The interior inverse scattering problem for a two-layered cavity using the Bayesian method

Yin Yunwen; Yin Weishi; Meng Pinchao; Liu Hongyu

doi:10.3934/ipi.2021069

Article Contents

2022, Volume 16, Issue 4: 673-690. Doi: 10.3934/ipi.2021069

This issue Previous Article Quasiconformal model with CNN features for large deformation image registration Next Article The domain derivative for semilinear elliptic inverse obstacle problems

The interior inverse scattering problem for a two-layered cavity using the Bayesian method

1.
School of Mathematics and Statistics, Changchun University of Science and Technology, Changchun, 130022, China
2.
Department of Mathematics, City University of Hong Kong, Kowloon, Hong Kong SAR, China

^* Corresponding author: Weishi Yin
^* Corresponding author: Weishi Yin

Received: June 2021

Revised: September 2021

Early access: November 2021

Published: August 2022

Abstract / Introduction Full Text(HTML) Figure(9) Related Papers Cited by

Abstract

In this paper, the Bayesian method is proposed for the interior inverse scattering problem to reconstruct the interface of a two-layered cavity. The scattered field is measured by the point sources located on a closed curve inside the interior interface. The well-posedness of the posterior distribution in the Bayesian framework is proved. The Markov Chain Monte Carlo algorithm is employed to explore the posterior density. Some numerical experiments are presented to demonstrate the effectiveness of the proposed method.

Keywords:

Mathematics Subject Classification: Primary: 78A46, 35R30, 65N21; Secondary: 62F15.

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Figure 1. A schematic illustration of the two-layered cavity scattering problem

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Figure 2. Recoveries of the circle-shaped interface $ \Gamma_{0} $ by the Bayesian method with $ \sigma = 0.005,0.01,0.05 $, respectively

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Figure 3. The Markov chains of the coefficients $ q_{0},a_{1} $ and $ b_{1} $ with $ \sigma = 0.005,0.01,0.05 $, respectively

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Figure 4. Recoveries of the peanut-shaped interface $ \Gamma_{0} $ by the Bayesian method with $ \sigma = 0.005,0.01,0.05 $, respectively

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Figure 5. The Markov chains of the coefficients $ q_{0},a_{1} $ and $ b_{1} $ with $ \sigma = 0.005,0.01,0.05 $, respectively

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Figure 6. Recoveries of the peanut-shaped interface $ \Gamma_{0} $ by the Bayesian method with $ \rho = 0.3,0.2,0.1 $, respectively

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Figure 7. The Markov chains of the coefficients $ q_{0},a_{1} $ and $ b_{1} $ with $ \rho = 0.3,0.2,0.1 $, respectively

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Figure 8. Recoveries of the peanut-shaped interface $ \Gamma_{0} $ by the Bayesian method with $ \gamma_{1},\gamma_{2} $ and $ \gamma_{3} $, respectively

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Figure 9. The Markov chains of the coefficients $ q_{0},a_{1} $ and $ b_{1} $ with $ \gamma_{1},\gamma_{2} $ and $ \gamma_{3} $, respectively

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