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
References:
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H. Haddar and P. Monk, The linear sampling method for solving the electromagnetic inverse medium problem, Inverse Problems, 18 (2002), 891-906.  doi: 10.1088/0266-5611/18/3/323.  Google Scholar

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J. Huang, Z. Deng and L. Xu, Bayesian approach for inverse interior scattering problems with limited aperture., Applicable Analysis, in press, (2020). Google Scholar

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M. A. IglesiasY. Lu and A. M. Stuart, A Bayesian level set method for geometric inverse problems, Interfaces and Free Boundaries, 18 (2016), 181-217.  doi: 10.4171/IFB/362.  Google Scholar

[30]

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[32]

L. Jiang and N. Ou, Bayesian inference using intermediate distribution based on coarse multiscale model for time fractional diffusion equations, SIAM Journal on Multiscale Modeling Simulation, 16 (2018), 327-355.  doi: 10.1137/17M1110535.  Google Scholar

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A. Kirsch, The MUSIC-algorithm and the factorization method in inverse scattering theory for inhomogeneous media, Inverse Problems, 18 (2002), 1025-1040.  doi: 10.1088/0266-5611/18/4/306.  Google Scholar

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[39]

Z. LiZ. Deng and J. Sun, Extended-sampling-Bayesian method for limited aperture inverse scattering problems, SIAM Journal on Imaging Sciences, 13 (2020), 422-444.  doi: 10.1137/19M1270501.  Google Scholar

[40]

Z. Li, Y. Liu, J. Sun and L. Xu, Quality-Bayesian approach to inverse acoustic source problems with partial data, SIAM Journal on Scientific Computing, 43 (2021), A1062–A1080. doi: 10.1137/20M132345X.  Google Scholar

[41]

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J. Liu, Y. Liu and J. Sun, An inverse medium problem using Stekloff eigenvalues and a Bayesian approach, Inverse Problems, 35 (2019), 094004, 20 pp. doi: 10.1088/1361-6420/ab1be9.  Google Scholar

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J. Martin, L. C. Wilcox, C. Burstedde and O. Ghattas., A stochastic Newton MCMC method for large-scale statistical inverse problems with application to seismic inversion, SIAM Journal on Scientific Computing, 34 (2012), A1460–A1487. doi: 10.1137/110845598.  Google Scholar

[44]

F. Natterer and F. Wiibbelmg, A propagation-backpropagation method for ultrasound tomography, Inverse Problems, 11 (1995), 1225-1232.  doi: 10.1088/0266-5611/11/6/007.  Google Scholar

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L. RoininenJ. M. J. Huttunen and S. Lasanen, Whittle-Matérn priors for Bayesian statistical inversion with applications in electrical impedance tomography, Inverse Problems and Imaging, 8 (2014), 561-586.  doi: 10.3934/ipi.2014.8.561.  Google Scholar

[49]

A. M. Stuart, Inverse problems: A Bayesian perspective, Acta Numerica, 19 (2010), 451-559.  doi: 10.1017/S0962492910000061.  Google Scholar

[50]

X.-C. Tai and H. Li, A piecewise constant level set method for elliptic inverse problems, Applied Numerical Mathematics, 57 (2007), 686-696.  doi: 10.1016/j.apnum.2006.07.010.  Google Scholar

[51]

M. Vögler, Reconstruction of the three-dimensional refractive index in electromagnetic scattering by using a propagation-backpropagation method, Inverse Problems, 19 (2003), 739-753.  doi: 10.1088/0266-5611/19/3/316.  Google Scholar

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X. YangZ. Deng and J. Wang, An ensemble Kalman filter approach based on level set parameterization for acoustic source identification using multiple frequency information, Communications in Mathematical Research, 36 (2020), 211-228.  doi: 10.4208/cmr.2020-0011.  Google Scholar

show all references

References:
[1]

G. BaoS. Hou and P. Li, Recent studies on inverse medium scattering problems, Modeling and Computations in Electromagnetics, Lect. Notes Comput. Sci. Eng., Springer, Berlin, 59 (2008), 165-186.  doi: 10.1007/978-3-540-73778-0_6.  Google Scholar

[2]

G. Bao and P. Li, Inverse medium scattering problems for electromagnetic waves, SIAM Journal on Applied Mathematics, 65 (2005), 2049-2066.  doi: 10.1137/040607435.  Google Scholar

[3]

G. Bao and P. J. Li, Shape reconstruction of inverse medium scattering for the Helmholtz equation, Computational Methods for Applied Inverse Problems, Inverse Ill-posed Probl. Ser., Walter de Gruyter, Berlin, 56 (2012), 283-305.   Google Scholar

[4]

G. Bao, P. Li, J. Lin and F. Triki, Inverse scattering problems with multi-frequencies, Inverse Problems, 31 (2015), 093001, 21 pp. doi: 10.1088/0266-5611/31/9/093001.  Google Scholar

[5]

T. Bui-Thanh and O. Ghattas, An analysis of infinite dimensional Bayesian inverse shape acoustic scattering and its numerical approximation, SIAM/ASA Journal on Uncertainty Quantification, 2 (2014), 203-222.  doi: 10.1137/120894877.  Google Scholar

[6]

T. Bui-Thanh, O. Ghattas, J. Martin and G. Stadler, A computational framework for infinite-dimensional Bayesian inverse problems part i: The linearized case, with application to global seismic inversion, SIAM Journal on Scientific Computing, 35 (2013), A2494–A2523. doi: 10.1137/12089586X.  Google Scholar

[7]

T. Bui-Thanh and Q. P. Nguyen, FEM-based discretization-invariant MCMC methods for PDE-constrained Bayesian inverse problems, Inverse Problems and Imaging, 10 (2016), 943-975. doi: 10.3934/ipi.2016028.  Google Scholar

[8]

M. Cheney, The linear sampling method and the MUSIC algorithm, Inverse Problems, 17 (2001), 591-595.  doi: 10.1088/0266-5611/17/4/301.  Google Scholar

[9]

H. Haddar and P. Monk, The linear sampling method for solving the electromagnetic inverse medium problem, Inverse Problems, 18 (2002), 891-906.  doi: 10.1088/0266-5611/18/3/323.  Google Scholar

[10]

D. Colton, H. Haddar and M. Piana, The linear sampling method in inverse electromagnetic scattering theory, Inverse Problems, 19 (2003), S105–S137. doi: 10.1088/0266-5611/19/6/057.  Google Scholar

[11]

D. Colton and R. Kress, Integral Equation Methods in Scattering Theory, A Wiley-Interscience Publication. John Wiley & Sons, Inc., New York, 1983.  Google Scholar

[12]

D. Colton and R. Kress, Inverse Acoustic and Electromagnetic Scattering Theory, Applied Mathematical Sciences, Vol. 93, Springer-Verlag, Berlin, 1998. doi: 10.1007/978-3-662-03537-5.  Google Scholar

[13]

S. L. CotterG. O. RobertsA. M. Stuart and D. White, MCMC methods for functions modifying old algorithms to make them faster, Statistical Science, 28 (2013), 424-446.  doi: 10.1214/13-STS421.  Google Scholar

[14]

T. CuiK. J. H. Law and Y. M. Marzouk, Dimension-independent likelihood-informed MCMC, Journal of Computational Physics, 304 (2016), 109-137.  doi: 10.1016/j.jcp.2015.10.008.  Google Scholar

[15]

M. Dashti and A. M. Stuart, The Bayesian approach to inverse problems, Handbook of Uncertainty Quantification, Springer, Cham, 1, 2, 3 (2017), 311-428.   Google Scholar

[16]

Z. Deng, X. Yang and J. Huang, A parametric Bayesian level set approach for acoustic source identification using multiple frequency information, preprint, (2019), arXiv: 1907.08660. Google Scholar

[17]

O. Dorn and D. Lesselier, Level set methods for inverse scattering - some recent developments, Inverse Problems, 25 (2009), 125001, 11 pp. doi: 10.1088/0266-5611/25/12/125001.  Google Scholar

[18]

O. DornE. L. Miller and C. M. Rappaport, A shape reconstruction method for electromagnetic tomography using adjoint fields and level sets, Inverse Problems, 16 (2000), 1119-1156.  doi: 10.1088/0266-5611/16/5/303.  Google Scholar

[19]

M. M. DunlopM. A. Iglesias and A. M. Stuart, Hierarchical Bayesian level set inversion, Statistics and Computing, 27 (2017), 1555-1584.  doi: 10.1007/s11222-016-9704-8.  Google Scholar

[20]

M. M. Dunlop and A. M. Stuart, The Bayesian formulation of EIT: Analysis and algorithms, Inverse Problems and Imaging, 10 (2016), 1007-1036.  doi: 10.3934/ipi.2016030.  Google Scholar

[21]

S. Fadil, A level-set approach for inverse problems involving obstacles, Optimisation and Calculus of Variations, 1 (1996), 17-23.  doi: 10.1051/cocv:1996101.  Google Scholar

[22]

Z. Feng and J. Li, An adaptive independence sampler MCMC algorithm for Bayesian inferences of functions, SIAM Journal on Scientific Computing, 40 (2018), A1301–A1321. doi: 10.1137/15M1021751.  Google Scholar

[23]

H. GengT. Yin and L. Xu, A priori error estimates of the DtN-FEM for the transmission problem in acoustics, Journal of Computational and Applied Mathematics, 313 (2017), 1-17.  doi: 10.1016/j.cam.2016.09.004.  Google Scholar

[24]

F. K. GruberE. A. Marengo and A. J. Devaney, Time-reversal imaging with multiple signal classification considering multiple scattering between the targets, Journal of the Acoustical Society of America, 115 (2004), 3042-3047.   Google Scholar

[25]

T. Hohage, On the numerical solution of a three-dimensional inverse medium scattering problem, Inverse Problems, 17 (2001), 1743-1763.  doi: 10.1088/0266-5611/17/6/314.  Google Scholar

[26]

S. HouK. Solna and H. Zhao, A direct imaging algorithm for extended targets, Inverse Problems, 22 (2006), 1151-1178.  doi: 10.1088/0266-5611/22/4/003.  Google Scholar

[27]

G. C. HsiaoN. NigamJ. E. Pasciak and L. Xu, Error analysis of the DtN-FEM for the scattering problem in acoustics via Fourier analysis, Journal of Computational and Applied Mathematics, 235 (2011), 4949-4965.  doi: 10.1016/j.cam.2011.04.020.  Google Scholar

[28]

J. Huang, Z. Deng and L. Xu, Bayesian approach for inverse interior scattering problems with limited aperture., Applicable Analysis, in press, (2020). Google Scholar

[29]

M. A. IglesiasY. Lu and A. M. Stuart, A Bayesian level set method for geometric inverse problems, Interfaces and Free Boundaries, 18 (2016), 181-217.  doi: 10.4171/IFB/362.  Google Scholar

[30]

K. ItoK. Kunisch and Z. Li, Level-set function approach to an inverse interface problem, Inverse Problems, 17 (2001), 1225-1242.  doi: 10.1088/0266-5611/17/5/301.  Google Scholar

[31]

J. JiaS. YueJ. Peng and J. Gao, Infinite-dimensional Bayesian approach for inverse scattering problems of a fractional Helmholtz equation, Journal of Functional Analysis, 275 (2018), 2299-2332.  doi: 10.1016/j.jfa.2018.08.002.  Google Scholar

[32]

L. Jiang and N. Ou, Bayesian inference using intermediate distribution based on coarse multiscale model for time fractional diffusion equations, SIAM Journal on Multiscale Modeling Simulation, 16 (2018), 327-355.  doi: 10.1137/17M1110535.  Google Scholar

[33]

J. Kaipio and E. Somersalo, Statistical and Computational Inverse Problems, Applied Mathematical Sciences, 160. Springer-Verlag, New York, 2005.  Google Scholar

[34]

A. Kirsch, The MUSIC-algorithm and the factorization method in inverse scattering theory for inhomogeneous media, Inverse Problems, 18 (2002), 1025-1040.  doi: 10.1088/0266-5611/18/4/306.  Google Scholar

[35] A. Kirsch and N. Grinberg, The Factorization Method for Inverse Problems, Oxford Lecture Series in Mathematics and its Applications, Vol. 36, Oxford University Press, Oxford, 2008.   Google Scholar
[36]

R. Kress, Newtons method for inverse obstacle scattering meets the method of least squares, Inverse Problems, 19 (2003), S91–S104. doi: 10.1088/0266-5611/19/6/056.  Google Scholar

[37]

R. Kress and W. Rundell, A quasi-Newton method in inverse obstacle scattering, Inverse Problems, 10 (1994), 114-1157.  doi: 10.1088/0266-5611/10/5/011.  Google Scholar

[38]

J. Li, A note on the Karhunen-Loève expansions for infinite-dimensional Bayesian inverse problems, Statistics Probability Letters, 106 (2015), 1-4.  doi: 10.1016/j.spl.2015.06.025.  Google Scholar

[39]

Z. LiZ. Deng and J. Sun, Extended-sampling-Bayesian method for limited aperture inverse scattering problems, SIAM Journal on Imaging Sciences, 13 (2020), 422-444.  doi: 10.1137/19M1270501.  Google Scholar

[40]

Z. Li, Y. Liu, J. Sun and L. Xu, Quality-Bayesian approach to inverse acoustic source problems with partial data, SIAM Journal on Scientific Computing, 43 (2021), A1062–A1080. doi: 10.1137/20M132345X.  Google Scholar

[41]

F. LindgrenH. Rue and J. Lindström, An explicit link between Gaussian fields and Gaussian Markov random fields: The stochastic partial differential equation approach, Journal of the Royal Statistical Society, 73 (2011), 423-498.  doi: 10.1111/j.1467-9868.2011.00777.x.  Google Scholar

[42]

J. Liu, Y. Liu and J. Sun, An inverse medium problem using Stekloff eigenvalues and a Bayesian approach, Inverse Problems, 35 (2019), 094004, 20 pp. doi: 10.1088/1361-6420/ab1be9.  Google Scholar

[43]

J. Martin, L. C. Wilcox, C. Burstedde and O. Ghattas., A stochastic Newton MCMC method for large-scale statistical inverse problems with application to seismic inversion, SIAM Journal on Scientific Computing, 34 (2012), A1460–A1487. doi: 10.1137/110845598.  Google Scholar

[44]

F. Natterer and F. Wiibbelmg, A propagation-backpropagation method for ultrasound tomography, Inverse Problems, 11 (1995), 1225-1232.  doi: 10.1088/0266-5611/11/6/007.  Google Scholar

[45]

S. Osher and J. A. Sethian, Fronts propagating with curvature-dependent speed: Lgorithms based on Hamilton-Jacobi formulations, Journal of Computational Physics, 79 (1988), 12-49.  doi: 10.1016/0021-9991(88)90002-2.  Google Scholar

[46]

N. Petra, J. Martin, G. Stadler and O. Ghattas, A computational framework for infinite-dimensional Bayesian inverse problems, part II: Stochastic Newton MCMC with application to ice sheet flow inverse problems, SIAM Journal on Scientific Computing, 36 (2014), A1525–A1555. doi: 10.1137/130934805.  Google Scholar

[47]

R. Potthast, A new non-iterative singular sources method for the reconstruction of piecewise constant media, Numerische Mathematik, 98 (2004), 703-730.  doi: 10.1007/s00211-004-0524-y.  Google Scholar

[48]

L. RoininenJ. M. J. Huttunen and S. Lasanen, Whittle-Matérn priors for Bayesian statistical inversion with applications in electrical impedance tomography, Inverse Problems and Imaging, 8 (2014), 561-586.  doi: 10.3934/ipi.2014.8.561.  Google Scholar

[49]

A. M. Stuart, Inverse problems: A Bayesian perspective, Acta Numerica, 19 (2010), 451-559.  doi: 10.1017/S0962492910000061.  Google Scholar

[50]

X.-C. Tai and H. Li, A piecewise constant level set method for elliptic inverse problems, Applied Numerical Mathematics, 57 (2007), 686-696.  doi: 10.1016/j.apnum.2006.07.010.  Google Scholar

[51]

M. Vögler, Reconstruction of the three-dimensional refractive index in electromagnetic scattering by using a propagation-backpropagation method, Inverse Problems, 19 (2003), 739-753.  doi: 10.1088/0266-5611/19/3/316.  Google Scholar

[52]

X. YangZ. Deng and J. Wang, An ensemble Kalman filter approach based on level set parameterization for acoustic source identification using multiple frequency information, Communications in Mathematical Research, 36 (2020), 211-228.  doi: 10.4208/cmr.2020-0011.  Google Scholar

Figure 1.  The geometry setting for the scattering problem
Figure 2.  Samples from the prior with $ \alpha = 2,3,4 $, for $ \tau = 10 $
Figure 3.  Samples from the prior with inverse length scale $ \tau = 10,\frac{20}{3},5 $, for $ \alpha = 3 $
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 $
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 $
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 $
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 $
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 $
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 $
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 $
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 $
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 $
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 $
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 $
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 $
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
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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