Inverse Scattering Problem

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Otmar Scherzer - One of the best experts on this subject based on the ideXlab platform.

Michael V Klibanov - One of the best experts on this subject based on the ideXlab platform.

  • convexification method for an Inverse Scattering Problem and its performance for experimental backscatter data for buried targets
    Siam Journal on Imaging Sciences, 2019
    Co-Authors: Michael V Klibanov, Aleksandr E Kolesov, Dinhliem Nguyen
    Abstract:

    We present in this paper a novel numerical reconstruction method for solving a three-dimensional Inverse Scattering Problem with Scattering data generated by a single direction of the incident plan...

  • convexification of a 3 d coefficient Inverse Scattering Problem
    Computers & Mathematics With Applications, 2019
    Co-Authors: Michael V Klibanov, Aleksandr E Kolesov
    Abstract:

    Abstract A version of the so-called “convexification” numerical method for a coefficient Inverse Scattering Problem for the 3D Helmholtz equation is developed analytically and tested numerically. BackScattering data are used, which result from a single direction of the propagation of the incident plane wave on an interval of frequencies. The method converges globally. The idea is to construct a weighted Tikhonov-like functional. The key element of this functional is the presence of the so-called Carleman Weight Function (CWF). This is the function which is involved in the Carleman estimate for the Laplace operator. This functional is strictly convex on any appropriate ball in a Hilbert space for an appropriate choice of the parameters of the CWF. Thus, both the absence of local minima and convergence of minimizers to the exact solution are guaranteed. Numerical tests demonstrate a good performance of the resulting algorithm. Unlikeprevious the so-called tail functions globally convergent method, we neither do not impose the smallness assumption of the interval of wavenumbers, nor we do not iterate with respect to the so-called tail functions.

  • uniqueness of a 3 d coefficient Inverse Scattering Problem without the phase information
    Inverse Problems, 2017
    Co-Authors: Michael V Klibanov, Vladimir G Romanov
    Abstract:

    We use a new method to prove the uniqueness theorem for a coefficient Inverse Scattering Problem without the phase information for the 3-D Helmholtz equation. We consider the case when only the modulus of the scattered wave field is measured and the phase is not measured. The spatially distributed refractive index is the subject of interest in this Problem. Applications of this Problem are in imaging of nanostructures and biological cells.

  • a phaseless Inverse Scattering Problem for the 3 d helmholtz equation
    Inverse Problems and Imaging, 2017
    Co-Authors: Michael V Klibanov
    Abstract:

    An Inverse Scattering Problem for the 3-D Helmholtz equation is considered. Only the modulus of the complex valued scattered wave field is assumed to be measured and the phase is not measured. This Problem naturally arises in the lensless quality control of fabricated nanostructures. Uniqueness theorem is proved.

  • nanostructures imaging via numerical solution of a 3 d Inverse Scattering Problem without the phase information
    Applied Numerical Mathematics, 2016
    Co-Authors: Michael V Klibanov, Loc Hoang Nguyen, Kejia Pan
    Abstract:

    Abstract Inverse Scattering Problems without the phase information arise in imaging of nanostructures, whose sizes are hundreds of nanometers, as well as in imaging of biological cells. The governing equation is the 3-D generalized Helmholtz equation with the unknown coefficient, which represents the spatially distributed dielectric constant. It is assumed in the classical Inverse Scattering Problem that both the modulus and the phase of the complex valued scattered wave field are measured outside of a scatterer. Unlike this, it is assumed here that only the modulus of the complex valued scattered wave field is measured on a certain interval of frequencies. The phase is not measured. In this paper a substantially modified reconstruction procedure of [25] is developed and numerically implemented. Ranges of parameters, which are realistic for imaging of nanostructures, are used in numerical examples. Note that numerical studies were not carried out in [25] .

Peter Elbau - One of the best experts on this subject based on the ideXlab platform.

Julian Chaubell - One of the best experts on this subject based on the ideXlab platform.

  • Inverse Scattering Problem for optical coherence tomography
    Optics Letters, 2003
    Co-Authors: Oscar P Bruno, Julian Chaubell
    Abstract:

    We deal with the imaging Problem of determining the internal structure of a body from backscattered laser light and low-coherence interferometry. Specifically, using the interference fringes that result when the backScattering of low-coherence light is made to interfere with the reference beam, we obtain maps detailing the values of the refractive index within the sample. Our approach accounts fully for the statistical nature of the coherence phenomenon; the numerical experiments that we present, which show image reconstructions of high quality, were obtained under noise floors exceeding those present for various experimental setups reported in the literature.

Peter Yuditskii - One of the best experts on this subject based on the ideXlab platform.

  • Inverse Scattering Problem for a special class of canonical systems and non-linear Fourier integral. Part I: asymptotics of eigenfunctions
    Operator Theory: Advances and Applications, 2009
    Co-Authors: Stanislas Kupin, Alexander Volberg, Franz Peherstorfer, Peter Yuditskii
    Abstract:

    An original approach to the Inverse Scattering for Jacobi matrices was recently suggested in Volberg-Yuditskii [2002]. The authors considered quite sophisticated spectral sets (including Cantor sets of positive Lebesgue measure), however they did not take into account the mass point spectrum. This paper follows similar lines for the continuous setting with an absolutely continuous spectrum on the half-axis and a pure point spectrum on the negative half-axis satisfying the Blaschke condition. This leads us to the solution of the Inverse Scattering Problem for a class of canonical systems that generalizes the case of Sturm-Liouville (Schrodinger) operator.

  • on the Inverse Scattering Problem for jacobi matrices with the spectrum on an interval a finite system of intervals or a cantor set of positive length
    arXiv: Spectral Theory, 2001
    Co-Authors: Alexander Volberg, Peter Yuditskii
    Abstract:

    Solving Inverse Scattering Problem for a discrete Sturm-Liouville operator with the fast decreasing potential one gets reflection coefficients $s_\pm$ and invertible operators $I+H_{s_\pm}$, where $ H_{s_\pm}$ is the Hankel operator related to the symbol $s_\pm$. The Marchenko-Fadeev theorem (in the continuous case) and the Guseinov theorem (in the discrete case), guarantees the uniqueness of solution of the Inverse Scattering Problem. In this article we asks the following natural question --- can one find a precise condition guaranteeing that the Inverse Scattering Problem is uniquely solvable and that operators $I+H_{s_\pm}$ are invertible? Can one claim that uniqueness implies invertibility or vise versa? Moreover we are interested here not only in the case of decreasing potential but also in the case of asymptotically almost periodic potentials. So we merege here two mostly developed cases of Inverse Problem for Sturm-Liouville operators: the Inverse Problem with (almost) periodic potential and the Inverse Problem with the fast decreasing potential.

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