# American Institute of Mathematical Sciences

doi: 10.3934/dcdsb.2022050
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## Factorization method for inverse time-harmonic elastic scattering with a single plane wave

 1 Department of Applied Mathematics, Beijing Computational Science Research Center, Beijing, 100193, China 2 School of Mathematical Sciences and LPMC, Nankai Universtiy, Tianjin, 300071, China

*Corresponding author: Guanghui Hu

Received  September 2021 Revised  January 2022 Early access March 2022

Fund Project: This work is supported by NSFC 12071236 and NSAF U1930402

This paper is concerned with the factorization method with a single far-field pattern to recover an arbitrary convex polygonal scatterer/source in linear elasticity. The approach also applies to the compressional (resp. shear) part of the far-field pattern excited by a single compressional (resp. shear) plane wave. The one-wave factorization is based on the scattering data for a priori given testing scatterers. It can be regarded as a domain-defined sampling method and does not require forward solvers. We derive the spectral system of the far-field operator for rigid disks and show that, using testing disks, the one-wave factorization method can be justified independently of the classical factorization method.

Citation: Guanqiu Ma, Guanghui Hu. Factorization method for inverse time-harmonic elastic scattering with a single plane wave. Discrete and Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2022050
##### References:
 [1] C. Alves and R. Kress, On the far-filed operator in elastic obstacle scattering, IMA J. Appl. Math., 67 (2002), 1-21.  doi: 10.1093/imamat/67.1.1. [2] H. Ammari, E. Bretin, J. Garnier, H. Kang, H. Lee and A. Wahab, Mathematical Methods in Elasticity Imaging, Volume 52, Princeton Series in Applied Mathematics, Princeton University Press, 2015. doi: 10.1515/9781400866625. [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 Pure Appl, 117 (2018), 263-301.  doi: 10.1016/j.matpur.2018.01.007. [5] E. Blåsten and Y. Lin, Radiating and non-radiating sources in elasticity, Inverse Problems, 35 (2019), 015005, 16 pp. doi: 10.1088/1361-6420/aae99e. [6] M. Bonnet and A. Constantinescu, Inverse problems in elasticity, Inverse Problems, 21 (2015), 1-50.  doi: 10.1088/0266-5611/21/2/R01. [7] A. Charalambopoulos, D. Gintides and K. Kiriaki, The linear sampling method for non-absorbing penetrable elastic bodies, Inverse Problems, 19 (2003), 549-561.  doi: 10.1088/0266-5611/19/3/305. [8] A. Charalambopoulos, A. Kirsch, K. Anagnostopoulos, D. Gintides and K. Kiriaki, The factorization method in inverse elastic scattering from penetrable bodies, Inverse Problems, 23 (2007), 27-51.  doi: 10.1088/0266-5611/23/1/002. [9] Z. Cheng and G. Huang, Reverse time migration for extended obstacles: Elastic waves (in Chinese), Science China Mathematics, 45 (2015), 1103-1114. [10] D. Colton and R. Kress, Inverse Acoustic and Electromagnetic Scattering Theory, 4th edition, Springer, Berlin, 2019. doi: 10.1007/978-3-030-30351-8. [11] S. Das, S. Banerjee and T. Kundu, Elastic wave scattering in a solid half-space with a circular cylindrical hole using the Distributed Point Source Method, Int. J. Solids Struct, 45 (2008), 4498-4508. [12] 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. [13] D. Gintides and M. Sini, Identification of obstacles using only the scattered P-waves or the scattered S-waves, Inverse Problems Imaging, 6 (2012), 39-55.  doi: 10.3934/ipi.2012.6.39. [14] P. Hahner and G. C. Hsiao, Uniqueness theorems in inverse obstacle scattering of elastic waves, Inverse Problems, 9 (1993), 525-534. [15] G. Hu, A. Kirsch and M. Sini, Some inverse problems arising from elastic scattering by rigid obstacles, Inverse Problems, 29 (2013), 015009, 21 pp. doi: 10.1088/0266-5611/29/1/015009. [16] M. Ikehata, Enclosing a polygonal cavity in a two-dimensional bounded domain from Cauchy data, Inverse Problems, 15 (1999), 1231-1241.  doi: 10.1088/0266-5611/15/5/308. [17] M. Ikehata and H. Itou, Extracting the support function of a cavity in an isotropic elastic body from a single set of boundary data, Inverse Problems, 25 (2009), 105005, 21 pp. doi: 10.1088/0266-5611/25/10/105005. [18] X. Ji and X. Liu, Inverse elastic scattering problems with phaseless far field data, Inverse Problems, 35 (2019), 114004, 39 pp. doi: 10.1088/1361-6420/ab2a35. [19] 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. [20] A. Kirsch and N. Grinberg, The Factorization Method for Inverse Problems, Oxford University Press, NewYork, 2008. [21] V. D. Kupradze, et. al., Three-Dimensional Problems of the Mathematical Theory of Elasticity and Thermoelasticity, Amsterdam: North-Holland, 1979. [22] S. Kusiak, R. Potthast and J. Sylvester, A range test for determining scatterers with unknown physical properties, Inverse Problems, 19 (2003), 533-547.  doi: 10.1088/0266-5611/19/3/304. [23] 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. [24] Y. Lin, G. Nakamura, R. Potthast and H. Wang, Duality between range and no-response tests and its application for inverse problems, Inverse Probl. Imaging, 15 (2021), 367-386.  doi: 10.3934/ipi.2020072. [25] J. Liu, X. Liu and J. Sun, Extended sampling method for inverse elastic scattering problems using one incident wave, SIAM J. Imaging Sci., 12 (2019), 874-892.  doi: 10.1137/19M1237788. [26] J. Liu and J. Sun, Extended sampling method in inverse scattering, Inverse Problems, 34 (2018), 085007, 17 pp. doi: 10.1088/1361-6420/aaca90. [27] D. R. Luke and R. Potthast, The no response test - a sampling method for inverse scattering problems, SIAM J. Appl. Math., 63 (2003), 1292-1312.  doi: 10.1137/S0036139902406887. [28] G. Ma and G. Hu, Factorization method with one plane wave: From model-driven and data-driven perspectives, Inverse Problems, 38 (2022), 015003, 26 pp. doi: 10.1088/1361-6420/ac38b5. [29] G. Nakamura and R. Potthast, Inverse Modeling - An Introduction to the Theory and Methods of Inverse Problems and Data Assimilation, IOP Expanding Physics. IOP Publishing, Bristol, 2015.

show all references

##### References:
 [1] C. Alves and R. Kress, On the far-filed operator in elastic obstacle scattering, IMA J. Appl. Math., 67 (2002), 1-21.  doi: 10.1093/imamat/67.1.1. [2] H. Ammari, E. Bretin, J. Garnier, H. Kang, H. Lee and A. Wahab, Mathematical Methods in Elasticity Imaging, Volume 52, Princeton Series in Applied Mathematics, Princeton University Press, 2015. doi: 10.1515/9781400866625. [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 Pure Appl, 117 (2018), 263-301.  doi: 10.1016/j.matpur.2018.01.007. [5] E. Blåsten and Y. Lin, Radiating and non-radiating sources in elasticity, Inverse Problems, 35 (2019), 015005, 16 pp. doi: 10.1088/1361-6420/aae99e. [6] M. Bonnet and A. Constantinescu, Inverse problems in elasticity, Inverse Problems, 21 (2015), 1-50.  doi: 10.1088/0266-5611/21/2/R01. [7] A. Charalambopoulos, D. Gintides and K. Kiriaki, The linear sampling method for non-absorbing penetrable elastic bodies, Inverse Problems, 19 (2003), 549-561.  doi: 10.1088/0266-5611/19/3/305. [8] A. Charalambopoulos, A. Kirsch, K. Anagnostopoulos, D. Gintides and K. Kiriaki, The factorization method in inverse elastic scattering from penetrable bodies, Inverse Problems, 23 (2007), 27-51.  doi: 10.1088/0266-5611/23/1/002. [9] Z. Cheng and G. Huang, Reverse time migration for extended obstacles: Elastic waves (in Chinese), Science China Mathematics, 45 (2015), 1103-1114. [10] D. Colton and R. Kress, Inverse Acoustic and Electromagnetic Scattering Theory, 4th edition, Springer, Berlin, 2019. doi: 10.1007/978-3-030-30351-8. [11] S. Das, S. Banerjee and T. Kundu, Elastic wave scattering in a solid half-space with a circular cylindrical hole using the Distributed Point Source Method, Int. J. Solids Struct, 45 (2008), 4498-4508. [12] 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. [13] D. Gintides and M. Sini, Identification of obstacles using only the scattered P-waves or the scattered S-waves, Inverse Problems Imaging, 6 (2012), 39-55.  doi: 10.3934/ipi.2012.6.39. [14] P. Hahner and G. C. Hsiao, Uniqueness theorems in inverse obstacle scattering of elastic waves, Inverse Problems, 9 (1993), 525-534. [15] G. Hu, A. Kirsch and M. Sini, Some inverse problems arising from elastic scattering by rigid obstacles, Inverse Problems, 29 (2013), 015009, 21 pp. doi: 10.1088/0266-5611/29/1/015009. [16] M. Ikehata, Enclosing a polygonal cavity in a two-dimensional bounded domain from Cauchy data, Inverse Problems, 15 (1999), 1231-1241.  doi: 10.1088/0266-5611/15/5/308. [17] M. Ikehata and H. Itou, Extracting the support function of a cavity in an isotropic elastic body from a single set of boundary data, Inverse Problems, 25 (2009), 105005, 21 pp. doi: 10.1088/0266-5611/25/10/105005. [18] X. Ji and X. Liu, Inverse elastic scattering problems with phaseless far field data, Inverse Problems, 35 (2019), 114004, 39 pp. doi: 10.1088/1361-6420/ab2a35. [19] 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. [20] A. Kirsch and N. Grinberg, The Factorization Method for Inverse Problems, Oxford University Press, NewYork, 2008. [21] V. D. Kupradze, et. al., Three-Dimensional Problems of the Mathematical Theory of Elasticity and Thermoelasticity, Amsterdam: North-Holland, 1979. [22] S. Kusiak, R. Potthast and J. Sylvester, A range test for determining scatterers with unknown physical properties, Inverse Problems, 19 (2003), 533-547.  doi: 10.1088/0266-5611/19/3/304. [23] 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. [24] Y. Lin, G. Nakamura, R. Potthast and H. Wang, Duality between range and no-response tests and its application for inverse problems, Inverse Probl. Imaging, 15 (2021), 367-386.  doi: 10.3934/ipi.2020072. [25] J. Liu, X. Liu and J. Sun, Extended sampling method for inverse elastic scattering problems using one incident wave, SIAM J. Imaging Sci., 12 (2019), 874-892.  doi: 10.1137/19M1237788. [26] J. Liu and J. Sun, Extended sampling method in inverse scattering, Inverse Problems, 34 (2018), 085007, 17 pp. doi: 10.1088/1361-6420/aaca90. [27] D. R. Luke and R. Potthast, The no response test - a sampling method for inverse scattering problems, SIAM J. Appl. Math., 63 (2003), 1292-1312.  doi: 10.1137/S0036139902406887. [28] G. Ma and G. Hu, Factorization method with one plane wave: From model-driven and data-driven perspectives, Inverse Problems, 38 (2022), 015003, 26 pp. doi: 10.1088/1361-6420/ac38b5. [29] G. Nakamura and R. Potthast, Inverse Modeling - An Introduction to the Theory and Methods of Inverse Problems and Data Assimilation, IOP Expanding Physics. IOP Publishing, Bristol, 2015.
Illustration of a convex polygonal source term where $O$ is corner point of $D$
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