doi: 10.3934/ipi.2021052
Online First

Online First articles are published articles within a journal that have not yet been assigned to a formal issue. This means they do not yet have a volume number, issue number, or page numbers assigned to them, however, they can still be found and cited using their DOI (Digital Object Identifier). Online First publication benefits the research community by making new scientific discoveries known as quickly as possible.

Readers can access Online First articles via the “Online First” tab for the selected journal.

Fourier method for reconstructing elastic body force from the coupled-wave field

1. 

National Key Laboratory of Science and Technology on Advanced Composites in Special Environments, Harbin Institute of Technology, Harbin 150080, China

2. 

Key Laboratory of Micro-systems and Micro-structures Manufacturing Ministry of Education, Harbin Institute of Technology, Harbin 150080, China

3. 

School of Mathematics, Harbin Institute of Technology, Harbin 150001, China

4. 

Department of Mathematics, Hong Kong Baptist University, Kowloon, Hong Kong SAR, China

* Corresponding author: Jiaqi Zhu and Minghui Song

Received  March 2021 Revised  March 2021 Early access July 2021

This paper is concerned with the inverse source problem of the time-harmonic elastic waves. A novel non-iterative reconstruction scheme is proposed for determining the elastic body force by using the multi-frequency Fourier expansion. The key ingredient of the approach is to choose appropriate admissible frequencies and establish an relationship between the Fourier coefficients and the coupled-wave field of compressional wave and shear wave. Both theoretical justifications and numerical examples are presented to verify the validity and robustness of the proposed method.

Citation: Xianchao Wang, Jiaqi Zhu, Minghui Song, Wei Wu. Fourier method for reconstructing elastic body force from the coupled-wave field. Inverse Problems & Imaging, doi: 10.3934/ipi.2021052
References:
[1]

B. AbdelazizA. El Badia and A. El Hajj, Direct algorithm for multipolar sources reconstruction, J. Math. Anal. Appl., 428 (2015), 306-336.  doi: 10.1016/j.jmaa.2015.03.013.  Google Scholar

[2]

R. Albanese and P. B. Monk, The inverse source problem for Maxwell's equations, Inverse Probl., 22 (2006), 1023-1035.  doi: 10.1088/0266-5611/22/3/018.  Google Scholar

[3]

C. J. S. Alves, N. F. M. Martins and N. C. Roberty, Identification and reconstruction of elastic body forces, Inverse Probl., 30 (2014), 055015. doi: 10.1088/0266-5611/30/5/055015.  Google Scholar

[4]

H. Ammari, E. Bretin, J. Garnier, H. Kang, H. Lee and A. Wahab, Mathematical Methods in Elasticity Imaging, Vol. 52, Princeton University Press, 2015. doi: 10.1515/9781400866625.  Google Scholar

[5]

T. S. AngelA. Kirsch and R. E. Kleinmann, Antenna control and optimization, Proc. IEEE, 79 (1991), 1559-1568.  doi: 10.1109/5.104230.  Google Scholar

[6]

T. Arens, Linear sampling methods for 2D inverse elastic wave scattering, Inverse Probl., 17 (2001), 1445-1464.  doi: 10.1088/0266-5611/17/5/314.  Google Scholar

[7]

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

[8]

E. Blåsten, Nonradiating sources and transmission eigenfunctions vanish at corners and edges, SIAM J. Math. Anal., 50 (2018), 6255-6270.  doi: 10.1137/18M1182048.  Google Scholar

[9]

E. Blåsten and Y. -H. Lin, Radiating and non-radiating sources in elasticity, Inverse Probl., 35 (2019), 015005. doi: 10.1088/1361-6420/aae99e.  Google Scholar

[10]

E. Blåsten, H. Liu and J. Xiao, On an electromagnetic problem in a corner and its applications, Analysis & PDE, (2020), in press. Google Scholar

[11]

E. Blåsten and H. Liu, Scattering by curvatures, radiationless sources, transmission eigenfunctions and inverse scattering problems, SIAM J. Math. Anal., (2021), in press. Google Scholar

[12]

N. Bleistein and J. K. Cohen, Nonuniqueness in the inverse source problem in acoustics and electromagnetics, J. Math. Phys., 18 (1977), 194-201.  doi: 10.1063/1.523256.  Google Scholar

[13]

P. Bolan, 3D Shepp-Logan Phantom, MATLAB Central File Exchange, (2021). Available from: https://www.mathworks.com/matlabcentral/fileexchange/50974-3d-shepp-logan-phantom. Google Scholar

[14]

X. Cao and H. Liu, Determining a fractional Helmholtz system with unknown source and medium parameter, Commun. Math. Sci., 17 (2019), 1861-1876.  doi: 10.4310/CMS.2019.v17.n7.a5.  Google Scholar

[15]

D. Colton and R. Kress, Inverse Acoustic and Electromagnetic Scattering Theory, 4th edition, Springer, Cham, 2019. doi: 10.1007/978-3-030-30351-8.  Google Scholar

[16]

Y. DengH. Liu and G. Uhlmann, On an inverse boundary problem arising in brain imaging, J. Diff. Equations, 267 (2019), 2471-2502.  doi: 10.1016/j.jde.2019.03.019.  Google Scholar

[17]

M. Eller and N. P. Valdivia, Acoustic source identification using multiple frequency information, Inverse Probl., 25 (2009), 115005. doi: 10.1088/0266-5611/25/11/115005.  Google Scholar

[18]

R. Griesmaier and C. Schmiedecke, A factorization method for multifrequency inverse source problems with sparse far field measurements, SIAM J. Imaging Sci., 10 (2017), 2119-2139.  doi: 10.1137/17M111290X.  Google Scholar

[19]

S. Kusiak and J. Sylvester, The scattering support, Commun. Pur. Appl. Math., 56 (2003), 1525-1548.  doi: 10.1002/cpa.3038.  Google Scholar

[20]

J. LiH. Liu and S. Ma, Determining a random Schrödinger equation with unknown source and potential, SIAM J. Math. Anal., 51 (2019), 3465-3491.  doi: 10.1137/18M1225276.  Google Scholar

[21]

J. LiH. Liu and S. Ma, Determining a random Schrödinger operator: both potential and source are random, Commun. Math. Phys., 381 (2021), 527-556.  doi: 10.1007/s00220-020-03889-9.  Google Scholar

[22]

J. LiH. Liu and H. Sun, On a gesture-computing technique using electromagnetic waves, Inverse Probl. Imag., 12 (2018), 677-696.  doi: 10.3934/ipi.2018029.  Google Scholar

[23]

H. Liu and G. Uhlmann, Determining both sound speed and internal source in thermo- and photo-acoustic tomography, Inverse Probl., 31 (2015), 105005. doi: 10.1088/0266-5611/31/10/105005.  Google Scholar

[24]

H. LiuY. Wang and C. Yang, Mathematical design of a novel gesture-based instruction/input device using wave detection, SIAM J. Imaging Sci., 9 (2016), 822-841.  doi: 10.1137/16M1063551.  Google Scholar

[25]

G. WangF. MaY. Guo and J. Li, Solving the multi-frequency electromagnetic inverse source problem by the Fourier method, J. Differ. Equations, 265 (2018), 417-443.  doi: 10.1016/j.jde.2018.02.036.  Google Scholar

[26]

X. Wang, Y. Guo, D. Zhang and H. Liu, Fourier method for recovering acoustic sources from multi-frequency far-field data, Inverse Probl., 33 (2017), 035001. doi: 10.1088/1361-6420/aa573c.  Google Scholar

[27]

X. Wang, Y. Guo, J. Li and H. Liu, Mathematical design of a novel input/instruction device using a moving emitter, Inverse Probl., 33 (2017), 105009. doi: 10.1088/1361-6420/aa873f.  Google Scholar

[28]

D. Zhang and Y. Guo, Fourier method for solving the multi-frequency inverse source problem for the Helmholtz equation, Inverse Probl., 31 (2015), 035007. doi: 10.1088/0266-5611/31/3/035007.  Google Scholar

[29]

D. ZhangY. GuoJ. Li and H. Liu, Locating multiple multipolar acoustic sources using the direct sampling method, Comm. Comput. Phys., 25 (2019), 1328-1356.  doi: 10.4208/cicp.oa-2018-0020.  Google Scholar

show all references

References:
[1]

B. AbdelazizA. El Badia and A. El Hajj, Direct algorithm for multipolar sources reconstruction, J. Math. Anal. Appl., 428 (2015), 306-336.  doi: 10.1016/j.jmaa.2015.03.013.  Google Scholar

[2]

R. Albanese and P. B. Monk, The inverse source problem for Maxwell's equations, Inverse Probl., 22 (2006), 1023-1035.  doi: 10.1088/0266-5611/22/3/018.  Google Scholar

[3]

C. J. S. Alves, N. F. M. Martins and N. C. Roberty, Identification and reconstruction of elastic body forces, Inverse Probl., 30 (2014), 055015. doi: 10.1088/0266-5611/30/5/055015.  Google Scholar

[4]

H. Ammari, E. Bretin, J. Garnier, H. Kang, H. Lee and A. Wahab, Mathematical Methods in Elasticity Imaging, Vol. 52, Princeton University Press, 2015. doi: 10.1515/9781400866625.  Google Scholar

[5]

T. S. AngelA. Kirsch and R. E. Kleinmann, Antenna control and optimization, Proc. IEEE, 79 (1991), 1559-1568.  doi: 10.1109/5.104230.  Google Scholar

[6]

T. Arens, Linear sampling methods for 2D inverse elastic wave scattering, Inverse Probl., 17 (2001), 1445-1464.  doi: 10.1088/0266-5611/17/5/314.  Google Scholar

[7]

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

[8]

E. Blåsten, Nonradiating sources and transmission eigenfunctions vanish at corners and edges, SIAM J. Math. Anal., 50 (2018), 6255-6270.  doi: 10.1137/18M1182048.  Google Scholar

[9]

E. Blåsten and Y. -H. Lin, Radiating and non-radiating sources in elasticity, Inverse Probl., 35 (2019), 015005. doi: 10.1088/1361-6420/aae99e.  Google Scholar

[10]

E. Blåsten, H. Liu and J. Xiao, On an electromagnetic problem in a corner and its applications, Analysis & PDE, (2020), in press. Google Scholar

[11]

E. Blåsten and H. Liu, Scattering by curvatures, radiationless sources, transmission eigenfunctions and inverse scattering problems, SIAM J. Math. Anal., (2021), in press. Google Scholar

[12]

N. Bleistein and J. K. Cohen, Nonuniqueness in the inverse source problem in acoustics and electromagnetics, J. Math. Phys., 18 (1977), 194-201.  doi: 10.1063/1.523256.  Google Scholar

[13]

P. Bolan, 3D Shepp-Logan Phantom, MATLAB Central File Exchange, (2021). Available from: https://www.mathworks.com/matlabcentral/fileexchange/50974-3d-shepp-logan-phantom. Google Scholar

[14]

X. Cao and H. Liu, Determining a fractional Helmholtz system with unknown source and medium parameter, Commun. Math. Sci., 17 (2019), 1861-1876.  doi: 10.4310/CMS.2019.v17.n7.a5.  Google Scholar

[15]

D. Colton and R. Kress, Inverse Acoustic and Electromagnetic Scattering Theory, 4th edition, Springer, Cham, 2019. doi: 10.1007/978-3-030-30351-8.  Google Scholar

[16]

Y. DengH. Liu and G. Uhlmann, On an inverse boundary problem arising in brain imaging, J. Diff. Equations, 267 (2019), 2471-2502.  doi: 10.1016/j.jde.2019.03.019.  Google Scholar

[17]

M. Eller and N. P. Valdivia, Acoustic source identification using multiple frequency information, Inverse Probl., 25 (2009), 115005. doi: 10.1088/0266-5611/25/11/115005.  Google Scholar

[18]

R. Griesmaier and C. Schmiedecke, A factorization method for multifrequency inverse source problems with sparse far field measurements, SIAM J. Imaging Sci., 10 (2017), 2119-2139.  doi: 10.1137/17M111290X.  Google Scholar

[19]

S. Kusiak and J. Sylvester, The scattering support, Commun. Pur. Appl. Math., 56 (2003), 1525-1548.  doi: 10.1002/cpa.3038.  Google Scholar

[20]

J. LiH. Liu and S. Ma, Determining a random Schrödinger equation with unknown source and potential, SIAM J. Math. Anal., 51 (2019), 3465-3491.  doi: 10.1137/18M1225276.  Google Scholar

[21]

J. LiH. Liu and S. Ma, Determining a random Schrödinger operator: both potential and source are random, Commun. Math. Phys., 381 (2021), 527-556.  doi: 10.1007/s00220-020-03889-9.  Google Scholar

[22]

J. LiH. Liu and H. Sun, On a gesture-computing technique using electromagnetic waves, Inverse Probl. Imag., 12 (2018), 677-696.  doi: 10.3934/ipi.2018029.  Google Scholar

[23]

H. Liu and G. Uhlmann, Determining both sound speed and internal source in thermo- and photo-acoustic tomography, Inverse Probl., 31 (2015), 105005. doi: 10.1088/0266-5611/31/10/105005.  Google Scholar

[24]

H. LiuY. Wang and C. Yang, Mathematical design of a novel gesture-based instruction/input device using wave detection, SIAM J. Imaging Sci., 9 (2016), 822-841.  doi: 10.1137/16M1063551.  Google Scholar

[25]

G. WangF. MaY. Guo and J. Li, Solving the multi-frequency electromagnetic inverse source problem by the Fourier method, J. Differ. Equations, 265 (2018), 417-443.  doi: 10.1016/j.jde.2018.02.036.  Google Scholar

[26]

X. Wang, Y. Guo, D. Zhang and H. Liu, Fourier method for recovering acoustic sources from multi-frequency far-field data, Inverse Probl., 33 (2017), 035001. doi: 10.1088/1361-6420/aa573c.  Google Scholar

[27]

X. Wang, Y. Guo, J. Li and H. Liu, Mathematical design of a novel input/instruction device using a moving emitter, Inverse Probl., 33 (2017), 105009. doi: 10.1088/1361-6420/aa873f.  Google Scholar

[28]

D. Zhang and Y. Guo, Fourier method for solving the multi-frequency inverse source problem for the Helmholtz equation, Inverse Probl., 31 (2015), 035007. doi: 10.1088/0266-5611/31/3/035007.  Google Scholar

[29]

D. ZhangY. GuoJ. Li and H. Liu, Locating multiple multipolar acoustic sources using the direct sampling method, Comm. Comput. Phys., 25 (2019), 1328-1356.  doi: 10.4208/cicp.oa-2018-0020.  Google Scholar

Figure 1.  Contour plots of the reconstructed source function with different measured directions, where the dotted red line denotes the angle of observation directions. (a) Exact $ F_1 $, (b) exact $ F_2 $, (c) reconstructed $ F_1 $, (d) reconstructed $ F_2 $
Figure 2.  Contour plots of reconstructed source function with limit-view data, where the dotted red line denotes the angle of observation directions. Left column: reconstructed $ F_1 $, right column: reconstructed $ F_2 $
Figure 3.  Slice plots of the exact source functions, where the left column is sliced at $ x = 0 $; center column is sliced at $ y = 0 $; right column is sliced at $ z = 0 $. Top row: exact $ F^{(1)} $; center row: exact $ F^{(2)} $; bottom row: exact $ F^{(3)} $
Figure 4.  Slice plots of the reconstructed source functions, where the left column is sliced at $ x = 0 $; center column is sliced at $ y = 0 $; right column is sliced at $ z = 0 $. Top row: reconstructed $ F^{(1)} $; center row: reconstructed $ F^{(2)} $; bottom row: reconstructed $ F^{(3)} $
[1]

Xiaoli Feng, Meixia Zhao, Peijun Li, Xu Wang. An inverse source problem for the stochastic wave equation. Inverse Problems & Imaging, , () : -. doi: 10.3934/ipi.2021055

[2]

Jianli Xiang, Guozheng Yan. The uniqueness of the inverse elastic wave scattering problem based on the mixed reciprocity relation. Inverse Problems & Imaging, 2021, 15 (3) : 539-554. doi: 10.3934/ipi.2021004

[3]

Lekbir Afraites, Chorouk Masnaoui, Mourad Nachaoui. Shape optimization method for an inverse geometric source problem and stability at critical shape. Discrete & Continuous Dynamical Systems - S, 2021  doi: 10.3934/dcdss.2021006

[4]

Lekbir Afraites. A new coupled complex boundary method (CCBM) for an inverse obstacle problem. Discrete & Continuous Dynamical Systems - S, 2021  doi: 10.3934/dcdss.2021069

[5]

Zhousheng Ruan, Sen Zhang, Sican Xiong. Solving an inverse source problem for a time fractional diffusion equation by a modified quasi-boundary value method. Evolution Equations & Control Theory, 2018, 7 (4) : 669-682. doi: 10.3934/eect.2018032

[6]

Loc H. Nguyen, Qitong Li, Michael V. Klibanov. A convergent numerical method for a multi-frequency inverse source problem in inhomogenous media. Inverse Problems & Imaging, 2019, 13 (5) : 1067-1094. doi: 10.3934/ipi.2019048

[7]

Huai-An Diao, Peijun Li, Xiaokai Yuan. Inverse elastic surface scattering with far-field data. Inverse Problems & Imaging, 2019, 13 (4) : 721-744. doi: 10.3934/ipi.2019033

[8]

John C. Schotland, Vadim A. Markel. Fourier-Laplace structure of the inverse scattering problem for the radiative transport equation. Inverse Problems & Imaging, 2007, 1 (1) : 181-188. doi: 10.3934/ipi.2007.1.181

[9]

Armin Lechleiter, Tobias Rienmüller. Factorization method for the inverse Stokes problem. Inverse Problems & Imaging, 2013, 7 (4) : 1271-1293. doi: 10.3934/ipi.2013.7.1271

[10]

Peijun Li, Ganghua Yuan. Increasing stability for the inverse source scattering problem with multi-frequencies. Inverse Problems & Imaging, 2017, 11 (4) : 745-759. doi: 10.3934/ipi.2017035

[11]

Kenichi Sakamoto, Masahiro Yamamoto. Inverse source problem with a final overdetermination for a fractional diffusion equation. Mathematical Control & Related Fields, 2011, 1 (4) : 509-518. doi: 10.3934/mcrf.2011.1.509

[12]

Yuxuan Gong, Xiang Xu. Inverse random source problem for biharmonic equation in two dimensions. Inverse Problems & Imaging, 2019, 13 (3) : 635-652. doi: 10.3934/ipi.2019029

[13]

Shumin Li, Masahiro Yamamoto, Bernadette Miara. A Carleman estimate for the linear shallow shell equation and an inverse source problem. Discrete & Continuous Dynamical Systems, 2009, 23 (1&2) : 367-380. doi: 10.3934/dcds.2009.23.367

[14]

Gen Nakamura, Michiyuki Watanabe. An inverse boundary value problem for a nonlinear wave equation. Inverse Problems & Imaging, 2008, 2 (1) : 121-131. doi: 10.3934/ipi.2008.2.121

[15]

Renjun Duan, Shuangqian Liu. Cauchy problem on the Vlasov-Fokker-Planck equation coupled with the compressible Euler equations through the friction force. Kinetic & Related Models, 2013, 6 (4) : 687-700. doi: 10.3934/krm.2013.6.687

[16]

Jingzhi Li, Jun Zou. A direct sampling method for inverse scattering using far-field data. Inverse Problems & Imaging, 2013, 7 (3) : 757-775. doi: 10.3934/ipi.2013.7.757

[17]

Xiaoqiang Dai, Wenke Li. Non-global solution for visco-elastic dynamical system with nonlinear source term in control problem. Electronic Research Archive, 2021, 29 (6) : 4087-4098. doi: 10.3934/era.2021073

[18]

Ming Chen, Chongchao Huang. A power penalty method for the general traffic assignment problem with elastic demand. Journal of Industrial & Management Optimization, 2014, 10 (4) : 1019-1030. doi: 10.3934/jimo.2014.10.1019

[19]

Fei Meng, Xiao-Ping Yang. Elastic limit and vanishing external force for granular systems. Kinetic & Related Models, 2019, 12 (1) : 159-176. doi: 10.3934/krm.2019007

[20]

Mohsen Tadi. A computational method for an inverse problem in a parabolic system. Discrete & Continuous Dynamical Systems - B, 2009, 12 (1) : 205-218. doi: 10.3934/dcdsb.2009.12.205

2020 Impact Factor: 1.639

Article outline

Figures and Tables

[Back to Top]