Scattering in a partially open waveguide: The inverse problem

Laurent Bourgeois; Jean-François Fritsch; Arnaud Recoquillay

doi:10.3934/ipi.2022052

Article Contents

2023, Volume 17, Issue 2: 463-489. Doi: 10.3934/ipi.2022052

This issue Previous Article Existence and uniqueness in anisotropic conductivity reconstruction with Faraday's law Next Article Super-localisation of a point-like emitter in a resonant environment: Correction of the mirage effect

Scattering in a partially open waveguide: The inverse problem

1.
Laboratoire Poems, CNRS, INRIA, Ensta Paris, Institut Polytechnique de Paris, 828 Boulevard des Maréchaux, 91762 Palaiseau, France
2.
Université Paris-Saclay, CEA, List F-91120, Palaiseau, France

^* Corresponding author: Laurent Bourgeois
^* Corresponding author: Laurent Bourgeois

Received: March 2022

Revised: September 2022

Early access: October 2022

Published: April 2023

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Abstract

In this paper we consider an inverse scattering problem which consists in retrieving obstacles in a partially embedded waveguide in the acoustic case, the measurements being located on the accessible part of the structure. Such accessible part can be considered as a closed waveguide (with a finite cross section), while the embedded part can be considered as an open waveguide (with an infinite cross section). We propose an approximate model of the open waveguide by using Perfectly Matched Layers in order to simplify the resolution of the inverse problem, which is based on a modal formulation of the Linear Sampling Method. Some numerical results show the efficiency of our approach. This paper can be viewed as a continuation of the article [11], which was focused on the forward problem.

Keywords:

Mathematics Subject Classification: Primary: 35J05, 35R25, 35R30, 35R35, 47G10.

Citation:

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Figure 1. Original configuration

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Figure 2. Configuration with PMLs

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Figure 3. Choice of the near-field equation. Top left: near-field equation (46). Top right: near-field equation (47). Bottom: wrong near-field equation associated with the uniform closed waveguide

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Figure 4. Restriction to $ N $ terms in the right-hand side of the near-field equation (46), with $ N \neq \tilde{N} $. Left: $ N = \tilde{N} + 1 $. Right: $ N = \tilde{N} - 1 $

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Figure 5. Influence of the distance between the defects and the interface. Left: one obstacle. Right: two obstacles

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Figure 6. Impact of the amplitude of noise. Top left: 1% noise. Top right: 10% noise. Bottom: 20% noise

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Figure 7. Impact of the frequency. Top left: $ N = 7 $. Top right: $ N = 13 $. Bottom: $ N = 26 $

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Figure 8. One obstacle at least is located in the sheath

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Figure 9. Contour $ C_R $ for the residue theorem

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References

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