Imaging junctions of waveguides

Laurent Bourgeois; Jean-François Fritsch; Arnaud Recoquillay

doi:10.3934/ipi.2020065

Article Contents

2021, Volume 15, Issue 2: 285-314. Doi: 10.3934/ipi.2020065

This issue Previous Article Inverse scattering and stability for the biharmonic operator Next Article A new variational approach based on level-set function for convex hull problem with outliers

Imaging junctions of waveguides

1.
Laboratoire POEMS, ENSTA Paris, 828 Boulevard des Maréchaux, 91120 Palaiseau, France
2.
Université Paris-Saclay, CEA, LIST, F-91120 Palaiseau, France

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

Received: March 31, 2020

Revised: August 31, 2020

Early access: November 2020

Published: April 2021

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Abstract

In this paper we address the identification of defects by the Linear Sampling Method in half-waveguides which are related to each other by junctions. Firstly a waveguide which is characterized by an abrupt change of properties is considered, secondly the more difficult case of several half-waveguides related to each other by a junction of complex geometry. Our approach is illustrated by some two-dimensional numerical experiments.

Keywords:

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

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Figure 1. Waveguide with an abrupt change of properties

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Figure 2. Obstacles within the waveguide

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Figure 15. Exact data on the section $ \Sigma^0 $. Left: $ R^0 = 1.1 $. Right: $ R^0 = 3 $

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Figure 3. A waveguide with a transition zone (the domain $ B_R $ is hatched)

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Figure 4. A junction of three half-waveguides (the domain $ B $ is hatched)

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Figure 5. Full-scattering, $ \kappa = 40 $ ($ P = 13 $) and $ \tilde{\kappa} = 60 $ ($ \tilde{P} = 20 $). Top left: obstacle 3 and exact data. Top right: obstacle 3 and noisy data. Bottom left: obstacle 4 and exact data. Bottom right: obstacle 4 and noisy data

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Figure 6. Back-scattering for obstacle 1. Top left: $ \kappa = 40 $ ($ P = 13 $) and $ \tilde{\kappa} = 20 $ ($ \tilde{P} = 7 $), exact data. Top right: $ \kappa = 40 $ and $ \tilde{\kappa} = 20 $, noisy data. Middle left: $ \kappa = \tilde{\kappa} = 40 $ ($ P = \tilde{P} = 13 $), exact data. Middle right: $ \kappa = \tilde{\kappa} = 40 $, noisy data. Bottom left: $ \kappa = 40 $ ($ P = 13 $) and $ \tilde{\kappa} = 60 $ ($ \tilde{P} = 20 $), exact data. Bottom right: $ \kappa = 40 $ and $ \tilde{\kappa} = 60 $, noisy data

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Figure 7. Back-scattering for obstacle 2. Top left: $ \kappa = 40 $ ($ P = 13 $) and $ \tilde{\kappa} = 20 $ ($ P = 7 $), exact data. Top right: $ \kappa = 40 $ and $ \tilde{\kappa} = 20 $, noisy data. Middle left: $ \kappa = \tilde{\kappa} = 40 $ ($ P = \tilde{P} = 13 $), exact data. Middle right: $ \kappa = \tilde{\kappa} = 40 $, noisy data. Bottom left: $ \kappa = 40 $ ($ P = 13 $) and $ \tilde{\kappa} = 60 $ ($ \tilde{P} = 20 $), exact data. Bottom right: $ \kappa = 40 $ and $ \tilde{\kappa} = 60 $, noisy data

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Figure 8. Back-scattering for obstacle 4. Top left: $ \kappa = 40 $ ($ P = 13 $) and $ \tilde{\kappa} = 20 $ ($ \tilde{P} = 7 $), exact data. Top right: $ \kappa = 40 $ and $ \tilde{\kappa} = 20 $, noisy data. Middle left: $ \kappa = \tilde{\kappa} = 40 $ ($ P = \tilde{P} = 13 $), exact data. Middle right: $ \kappa = \tilde{\kappa} = 40 $, noisy data. Bottom left: $ \kappa = 40 $ ($ P = 13 $) and $ \tilde{\kappa} = 60 $ ($ \tilde{P} = 20 $), exact data. Bottom right: $ \kappa = 40 $ and $ \tilde{\kappa} = 60 $, noisy data

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Figure 9. Full-scattering, obstacle 3, $ \kappa = \tilde{\kappa} = 40 $, $ h = 0.65 $ ($ P = 9 $) and $ \tilde{h} = 1 $ ($ \tilde{P} = 13 $). Left: exact data. Right: noisy data

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Figure 10. Back-scattering for obstacle 1, $ \kappa = \tilde{\kappa} = 30 $ and $ h>\tilde{h} $. Top left: $ h = 1 $ ($ P = 10 $) and $ \tilde{h} = 0.5 $ ($ \tilde{P} = 5 $), exact data. Top right: $ h = 1 $ and $ \tilde{h} = 0.5 $, noisy data. Bottom left: $ h = 1 $ ($ P = 10 $) and $ \tilde{h} = 0.75 $ ($ \tilde{P} = 8 $), exact data. Bottom right: $ h = 1 $ and $ \tilde{h} = 0.75 $, noisy data

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Figure 11. Back-scattering for obstacle 1, $ \kappa = \tilde{\kappa} = 30 $ and $ h<\tilde{h} $. Top left: $ h = 0.5 $ ($ P = 5 $) and $ \tilde{h} = 1 $ ($ \tilde{P} = 10 $), exact data. Top right: $ h = 0.5 $ and $ \tilde{h} = 1 $, noisy data. Bottom left: $ h = 0.75 $ ($ P = 8 $) and $ \tilde{h} = 1 $ ($ \tilde{P} = 10 $), exact data. Bottom right: $ h = 0.75 $ and $ \tilde{h} = 1 $, noisy data

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Figure 12. Back-scattering for obstacle 2, $ \kappa = \tilde{\kappa} = 30 $ and $ h>\tilde{h} $. Top left: $ h = 1 $ ($ P = 10 $) and $ \tilde{h} = 0.5 $ ($ \tilde{P} = 5 $), exact data. Top right: $ h = 1 $ and $ \tilde{h} = 0.5 $, noisy data. Bottom left: $ h = 1 $ ($ P = 10 $) and $ \tilde{h} = 0.75 $ ($ \tilde{P} = 8 $), exact data. Bottom right: $ h = 1 $ and $ \tilde{h} = 0.75 $, noisy data

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Figure 13. Back-scattering for obstacle 2, $ \kappa = \tilde{\kappa} = 30 $ and $ h<\tilde{h} $. Top left: $ h = 0.5 $ ($ P = 5 $) and $ \tilde{h} = 1 $ ($ \tilde{P} = 10 $), exact data. Top right: $ h = 0.5 $ and $ \tilde{h} = 1 $, noisy data. Bottom left: $ h = 0.75 $ ($ P = 8 $) and $ \tilde{h} = 1 $ ($ \tilde{P} = 10 $), exact data. Bottom right: $ h = 0.75 $ and $ \tilde{h} = 1 $, noisy data

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Figure 14. Data on a single half-waveguide. Top left: exact data on section $ \Sigma^0 $. Top right: noisy data on section $ \Sigma^0 $. Middle left: exact data on section $ \Sigma^1 $. Middle right: noisy data on section $ \Sigma^1 $. Bottom left: exact data on section $ \Sigma^2 $. Bottom right: noisy data on section $ \Sigma^2 $

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Figure 16. Top: data on two half-waveguides. Left: exact data on sections $ \Sigma^0 $ and $ \Sigma^1 $. Right: noisy data on sections $ \Sigma^0 $ and $ \Sigma^1 $. Bottom: data on three half-waveguides. Left: exact data on sections $ \Sigma^0 $, $ \Sigma^1 $ and $ \Sigma^2 $. Right: noisy data on sections $ \Sigma^0 $, $ \Sigma^1 $ and $ \Sigma^2 $

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