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

Rafael Ceja Ayala; Isaac Harris; Andreas Kleefeld

doi:10.3934/ipi.2023051

Article Contents

2024, Volume 18, Issue 3: 708-729. Doi: 10.3934/ipi.2023051

This issue Previous Article Recovering a function from spherical means in 3D using local data Next Article Uniqueness in inverse scattering problems with phaseless near-field data generated by superpositions of two incident plane waves at a fixed frequency

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

1.
Department of Mathematics Purdue University West Lafayette, IN 47907, USA
2.
Forschungszentrum Jülich GmbH, Jülich Supercomputing Centre, Wilhelm-Johnen-Straße, 52425 Jülich, Germany
3.
University of Applied Sciences Aachen, Faculty of Medical Engineering and Technomathematics, Heinrich-Mußmann-Str. 1, 52428 Jülich, Germany

^*Corresponding author: Isaac Harris
^*Corresponding author: Isaac Harris

Received: May 2023

Revised: September 2023

Early access: December 2023

Published: June 2024

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

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Abstract

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.

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

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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

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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

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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

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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

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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

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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

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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

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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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