Extended sampling method for interior inverse scattering problems

Fang Zeng

doi:10.3934/ipi.2020033

Article Contents

2020, Volume 14, Issue 4: 719-731. Doi: 10.3934/ipi.2020033

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Extended sampling method for interior inverse scattering problems

Fang Zeng^,

College of Mathematics and Statistics, Chongqing University, Chongqing 400044, China

^* Corresponding author: fzeng@cqu.edu.cn
^* Corresponding author: fzeng@cqu.edu.cn

Received: September 2019

Revised: February 2020

Published: May 2020

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Abstract

We consider an interior inverse scattering problem of reconstructing the shape of a cavity. The measurements are the scattered fields on a curve inside the cavity due to only one point source. In this paper, we employ the extending sampling method to reconstruct the cavity based on limited data. Numerical examples are provided to show the effectiveness of the method.

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Mathematics Subject Classification: Primary: 45Q05, 65N20, 74J20.

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Figure 1. Explicative figure. Relative location of the reference cavities $ B_z $ and $ D $

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Figure 2. Explicative figure. The reference cavities $ B_z $ marked with different sampling points $ z $ along different (left side) or same (right side) polar angle direction

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Figure 3. The contour plots of the indicator functions $ I_1(z) $ (left) and $ I_2(z) $ (right). Exact boundary $ \partial D $, measurement location $ C $ and point source $ x_0 $ are shown in red

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Figure 4. The contour plots of the indicator functions $ I_1(z) $ (left) and $ I_2(z) $ (right). Exact boundary $ \partial D $, measurement location $ C $ and point source $ x_0 $ are shown in red

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Figure 5. The contour plots of the indicator functions $ I_1(z) $ (left) and $ I_2(z) $ (right). Exact boundary $ \partial D $, measurement location $ C $ and point source $ x_0 $ are shown in red

Download: Full-size image PowerPoint slide

Figure 6. The contour plots of the indicator functions $ I_1(z) $ (left) and $ I_2(z) $ (right). Exact boundary $ \partial D $, measurement location $ C $ and point source $ x_0 $ are shown in red

Download: Full-size image PowerPoint slide

Figure 7. The contour plots of the indicator functions $ I_1(z) $ (left) and $ I_2(z) $ (right). Exact boundary $ \partial D $, measurement location $ C $ and point source $ x_0 $ are shown in red

Download: Full-size image PowerPoint slide

Figure 8. The contour plots of the indicator functions $ I_1(z) $ (left) and $ I_2(z) $ (right). Exact boundary $ \partial D $, measurement location $ C $ and point source $ x_0 $ are shown in red

Download: Full-size image PowerPoint slide

Figure 9. The contour plots of the indicator functions $ I_1(z) $ (left) and $ I_2(z) $ (right). Exact boundary $ \partial D $, measurement location $ C $ and point source $ x_0 $ are shown in red

Download: Full-size image PowerPoint slide

Figure 10. The contour plots of the indicator functions $ I_1(z) $ (left) and $ I_2(z) $ (right). Exact boundary $ \partial D $, measurement location $ C $ and point source $ x_0 $ are shown in red

Download: Full-size image PowerPoint slide

Figure 11. The contour plots of the indicator functions $ I_1(z) $ (left) and $ I_2(z) $ (right). Exact boundary $ \partial D $, measurement location $ C $ and point source $ x_0 $ are shown in red

Download: Full-size image PowerPoint slide

Figure 12. The contour plots of the indicator functions $ I_1(z) $ (left) and $ I_2(z) $ (right). Exact boundary $ \partial D $, measurement location $ C $ and point source $ x_0 $ are shown in red

Download: Full-size image PowerPoint slide

Figure 13. The contour plots of the indicator functions $ I_1(z) $ (left) and $ I_2(z) $ (right). Exact boundary $ \partial D $, measurement location $ C $ and point source $ x_0 $ are shown in red

Download: Full-size image PowerPoint slide

Figure 14. The contour plots of the indicator functions $ I_1(z) $ (left) and $ I_2(z) $ (right). Exact boundary $ \partial D $, measurement location $ C $ and point source $ x_0 $ are shown in red

Download: Full-size image PowerPoint slide

Figure 15. The contour plots of the indicator functions $ I_1(z) $. Exact boundaries $ \partial D $, measurement locations $ C $ and point sources $ x_0 $ are shown in red

Download: Full-size image PowerPoint slide

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