A Bayesian level set method for an inverse medium scattering problem in acoustics

Jiangfeng Huang; Zhiliang Deng; Liwei Xu

doi:10.3934/ipi.2021029

Article Contents

2021, Volume 15, Issue 5: 1077-1097. Doi: 10.3934/ipi.2021029

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A Bayesian level set method for an inverse medium scattering problem in acoustics

School of Mathematical Sciences, University of Electronic Science and Technology of China, Sichuan 611731, China

^* Corresponding author: Zhiliang Deng
^* Corresponding author: Zhiliang Deng

Received: April 30, 2020

Revised: February 28, 2021

Early access: May 2021

Published: October 2021

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Abstract

In this work, we are interested in the determination of the shape of the scatterer for the two dimensional time harmonic inverse medium scattering problems in acoustics. The scatterer is assumed to be a piecewise constant function with a known value inside inhomogeneities and its shape is represented by the level set functions for which we investigate the information using the Bayesian method. In the Bayesian framework, the solution of the geometric inverse problem is defined as a posterior probability distribution. The well-posedness of the posterior distribution is discussed and the Markov chain Monte Carlo (MCMC) method is applied to generate samples from the posterior distribution. Numerical experiments are presented to demonstrate the effectiveness of the proposed method.

Keywords:

Inverse medium scattering problems,
Helmholtz equations,
Bayesian level set method,
Markov chain Monte Carlo (MCMC) methods.

Mathematics Subject Classification: Primary: 65N21, 62F15, 78A46.

Citation:

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Figure 1. The geometry setting for the scattering problem

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Figure 2. Samples from the prior with $ \alpha = 2,3,4 $, for $ \tau = 10 $

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Figure 3. Samples from the prior with inverse length scale $ \tau = 10,\frac{20}{3},5 $, for $ \alpha = 3 $

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Figure 4. The Top: true scatterer, Second Row: the reconstructions of the scatterer for the regular Bayesian method with $ \tau = 20, 10, 5 $. Third Row: the reconstructions of the scatterer for the Bayesian level set method with $ \tau = 20, 10, 5 $

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Figure 5. The standard deviation for the regular Bayesian method (top block) and the Bayesian level set method (bottom block) with $ \tau = 20, 10, 5 $

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Figure 6. The trace plots, the autocorrelation functions of the data misfit and the relative error for the regular Bayesian method (top block) and the Bayesian level set method (bottom block) with $ \tau = 20, 10, 5 $

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Figure 7. The Top: true scatterer, Second Row: the reconstructions of the scatterer for the regular Bayesian method with $ \tau = 20, 10, 5 $. Third Row: the reconstructions of the scatterer for the Bayesian level set method with $ \tau = 20, 10, 5 $

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Figure 8. The standard deviation for the regular Bayesian method (top block) and the Bayesian level set method (bottom block) with $ \tau = 20, 10, 5 $

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Figure 9. The trace plots, the autocorrelation functions of the data misfit and the relative error for the regular Bayesian method (top block) and the Bayesian level set method (bottom block) with $ \tau = 20, 10, 5 $

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Figure 10. The Top: true scatterer, Second Row: the reconstructions of the scatterer for the regular Bayesian method with $ \tau = 10, \frac{20}{3}, 5 $. Third Row: the reconstructions of the scatterer for the Bayesian level set method with $ \tau = 10, \frac{20}{3}, 5 $

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Figure 11. The standard deviation for the regular Bayesian method (top block) and the Bayesian level set method (bottom block) with $ \tau = 10, \frac{20}{3}, 5 $

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Figure 12. The trace plots, the autocorrelation functions of the data misfit and the relative error for the regular Bayesian method (top block) and the Bayesian level set method (bottom block) with $ \tau = 10, 20/3, 5 $

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Figure 13. The Top: true scatterer, Second Row: the reconstructions of the scatterer for the regular Bayesian method with $ \alpha = 2, 3, 4 $. Third Row: the reconstructions of the scatterer for the Bayesian level set method with $ \alpha = 2, 3, 4 $

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Figure 14. The standard deviation for the regular Bayesian method (top block) and the Bayesian level set method (bottom block) with $ \alpha = 2, 3, 4 $

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Figure 15. The trace plots, the autocorrelation functions of the data misfit and the relative error for the regular Bayesian method (top block) and the Bayesian level set method (bottom block) with $ \alpha = 2, 3, 4 $

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Figure 16. The Top: true scatterer. Second Row: reconstruct the scatterer with single-frequency data (left) and multi-frequency data (right) by using the regular Bayesian method, respectively. Third Row: reconstruct the scatterer with single-frequency data (left) and multi-frequency data (right) by using the Bayesian level set method, respectively

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Algorithm 1: The pCN-MCMC algorithm.

1: Collect the scattered field measured data over all frequencies $ k_{m} $, $ m=1,\cdots, M $ and the incident direction $ \textbf{d}_{j} $, $ j=1,\cdots J $.
2: Set $ s=0 $. Choose an initial state $ \phi^{(0)}\in \mathcal{X} $.
3: For$ s=0 $ to $ N_{s} $ do
4:    Propose $ \psi^{(s)}=\sqrt{1-\beta^{2}}\phi^{(s)}+\beta\xi'^{(s)} $, $ \xi'^{(s)}\sim \mathcal{N}(0,\mathcal{C}_{\alpha,\tau}) $;
5:    Draw $ \theta\sim U[0,1] $
6:   Let $ a(\phi^{(s)},\psi^{(s)}):=\min\{1,\exp(\Phi(\phi^{(s)})-\Phi(\psi^{(s)}))\} $;
7:   if $ \theta\leq a $ then
8:      $ \phi^{(s+1)}= \psi^{(s)} $;
9:   else
10:       $ \phi^{(s+1)} = \phi^{(s)} $;
11:   end if
12: end for

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