# American Institute of Mathematical Sciences

October  2021, 15(5): 1077-1097. doi: 10.3934/ipi.2021029

## 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

Received  May 2020 Revised  March 2021 Published  October 2021 Early access  May 2021

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.

Citation: Jiangfeng Huang, Zhiliang Deng, Liwei Xu. A Bayesian level set method for an inverse medium scattering problem in acoustics. Inverse Problems & Imaging, 2021, 15 (5) : 1077-1097. doi: 10.3934/ipi.2021029
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The geometry setting for the scattering problem
Samples from the prior with $\alpha = 2,3,4$, for $\tau = 10$
Samples from the prior with inverse length scale $\tau = 10,\frac{20}{3},5$, for $\alpha = 3$
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 $\tau = 20, 10, 5$
The standard deviation for the regular Bayesian method (top block) and the Bayesian level set method (bottom block) with $\tau = 20, 10, 5$
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 $\tau = 20, 10, 5$
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 $\tau = 20, 10, 5$
The standard deviation for the regular Bayesian method (top block) and the Bayesian level set method (bottom block) with $\tau = 20, 10, 5$
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 $\tau = 20, 10, 5$
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 $\tau = 10, \frac{20}{3}, 5$
The standard deviation for the regular Bayesian method (top block) and the Bayesian level set method (bottom block) with $\tau = 10, \frac{20}{3}, 5$
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 $\tau = 10, 20/3, 5$
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 $\alpha = 2, 3, 4$
The standard deviation for the regular Bayesian method (top block) and the Bayesian level set method (bottom block) with $\alpha = 2, 3, 4$
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 $\alpha = 2, 3, 4$
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
 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
 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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