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Direct sampling method via Landweber iteration for an absorbing scatterer with a conductive boundary

  • *Corresponding author: Isaac Harris

    *Corresponding author: Isaac Harris 

The authors R. Ceja Ayala and I. Harris are supported in part by the NSF DMS grant [2107891]

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  • In this paper, we consider the inverse shape problem of recovering isotropic scatterers with a conductive boundary condition. Here, we assume that the measured far-field data is known at a fixed wave number. Motivated by recent work, we study a new direct sampling indicator based on the Landweber iteration and the factorization method. Therefore, we prove the connection between these reconstruction methods. The method studied here falls under the category of qualitative reconstruction methods where an imaging function is used to recover the absorbing scatterer. We prove stability of our new imaging function as well as derive a discrepancy principle for recovering the regularization parameter. The theoretical results are verified with numerical examples to show how the reconstruction performs by the new Landweber direct sampling method.

    Mathematics Subject Classification: Primary: 35R30; Secondary: 78A46.

    Citation:

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  • Figure 1.  Reconstruction using an interpolating polynomial of degree $ M = 4 $ of peanut region by the Landweber direct sampling method. Images left to right: reconstruction using equidistant points, singular values, and Gaussian quadrature points

    Figure 2.  Reconstruction using an interpolating polynomial of degree $ M = 6 $ of peanut region by the Landweber direct sampling method. Images left to right: reconstruction using equidistant points, singular values, and Gaussian quadrature points

    Figure 3.  Reconstruction using an interpolating polynomial of degree $ M = 4 $ of peanut region by the Landweber direct sampling method with $ 20\% $ noise. Images left to right: reconstruction using equidistant points, singular values, and Gaussian quadrature points

    Figure 4.  Reconstruction using an interpolating polynomial of degree $ M = 4 $ of kite scatterer by the Landweber direct sampling method with $ 10\% $ noise. Images left to right: reconstruction using equidistant points, singular values, and Gaussian quadrature points

    Figure 5.  Reconstruction using an interpolating polynomial of degree $ M = 4 $ of kite scatterer by the Landweber direct sampling method with $ 20\% $ noise. Images left to right: reconstruction using equidistant points, singular values, and Gaussian quadrature points

    Figure 6.  Reconstruction using an interpolating polynomial of degree $ M = 4 $ of circle scatterer by the Landweber iteration method with $ 15\% $ noise. Images left to right: reconstruction using equidistant points, singular values, and Gaussian quadrature points

    Figure 7.  Reconstruction using an interpolating polynomial of degree $ M = 4 $ of circle scatterer by the Landweber iteration method with $ 15\% $ noise. Images left to right: reconstruction using equidistant points, singular values, and Gaussian quadrature points

    Table 1.  Absolute error of the far-field with 64 equidistant incident directions and 64 evaluation for the disk with $ R = 1 $ and the parameters, $ \eta = 2+\mathrm{i} $, and $ n = 4+\mathrm{i} $ for varying number of faces (collocation nodes). The wave numbers are $ k = 2 $, $ k = 4 $, and $ k = 6 $

    $ N_f $ $ \varepsilon_2^{(N_f)} $ $ \varepsilon_4^{(N_f)} $ $ \varepsilon_6^{(N_f)} $
    10 0.82745 9.75548 74.46130
    20 0.01051 0.41988 3.07890
    40 0.00089 0.00556 0.03872
    80 0.00011 0.00018 0.00108
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