior Transmission Problem in the case when the medium has cavities, i.e. regions in which the index of refraction is the same as in the host medium. We address the configuration where the electromagnetic permeability is constant while the electric permittivity is variable and may be anisotropic. In this case, using appropriate reformulation of the Problem into a fourth order pde, we establish the Fredholm property for this Problem and show that Transmission eigenvalues exist and form a discrete set. Monotonicity properties of the first eigenvalue in terms of the permittivity and the size of the cavity are established.
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The interior Transmission Problem for regions with cavities
SIAM Journal on Mathematical Analysis, 2010Co-Authors: Fioralba Cakoni, David Colton, Houssem HaddarAbstract:We consider the interior Transmission Problem in the case when the inhomogeneous medium has cavities, i.e., regions in which the index of refraction is the same as the host medium. In this case we establish the Fredholm property for this Problem and show that Transmission eigenvalues exist and form a discrete set. We also derive Faber–Krahn-type inequalities for the Transmission eigenvalues.
Fioralba Cakoni - One of the best experts on this subject based on the ideXlab platform.
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new interior Transmission Problem applied to a single floquet bloch mode imaging of local perturbations in periodic media
Inverse Problems, 2019Co-Authors: Fioralba Cakoni, Houssem Haddar, Thiphong NguyenAbstract:This paper considers the imaging of local perturbations of an infinite penetrable periodic layer. A cell of this periodic layer consists of several bounded inhomogeneities situated in a known homogeneous media. We use a differential linear sampling method to reconstruct the support of perturbations without using the Green's function of the periodic layer nor reconstruct the periodic background inhomogeneities. The justification of this imaging method relies on the well-posedeness of a nonstandard interior Transmission Problem, which until now was an open Problem except for the special case when the local perturbation did not intersect the background inhomogeneities. The analysis of this new interior Transmission Problem is the main focus of this paper. We then complete the justification of our inversion method and present some numerical examples that confirm the theoretical behavior of the differential indicator function determining the reconstructable regions in the periodic layer.
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new interior Transmission Problem applied to a single floquet bloch mode imaging of local perturbations in periodic media
arXiv: Mathematical Physics, 2018Co-Authors: Fioralba Cakoni, Houssem Haddar, Thiphong NguyenAbstract:This paper considers the imaging of local perturbations of an infinite penetrable periodic layer. A cell of this periodic layer consists of several bounded inhomogeneities situated in a known homogeneous media. We use \mfied{a differential linear sampling method} to reconstruct the support of perturbations without using the Green's function of the periodic layer nor reconstruct the periodic background inhomogeneities. The justification of this imaging method relies on the well-posedeness of a nonstandard interior Transmission Problem, which until now was an open Problem except for the special case when the local perturbation didn't intersect the background inhomogeneities. The analysis of this new interior Transmission Problem is the main focus of this paper. We then complete the justification of our inversion method and present some numerical examples that confirm the theoretical behavior of the differential indicator function determining the reconstructable regions in the periodic layer.
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The interior Transmission Problem for regions with cavities
SIAM Journal on Mathematical Analysis, 2010Co-Authors: Fioralba Cakoni, David Colton, Houssem HaddarAbstract:We consider the interior Transmission Problem in the case when the inhomogeneous medium has cavities, i.e., regions in which the index of refraction is the same as the host medium. In this case we establish the Fredholm property for this Problem and show that Transmission eigenvalues exist and form a discrete set. We also derive Faber–Krahn-type inequalities for the Transmission eigenvalues.
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A variational approach for the solution of the electromagnetic interior Transmission Problem for anisotropic media
Inverse Problems & Imaging, 2007Co-Authors: Fioralba Cakoni, Houssem HaddarAbstract:The interior Transmission Problem plays a basic role in the study of inverse scattering Problems for inhomogeneous medium. In this paper we study the interior Transmission Problem for the Maxwell equations in the electromagnetic scattering Problem for an anisotropic inhomogeneous object. We use a variational approach which extends the method developed in [15] to the case when the index of refraction is less than one as well as for partially coated scatterers. In addition, we also describe the structure of the Transmission eigenvalues.
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Interior Transmission Problem for Anisotropic Media
Mathematical and Numerical Aspects of Wave Propagation WAVES 2003, 2003Co-Authors: Fioralba Cakoni, Houssem HaddarAbstract:In recent years there has been considerable interest in the inverse scattering Problem for anisotropic medium, particularly in the case of acoustic waves and electromagnetic waves. Due to the lack of uniqueness in determining the constitutive parameters, traditional methods for solving the inverse scattering Problems based on the use of weak scattering approximations or nonlinear optimization techniques are Problematic. On the other hand, since the support is uniquely determined [4], the recently developed linear sampling method [1], [2] for determining the support of the scatterer from the knowledge of the far field pattern at a fixed frequency is ideally suited to solving the inverse scattering Problem for anisotropic medium. The unique determination of the support and the linear sampling method for the scalar anisotropic scattering Problem were first considered in [4] and [1] respectively. The techniques used in the above papers are based on an analysis of a boundary value Problem called the interior Transmission Problem. This analysis is performed only in the case where the norm of the real part of the matrix A that describes the physical properties of the medium is greater than one. The purpose of this paper is to complete the study of the interior Transmission Problem in the case where the norm of Re(A) is less then one. Our results imply that the uniqueness result of [4] and the validity of the linear sampling method [1] are also valid for the case ∥Re(A)∥ ≤ 1.
Angelo Morro - One of the best experts on this subject based on the ideXlab platform.
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Uniqueness of the Solution to the Reflection-Transmission Problem in a Viscoelastic Layer
Mathematical and Numerical Aspects of Wave Propagation WAVES 2003, 2003Co-Authors: Angelo MorroAbstract:Uniqueness is investigated for the solution to the reflection-Transmission Problem in a viscoelastic layer sandwiched between elastic half spaces. The layer and the half-spaces are isotropic but are allowed to be uniaxially inhomogeneous.
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Existence and uniqueness in the reflection-Transmission Problem
Quarterly Journal of Mechanics and Applied Mathematics, 1999Co-Authors: Giacomo Caviglia, Angelo MorroAbstract:The sextic Stroh formalism is applied to the analysis of reflection and\nTransmission Problems for inhomogeneous plane waves at a plane interface\nbetween viscoelastic anisotropic half-spaces. The direction of a wave is\ntaken to be that of the time-averaged energy flux and hence\ncompatibility of the equations for reflection and Transmission\ncoefficients is investigated. Sufficient conditions are determined for\nexistence and uniqueness of the solution to the reflection-Transmission\nProblem. The occurrence of surface or interface waves is shown to\nprovide either incompatibility or non-uniqueness of the solution,\ndepending on the incident wave. The phenomenon of the Brewster angle,\nrelated to a zero reflection coefficient, is also pointed out.
Jaime E. Muñoz Rivera - One of the best experts on this subject based on the ideXlab platform.
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Transmission Problem in Thermoelasticity
Boundary Value Problems, 2011Co-Authors: Margareth S. Alves, Jaime E. Muñoz Rivera, Mauricio Sepúlveda, Octavio Vera VillagránAbstract:We show that the energy to the thermoelastic Transmission Problem decays exponentially as time goes to infinity. We also prove the existence, uniqueness, and regularity of the solution to the system.
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ANALYTICITY OF Transmission Problem TO THERMOELASTIC PLATES
Quarterly of Applied Mathematics, 2010Co-Authors: Hugo D. Fernández Sare, Jaime E. Muñoz RiveraAbstract:In this paper we consider an oscillation model to a plate comprised of two different thermoelastic materials; that is, we study a Transmission Problem to thermoelastic plates. Our main result is to prove that the corresponding semigroup associated to this Problem is of analytic type.
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Stability for a Transmission Problem in thermoelasticity with second sound
Journal of Thermal Stresses, 2008Co-Authors: Hugo D. Fernández Sare, Jaime E. Muñoz Rivera, Reinhard RackeAbstract:We consider a semilinear Transmission Problem for a coupling of an elastic and a thermoelastic material. The heat conduction is modeled by Cattaneo's law removing the physical paradox of infinite propagation speed of signals. The damped, totally hyperbolic system is shown to be exponentially stable, and the existence of a global attractor is shown.
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A nonlinear Transmission Problem with time dependent coefficients.
2007Co-Authors: Eugenio Cabanillas Lapa, Jaime E. Muñoz RiveraAbstract:In this paper we consider the nonlinear Transmission Problem for the wave equation with time dependent coefficients and linear internal damping. We prove global existence and exponential decay of solution. The result is achieved by considering energy like Lyapunov functionals and suitable unique continuation theorem for the wave equation.
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Transmission Problem in thermoelasticity with symmetry
IMA Journal of Applied Mathematics, 2003Co-Authors: Alfredo Marzocchi, Jaime E. Muñoz Rivera, Maria Grazia NasoAbstract:In this paper we show the existence, uniqueness and regularity of the solutions to the thermoelastic Transmission Problem. Moreover, when the solutions are symmetrical we show that the energy decays exponentially as time goes to infinity, no matter how small is the size of the thermoelastic part.
Karl-mikael Perfekt - One of the best experts on this subject based on the ideXlab platform.
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The Transmission Problem on a three-dimensional wedge
Archive for Rational Mechanics and Analysis, 2018Co-Authors: Karl-mikael PerfektAbstract:We consider the Transmission Problem for the Laplace equation on an infinite three-dimensional wedge, determining the complex parameters for which the Problem is well-posed, and characterizing the infinite multiplicity nature of the spectrum. This is carried out in two formulations leading to rather different spectral pictures. One formulation is in terms of square integrable boundary data, the other is in terms of finite energy solutions. We use the layer potential method, which requires the harmonic analysis of a non-commutative non-unimodular group associated with the wedge.
