Bayesian approach to inverse time-harmonic acoustic obstacle scattering with phaseless data generated by point source waves. (English) Zbl 07427314

Summary: This paper concerns the Bayesian approach to inverse acoustic scattering problems of inferring the position and shape of a sound-soft obstacle from phaseless far-field data generated by two-dimensional point source waves. Given the total number of obstacle parameters, the Markov chain Monte Carlo (MCMC) method is employed to reconstruct the boundary of the obstacle in a high-dimensional space, which usually leads to slow convergence and prohibitively high computational cost. We use the Gibbs sampling and preconditioned Crank-Nicolson (pCN) algorithm with random proposal variance to improve the convergence rate, and design an effective strategy for the surrogate model constructed by the generalized polynomial chaos (gPC) method to reduce the computational cost of MCMC. Numerical examples are provided to illustrate the effectiveness of the proposed method.


35R30 Inverse problems for PDEs
35P25 Scattering theory for PDEs
62F15 Bayesian inference
78A46 Inverse problems (including inverse scattering) in optics and electromagnetic theory
Full Text: DOI arXiv


[1] Colton, D.; Kress, R., Inverse Acoustic and Electromagnetic Scattering Theory, Vol. 93 (2013), Springer
[2] Cakoni, F.; Colton, D., A Qualitative Approach to Inverse Scattering Theory, Vol. 188 (2014), Springer Science & Business Media
[3] A. Kirsch, N. Grinberg, The Factorization Method for Inverse Problems. · Zbl 1222.35001
[4] Kirsch, A., An Introduction to the Mathematical Theory of Inverse Problems, Vol. 120 (2011), Springer Science & Business Media
[5] Nakamura, G.; Potthast, R., Inverse Modeling: An Introduction to the Theory and Methods of Inverse Problems and Data Assimilation (2015), IOP Publishing: IOP Publishing Bristol UK · Zbl 1346.65029
[6] Kress, R.; Rundell, W., Inverse obstacle scattering with modulus of the far field pattern as data, (Engl, H. W., Inverse Problems in Medical Imaging and Nondestructive Testing (1997), Springer: Springer Wien), 75-92 · Zbl 0880.65105
[7] Xu, X.; Zhang, B.; Zhang, H., Uniqueness in inverse scattering problems with phaseless far-field data at a fixed frequency, SIAM J. Appl. Math., 78, 3, 1737-1753 (2018) · Zbl 1394.78015
[8] Zhang, B.; Zhang, H., Fast imaging of scattering obstacles from phaseless far-field measurements at a fixed frequency, Inverse Problems, 34, 10, Article 104005 pp. (2018) · Zbl 1452.65311
[9] Zhang, D.; Guo, Y., Uniqueness results on phaseless inverse acoustic scattering with a reference ball, Inverse Problems, 34, 8, Article 085002 pp. (2018) · Zbl 1442.35553
[10] Yang, Z.; Gui, X.; Ming, J.; Hu, G., Bayesian approach to inverse time-harmonic acoustic scattering with phaseless far-field data, Inverse Problems, 36, 6, Article 065012 pp. (2020) · Zbl 1443.76205
[11] Klibanov, M. V., Phaseless inverse scattering problems in three dimensions, SIAM J. Appl. Math., 74, 2, 392-410 (2014) · Zbl 1293.35188
[12] Klibanov, M. V., A phaseless inverse scattering problem for the 3-d Helmholtz equation, Inverse Probl. Imaging, 11, 2, 263-276 (2017) · Zbl 1359.35227
[13] Ivanyshyn, O., Shape reconstruction of acoustic obstacles from the modulus of the far field pattern, Inverse Probl. Imaging, 1, 4, 609-622 (2007) · Zbl 1194.35502
[14] Ivanyshyn, O.; Kress, R., Identification of sound-soft 3D obstacles from phaseless data, Inverse Probl. Imaging, 4, 1, 131-149 (2010) · Zbl 1220.35194
[15] Bui-Thanh, T.; Ghattas, O., An analysis of infinite dimensional Bayesian inverse shape acoustic scattering and its numerical approximation, SIAM/ASA J. Uncertain. Quantif., 2, 1, 203-222 (2014) · Zbl 1306.65270
[16] Stuart, A. M., Inverse problems: a Bayesian perspective, Acta Numer., 19, 451-559 (2010) · Zbl 1242.65142
[17] Kaipio, J.; Somersalo, E., Statistical and Computational Inverse Problems, Vol. 160 (2006), Springer Science & Business Media
[18] Law, K.; Stuart, A. M.; Zygalakis, K., (Data Assimilation: A Mathematical Introduction. Data Assimilation: A Mathematical Introduction, Texts in Applied Mathematics, vol. 62 (2015), Springer International Publishing) · Zbl 1353.60002
[19] Iglesias, M. A.; Lin, K.; Stuart, A. M., Well-posed Bayesian geometric inverse problems arising in subsurface flow, Inverse Problems, 30, 11, Article 114001 pp. (2014) · Zbl 1304.35767
[20] Baussard, A.; PrĂ©mel, D.; Venard, O., A Bayesian approach for solving inverse scattering from microwave laboratory-controlled data, Inverse Problems, 17, 6, 1659 (2001) · Zbl 1005.78003
[21] Liu, J.; Liu, Y.; Sun, J., An inverse medium problem using Stekloff eigenvalues and a Bayesian approach, Inverse Problems, 35, 9, Article 094004 pp. (2019) · Zbl 1471.65188
[22] Wang, Y.; Ma, F.; Zheng, E., Bayesian method for shape reconstruction in the inverse interior scattering problem, Math. Probl. Eng. (2015)
[23] Brooks, S.; Gelman, A.; Jones, G.; Meng, X., Handbook of Markov Chain Monte Carlo (2011), CRC press · Zbl 1218.65001
[24] Gamerman, D.; Lopes, H. F., Markov Chain Monte Carlo: Stochastic Simulation for Bayesian Inference (2006), Chapman and Hall/CRC · Zbl 1137.62011
[25] Geyer, C. J., Practical Markov chain Monte Carlo, Statist. Sci., 7, 4, 473-483 (1992)
[26] Cotter, S. L.; Roberts, G. O.; Stuart, A. M.; White, D., MCMC methods for functions: Modifying old algorithms to make them faster, Statist. Sci., 28, 3, 424-446 (2013) · Zbl 1331.62132
[27] Geman, S.; Geman, D., Stochastic relaxation, Gibbs distributions, and the Bayesian restoration of images, IEEE Trans. Pattern Anal. Mach. Intell., 6, 721-741 (1984) · Zbl 0573.62030
[28] Liu, J. S., Monte Carlo Strategies in Scientific Computing (2008), Springer Science & Business Media · Zbl 1132.65003
[29] Ma, X.; Zabaras, N., An efficient Bayesian inference approach to inverse problems based on an adaptive sparse grid collocation method, Inverse Problems, 25, 3, Article 035013 pp. (2009) · Zbl 1161.62011
[30] Marzouk, Y.; Xiu, D., A stochastic collocation approach to Bayesian inference in inverse problems (2009) · Zbl 1364.62064
[31] Yan, L.; Guo, L., Stochastic collocation algorithms using l_1-minimization for bayesian solution of inverse problems, SIAM J. Sci. Comput., 37, 3, A1410-A1435 (2015) · Zbl 1328.65200
[32] Zhang, G.; Lu, D.; Ye, M.; Gunzburger, M.; Webster, C., An adaptive sparse-grid high-order stochastic collocation method for Bayesian inference in groundwater reactive transport modeling, Water Resour. Res., 49, 10, 6871-6892 (2013)
[33] Marzouk, Y. M.; Najm, H. N., Dimensionality reduction and polynomial chaos acceleration of Bayesian inference in inverse problems, J. Comput. Phys., 228, 6, 1862-1902 (2009) · Zbl 1161.65308
[34] Cui, T.; Marzouk, Y. M.; Willcox, K. E., Data-driven model reduction for the Bayesian solution of inverse problems, Internat. J. Numer. Methods Engrg., 102, 5, 966-990 (2015) · Zbl 1352.65445
[35] Liao, Q.; Li, J., An adaptive reduced basis ANOVA method for high-dimensional Bayesian inverse problems, J. Comput. Phys., 396, 364-380 (2019) · Zbl 1453.62587
[36] Lu, F.; Morzfeld, M.; Tu, X.; Chorin, A. J., Limitations of polynomial chaos expansions in the Bayesian solution of inverse problems, J. Comput. Phys., 282, 138-147 (2015) · Zbl 1351.62075
[37] Li, J.; Marzouk, Y. M., Adaptive construction of surrogates for the Bayesian solution of inverse problems, SIAM J. Sci. Comput., 36, 3, A1163-A1186 (2014) · Zbl 1415.65009
[38] Yan, L.; Zhou, T., Adaptive multi-fidelity polynomial chaos approach to Bayesian inference in inverse problems, J. Comput. Phys., 381, 110-128 (2019) · Zbl 1451.62033
[39] Rick, D., Probability: Theory and Examples (2010), Cambridge university press · Zbl 1202.60001
[40] Hastings, W. K., Monte Carlo sampling methods using Markov chains and their applications, Biometrika, 57, 1, 97-109 (1970) · Zbl 0219.65008
[41] Metropolis, N.; Rosenbluth, A. W.; Rosenbluth, M. N.; Teller, A. H.; Teller, E., Equation of state calculations by fast computing machines, J. Chem. Phys., 21, 6, 1087-1092 (1953) · Zbl 1431.65006
[42] Cameron, R. H.; Martin, W. T., The orthogonal development of non-linear functionals in series of Fourier-Hermite functionals, Ann. of Math., 48, 2, 385-392 (1947) · Zbl 0029.14302
[43] Robert, C. P.; Casella, G., Monte Carlo Statistical Methods (2004), Springer: Springer New York · Zbl 1096.62003
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.