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December  2018, 12(6): 1411-1428. doi: 10.3934/ipi.2018059

Inverse source problems in electrodynamics

Guanghui Hu 1, , Peijun Li 2, , Xiaodong Liu 3, and  Yue Zhao 4,,

1. 

Beijing Computational Science Research Center, Building 9, East Zone, ZPark Ⅱ, No.10 Xibeiwang East Road, Haidian District, Beijing 100193, China
2. 

Department of Mathematics, Purdue University, West Lafayette, Indiana 47907, USA
3. 

Institute of Applied Mathematics, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, China
4. 

School of Mathematics and Statistics, Central China Normal University, Wuhan 430079, China

* Corresponding author: Yue Zhao

Received  January 2018 Revised  July 2018 Published  October 2018

Fund Project: The research of PL was supported in part by the NSF grant DMS-1151308. The work of GH is supported by the NSFC grant (No. 11671028) and NSAF grant (No. U1530401). The research of XL is supported by the NNSF of China under grant 11571355 and the Youth Innovation Promotion Association, CAS.

This paper concerns inverse source problems for the time-dependent Maxwell equations. The electric current density is assumed to be the product of a spatial function and a temporal function. We prove uniqueness and stability in determining the spatial or temporal function from the electric field, which is measured on a sphere or at a point over a finite time interval.

Keywords: Stability, uniqueness, inverse source problem, Maxwell's equations.
Mathematics Subject Classification: Primary: 35R30; Secondary: 35Q61.
Citation: Guanghui Hu, Peijun Li, Xiaodong Liu, Yue Zhao. Inverse source problems in electrodynamics. Inverse Problems & Imaging, 2018, 12 (6) : 1411-1428. doi: 10.3934/ipi.2018059
References:
[1]

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

[2]

A. Alzaalig, G. Hu, X. Liu and J. Sun, Fast acoustic source imaging using multi-frequency sparse data, arXiv: 1712.02654v1, 2017. Google Scholar

[3]

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

[4]

H. Ammari, G. Bao and J. Flemming, An inverse source problem for Maxwell's equations in magnetoencephalography, SIAM J. Appl. Math., 62 (2002), 1369-1382.  doi: 10.1137/S0036139900373927.  Google Scholar

[5]

H. Ammari and J.-C. Nédélec, Low-frequency electromagnetic scattering, SIAM J. Math. Anal., 31 (2000), 836-861.  doi: 10.1137/S0036141098343604.  Google Scholar

[6]

Yu. E. Anikonov, J. Cheng and M. Yamamoto, A uniqueness result in an inverse hyperbolic problem with analyticity, European J. Appl. Math., 15 (2004), 533-543.  doi: 10.1017/S0956792504005649.  Google Scholar

[7]

S. Arridge, Optical tomography in medical imaging, Inverse Problems, 15 (1999), R41-R93.  doi: 10.1088/0266-5611/15/2/022.  Google Scholar

[8]

G. Bao, G. Hu, Y. Kian and T. Yin, Inverse source problems in elastodynamics, Inverse Problems, 34 (2018), 045009, 31pp. doi: 10.1088/1361-6420/aaaf7e.  Google Scholar

[9]

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

[10]

G. Bao, P. Li and Y. Zhao, Stability in the inverse source problem for elastic and electromagnetic waves with multi-frequencies, preprint. Google Scholar

[11]

G. Bao, J. Lin and F. Triki, A multi-frequency inverse source problem, J. Differential Equations, 249 (2010), 3443-3465.  doi: 10.1016/j.jde.2010.08.013.  Google Scholar

[12]

Z. Chen and J.-C. Nédélec, On Maxwell equations with the transparent boundary condition, J. Computational Mathematics, 26 (2008), 284-296.   Google Scholar

[13]

J. Cheng, V. Isakov and S. Lu, Increasing stability in the inverse source problem with many frequencies, J. Differential Equations, 260 (2016), 4786-4804.  doi: 10.1016/j.jde.2015.11.030.  Google Scholar

[14]

X. Deng, X. Cai and J. Zou, A parallel space-time domain decomposition method for unsteady source inversion problems, Inverse Probl. Imaging, 9 (2015), 1069-1091.  doi: 10.3934/ipi.2015.9.1069.  Google Scholar

[15]

X. Deng, X. Cai and J. Zou, Two-level space-time domain decomposition methods for three-dimensional unsteady inverse source problems, J. Sci. Comput., 67 (2016), 860-882.  doi: 10.1007/s10915-015-0109-1.  Google Scholar

[16]

A. Devaney and G. Sherman, Nonuniqueness in inverse source and scattering problems, IEEE Trans. Antennas Propag., 30 (1982), 1034-1042.  doi: 10.1109/TAP.1982.1142902.  Google Scholar

[17]

M. Eller and N. Valdivia, Acoustic source identification using multiple frequency information, Inverse Problems, 25 (2009), 115005, 20pp. doi: 10.1088/0266-5611/25/11/115005.  Google Scholar

[18]

L. B. Felsen and N. Marcuvitz, Radiation and Scattering of Waves, IEEE press, Prentice-Hall, 1973.  Google Scholar

[19]

K.-H. Hauer, L. Kühn and R. Potthast, On uniqueness and non-uniqueness for current reconstruction from magnetic fields, Inverse Problems, 21 (2005), 955-967.  doi: 10.1088/0266-5611/21/3/010.  Google Scholar

[20]

V. Isakov, Inverse Source Problems, AMS, Providence, RI, 1989. doi: 10.1090/surv/034.  Google Scholar

[21]

J. Jackson, Classical Electrodynamics, Second edition, John Wiley and Sons, Inc., New York-London-Sydney, 1975.  Google Scholar

[22]

M. V. Klibanov, Inverse problems and Carleman estimates, Inverse Problems, 8 (1992), 575-596.  doi: 10.1088/0266-5611/8/4/009.  Google Scholar

[23]

P. Li and G. Yuan, Stability on the inverse random source scattering problem for the one-dimensional Helmholtz equation, J. Math. Anal. Appl., 450 (2017), 872-887.  doi: 10.1016/j.jmaa.2017.01.074.  Google Scholar

[24]

P. Li and G. Yuan, Increasing stability for the inverse source scattering problem with multi-frequencies, Inverse Problems and Imaging, 11 (2017), 745-759.  doi: 10.3934/ipi.2017035.  Google Scholar

[25]

S. Li, Carleman estimates for second order hyperbolic systems in anisotropic cases and an inverse source problem. Part Ⅱ: an inverse source problem, Applicable Analysis, 94 (2015), 2287-2307.  doi: 10.1080/00036811.2014.986847.  Google Scholar

[26]

S. Li and M. Yamamoto, An inverse source problem for Maxwell's equations in anisotropic media, Applicable Analysis, 84 (2005), 1051-1067.  doi: 10.1080/00036810500047725.  Google Scholar

[27]

K. Liu, Y. Xu and J. Zou, A multilevel sampling method for detecting sources in a stratified ocean waveguide, J. Comput. Appl. Math., 309 (2017), 95-110.  doi: 10.1016/j.cam.2016.06.039.  Google Scholar

[28]

E. Marx and D. Maystre, Dyadic Green functions for the time-dependent wave equation, J. Math. Phys., 23 (1982), 1047-1056.  doi: 10.1063/1.525493.  Google Scholar

[29]

J.-C. Nédélec, Acoustic and Electromagnetic Equations: Integral Representations for Harmonic Problems, Springer, New York, 2000. Google Scholar

[30]

R. Nevels and J. Jeong, Time domain coupled field dyadic Green function solution for Maxwell's equations, IEEE Transactions on Antennas and Propagation, 56 (2008), 2761-2764.  doi: 10.1109/TAP.2008.927574.  Google Scholar

[31]

P. Ola, L. Päivärinta and E. Somersalo, An inverse boundary value problem in electrodynamics, Duke Math. J., 70 (1993), 617-653.  doi: 10.1215/S0012-7094-93-07014-7.  Google Scholar

[32]

A. G. Ramm and E. Somersalo, Electromagnetic inverse problem with surface measurements at low frequencies, Inverse Problems, 5 (1989), 1107-1116.  doi: 10.1088/0266-5611/5/6/016.  Google Scholar

[33]

V. G. Romanov and S. I. Kabanikhin, Inverse Problems for Maxwell's Equations, VSP, Utrecht, 1994. doi: 10.1515/9783110900101.  Google Scholar

[34]

M. Yamamoto, On an inverse problem of determining source terms in Maxwell's equations with a single measurement, Inverse Probl. Tomograph. Image Process, New York, Plenum Press, 15 (1998), 241–256.  Google Scholar

[35]

Y. Zhao and P. Li, Stability on the one-dimensional inverse source scattering problem in a two-layered medium, Applicable Analysis, to appear. Google Scholar

show all references

References:
[1]

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

[2]

A. Alzaalig, G. Hu, X. Liu and J. Sun, Fast acoustic source imaging using multi-frequency sparse data, arXiv: 1712.02654v1, 2017. Google Scholar

[3]

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

[4]

H. Ammari, G. Bao and J. Flemming, An inverse source problem for Maxwell's equations in magnetoencephalography, SIAM J. Appl. Math., 62 (2002), 1369-1382.  doi: 10.1137/S0036139900373927.  Google Scholar

[5]

H. Ammari and J.-C. Nédélec, Low-frequency electromagnetic scattering, SIAM J. Math. Anal., 31 (2000), 836-861.  doi: 10.1137/S0036141098343604.  Google Scholar

[6]

Yu. E. Anikonov, J. Cheng and M. Yamamoto, A uniqueness result in an inverse hyperbolic problem with analyticity, European J. Appl. Math., 15 (2004), 533-543.  doi: 10.1017/S0956792504005649.  Google Scholar

[7]

S. Arridge, Optical tomography in medical imaging, Inverse Problems, 15 (1999), R41-R93.  doi: 10.1088/0266-5611/15/2/022.  Google Scholar

[8]

G. Bao, G. Hu, Y. Kian and T. Yin, Inverse source problems in elastodynamics, Inverse Problems, 34 (2018), 045009, 31pp. doi: 10.1088/1361-6420/aaaf7e.  Google Scholar

[9]

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

[10]

G. Bao, P. Li and Y. Zhao, Stability in the inverse source problem for elastic and electromagnetic waves with multi-frequencies, preprint. Google Scholar

[11]

G. Bao, J. Lin and F. Triki, A multi-frequency inverse source problem, J. Differential Equations, 249 (2010), 3443-3465.  doi: 10.1016/j.jde.2010.08.013.  Google Scholar

[12]

Z. Chen and J.-C. Nédélec, On Maxwell equations with the transparent boundary condition, J. Computational Mathematics, 26 (2008), 284-296.   Google Scholar

[13]

J. Cheng, V. Isakov and S. Lu, Increasing stability in the inverse source problem with many frequencies, J. Differential Equations, 260 (2016), 4786-4804.  doi: 10.1016/j.jde.2015.11.030.  Google Scholar

[14]

X. Deng, X. Cai and J. Zou, A parallel space-time domain decomposition method for unsteady source inversion problems, Inverse Probl. Imaging, 9 (2015), 1069-1091.  doi: 10.3934/ipi.2015.9.1069.  Google Scholar

[15]

X. Deng, X. Cai and J. Zou, Two-level space-time domain decomposition methods for three-dimensional unsteady inverse source problems, J. Sci. Comput., 67 (2016), 860-882.  doi: 10.1007/s10915-015-0109-1.  Google Scholar

[16]

A. Devaney and G. Sherman, Nonuniqueness in inverse source and scattering problems, IEEE Trans. Antennas Propag., 30 (1982), 1034-1042.  doi: 10.1109/TAP.1982.1142902.  Google Scholar

[17]

M. Eller and N. Valdivia, Acoustic source identification using multiple frequency information, Inverse Problems, 25 (2009), 115005, 20pp. doi: 10.1088/0266-5611/25/11/115005.  Google Scholar

[18]

L. B. Felsen and N. Marcuvitz, Radiation and Scattering of Waves, IEEE press, Prentice-Hall, 1973.  Google Scholar

[19]

K.-H. Hauer, L. Kühn and R. Potthast, On uniqueness and non-uniqueness for current reconstruction from magnetic fields, Inverse Problems, 21 (2005), 955-967.  doi: 10.1088/0266-5611/21/3/010.  Google Scholar

[20]

V. Isakov, Inverse Source Problems, AMS, Providence, RI, 1989. doi: 10.1090/surv/034.  Google Scholar

[21]

J. Jackson, Classical Electrodynamics, Second edition, John Wiley and Sons, Inc., New York-London-Sydney, 1975.  Google Scholar

[22]

M. V. Klibanov, Inverse problems and Carleman estimates, Inverse Problems, 8 (1992), 575-596.  doi: 10.1088/0266-5611/8/4/009.  Google Scholar

[23]

P. Li and G. Yuan, Stability on the inverse random source scattering problem for the one-dimensional Helmholtz equation, J. Math. Anal. Appl., 450 (2017), 872-887.  doi: 10.1016/j.jmaa.2017.01.074.  Google Scholar

[24]

P. Li and G. Yuan, Increasing stability for the inverse source scattering problem with multi-frequencies, Inverse Problems and Imaging, 11 (2017), 745-759.  doi: 10.3934/ipi.2017035.  Google Scholar

[25]

S. Li, Carleman estimates for second order hyperbolic systems in anisotropic cases and an inverse source problem. Part Ⅱ: an inverse source problem, Applicable Analysis, 94 (2015), 2287-2307.  doi: 10.1080/00036811.2014.986847.  Google Scholar

[26]

S. Li and M. Yamamoto, An inverse source problem for Maxwell's equations in anisotropic media, Applicable Analysis, 84 (2005), 1051-1067.  doi: 10.1080/00036810500047725.  Google Scholar

[27]

K. Liu, Y. Xu and J. Zou, A multilevel sampling method for detecting sources in a stratified ocean waveguide, J. Comput. Appl. Math., 309 (2017), 95-110.  doi: 10.1016/j.cam.2016.06.039.  Google Scholar

[28]

E. Marx and D. Maystre, Dyadic Green functions for the time-dependent wave equation, J. Math. Phys., 23 (1982), 1047-1056.  doi: 10.1063/1.525493.  Google Scholar

[29]

J.-C. Nédélec, Acoustic and Electromagnetic Equations: Integral Representations for Harmonic Problems, Springer, New York, 2000. Google Scholar

[30]

R. Nevels and J. Jeong, Time domain coupled field dyadic Green function solution for Maxwell's equations, IEEE Transactions on Antennas and Propagation, 56 (2008), 2761-2764.  doi: 10.1109/TAP.2008.927574.  Google Scholar

[31]

P. Ola, L. Päivärinta and E. Somersalo, An inverse boundary value problem in electrodynamics, Duke Math. J., 70 (1993), 617-653.  doi: 10.1215/S0012-7094-93-07014-7.  Google Scholar

[32]

A. G. Ramm and E. Somersalo, Electromagnetic inverse problem with surface measurements at low frequencies, Inverse Problems, 5 (1989), 1107-1116.  doi: 10.1088/0266-5611/5/6/016.  Google Scholar

[33]

V. G. Romanov and S. I. Kabanikhin, Inverse Problems for Maxwell's Equations, VSP, Utrecht, 1994. doi: 10.1515/9783110900101.  Google Scholar

[34]

M. Yamamoto, On an inverse problem of determining source terms in Maxwell's equations with a single measurement, Inverse Probl. Tomograph. Image Process, New York, Plenum Press, 15 (1998), 241–256.  Google Scholar

[35]

Y. Zhao and P. Li, Stability on the one-dimensional inverse source scattering problem in a two-layered medium, Applicable Analysis, to appear. Google Scholar

Figure 1.  (left): A Gaussian-modulated sinusoidal pulse function $\chi$ with $\omega = 6$, $\sigma = 1.6$, $\tau = 3$; (right): Fourier spectrum of $\chi$
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