Factorization method for near-field inverse scattering problems in elastodynamics

Chun Liu; Guanghui Hu; Tao Yin; Bo Zhang

doi:10.3934/ipi.2025054

Article Contents

2026, Volume 23: 51-69. Doi: 10.3934/ipi.2025054

This volume Previous Article Fast reconstruction approaches for photoacoustic tomography with smoothing Sobolev/Matérn priors Next Article Stability of coefficients inverse problem of 3D viscoelastic system

Factorization method for near-field inverse scattering problems in elastodynamics

1.
School of Mathematical Sciences and LPMC, Nankai University, Tianjing 300071, China
2.
State Key Laboratory of Mathematical Sciences, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, China
3.
School of Mathematical Sciences, University of Chinese Academy of Sciences, Beijing 100049, China

Received: June 2025

Revised: October 2025

Early access: October 31, 2025

Published online: October 31, 2025

Abstract / Introduction Full Text(HTML) Figure(5) Related Papers Cited by

Abstract

Consider a time-harmonic elastic wave scattering problem with a bounded rigid obstacle embedded in a homogeneous and isotropic elastic medium. This work is concerned with the inverse problem of recovering the location and shape of the obstacle from near-field data generated by infinitely many incident point source waves at a fixed energy. An outgoing-to-incoming operator is defined for facilitating the factorization of the near-field operator. Necessary properties of the factorization of the composition of the outgoing-to-incoming operator and the near-field operator are shown to apply the range identity to construct an indicator function for the reconstruction. Numerical examples in 2D are presented to show the validity and accuracy of the inversion algorithm.

Keywords:

Mathematics Subject Classification: Primary: 35R30, 74J25.

Citation:

Full Text(HTML)

Figure 1. Reconstruction of the kite-shaped obstacle with $ \mathit{\boldsymbol{a}} = (\cos \alpha, \sin \alpha)^\top, \alpha = 2\pi/3 $.

Download: Full-size image PowerPoint slide

Figure 2. Reconstruction of the kite-shaped obstacle with different $ \alpha $.

Download: Full-size image PowerPoint slide

Figure 3. Reconstruction of the star-shaped obstacle with $ \mathit{\boldsymbol{a}} = (\cos \alpha, \sin \alpha)^\top, \alpha = 2\pi/3 $.

Download: Full-size image PowerPoint slide

Figure 4. Reconstruction of the star-shaped obstacle with different $ \alpha $.

Download: Full-size image PowerPoint slide

Figure 5. Reconstruction of two rigid obstacles with different $ \alpha $.

Download: Full-size image PowerPoint slide

Related Papers

Cited by

References

[1]	C. J. S. Alves and R. Kress, On the far-field operator in elastic obstacle scattering, IMA J. Appl. Math., 67 (2002), 1-21. doi: 10.1093/imamat/67.1.1.
[2]	H. Ammari, P. Calmon and E. Iakovleva, Direct elastic imaging of a ismall inclusion, SIAM J. Imaging Sci., 1 (2008), 169-187. doi: 10.1137/070696076.
[3]	T. Arens, Linear sampling methods for 2D inverse elastic wave scattering, Inverse Problems, 17 (2001), 1445-1464. doi: 10.1088/0266-5611/17/5/314.
[4]	G. Bao, G. Hu, J. Sun and T. Yin, Direct and inverse elastic scattering from anisotropic media, J. Math. Pures Appl., 117 (2018), 263-301. doi: 10.1016/j.matpur.2018.01.007.
[5]	Z. Cheng and H. Dong, Uniqueness and reconstruction method for inverse elastic wave scattering with phaseless data, Inverse Probl. Imaging, 18 (2024), 406-433. doi: 10.3934/ipi.2023038.
[6]	D. Colton and R. Kress, Inverse Acoustic and Electromagnetic Scattering Theory, 3$^{nd}$ edition, Springer, New York, 2013. doi: 10.1007/978-1-4614-4942-3.
[7]	J. Elschner and G. Hu, Uniqueness and factorization method for inverse elastic scattering with a single incoming wave, Inverse Problems, 35 (2019), 094002, 18 pp. doi: 10.1088/1361-6420/ab20be.
[8]	J. T. Fokkema and P. M. Berg, Elastodynamic diffraction by a periodic rough surface (stress-free boundary), J. Acoust. Soc. Am., 62 (1977), 1095-1101. doi: 10.1121/1.381642.
[9]	P. Hähner and G. C. Hsiao, Uniqueness theorems in inverse obstacle scattering of elastic waves, Inverse Problems, 9 (1993), 525-534. doi: 10.1088/0266-5611/9/5/002.
[10]	G. Hu, A. Kirsch and M. Sini, Some inverse problems arising from elastic scattering by rigid obstacles, Inverse Problems, 29 (2012), 015009, 21 pp. doi: 10.1088/0266-5611/29/1/015009.
[11]	G. Hu, J. Yang, B. Zhang and H. Zhang, Near-field imaging of scattering obstacles with the factorization method, Inverse Problems, 30 (2014), 095005, 25 pp. doi: 10.1088/0266-5611/30/9/095005.
[12]	K. Ito, B. Jin and J. Zou, A direct sampling method to an inverse medium scattering problem, Inverse Problems, 28 (2012), 025003, 11 pp. doi: 10.1088/0266-5611/28/2/025003.
[13]	X. Ji, X. Liu and Y. Xi, Direct sampling methods for inverse elastic scattering problems, Inverse Problems, 34 (2018), 035008, 22 pp. doi: 10.1088/1361-6420/aaa996.
[14]	A. Kirsch, Characterization of the shape of a scattering obstacle using the spectral data of the far field operator, Inverse Problems, 14 (1998), 1489-1512. doi: 10.1088/0266-5611/14/6/009.
[15]	A. Kirsch and N. Grinberg, The Factorization Method for Inverse Problems, Oxford University Press, Oxford, 2008.
[16]	V. D. Kupradze, Three-Dimensional Problems of the Mathematical Theory of Elasticity and Thermoelasticity, North-Holland, Amsterdam, 1979.
[17]	P. Li, Y. Wang, Z. Wang and Y. Zhao, Inverse obstacle scattering for elastic waves, Inverse Problems, 32 (2016), 115018, 24 pp. doi: 10.1088/0266-5611/32/11/115018.
[18]	P. Li and X. Yuan, Inverse obstacle scattering for elastic waves in three dimensions, Inverse Probl. Imaging, 13 (2019), 545-573. doi: 10.3934/ipi.2019026.
[19]	S. Li, J. Lv and Y. Wang, Factorization method for inverse elastic obstacle scattering with neumann boundary condition, J. Sci. Comput., 104 (2025), Paper No. 45, 37 pp. doi: 10.1007/s10915-025-02966-2.
[20]	X. Liu, S. Meng and B. Zhang, Modified sampling method with near field measurements, SIAM J. Appl. Math., 82 (202), 244-266. doi: 10.1137/21M1432235.
[21]	J. H. Rose, Elastic wave inverse scattering in nondestructive evaluation, PAGEOPH, 131 (1989), 715-739. doi: 10.1007/BF00876270.
[22]	J. W. C. Sherwood, Elastic wave propagation in a semi-infinite solid medium, Proc. Phys. Soc., 71 (1958), 207-219. doi: 10.1088/0370-1328/71/2/308.
[23]	J. Xiang and G. Yan, The uniqueness of the inverse elastic wave scattering problem based on the mixed reciprocity relation, Inverse Probl. Imaging, 15 (2021), 539-554. doi: 10.3934/ipi.2021004.
[24]	J. Xiang and G. Yan, The factorization method for a mixed inverse elastic scattering problem, IMA J. Appl. Math., 87 (2022), 407-437. doi: 10.1093/imamat/hxac010.
[25]	T. Yin, G. Hu, L. Xu and B. Zhang, Near-field imaging of obstacles with the factorization method: Fluid–solid interaction, Inverse Problems, 32 (2016), 015003, 29 pp. doi: 10.1088/0266-5611/32/1/015003.
[26]	J. Yue, M. Li, P. Li and X. Yuan, Numerical solution of an inverse obstacle scattering problem for elastic waves via the Helmholtz decomposition, Commun. Comput. Phys., 26 (2019), 809-837. doi: 10.4208/cicp.OA-2018-0194.