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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 and 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.

[2]

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

[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.

[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.

[5]

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

[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.

[7]

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

[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.

[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.

[10]

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

[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.

[12]

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

[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.

[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.

[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.

[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.

[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.

[18]

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

[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.

[20]

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

[21]

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

[22]

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

[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.

[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.

[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.

[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.

[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.

[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.

[29]

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

[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.

[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.

[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.

[33]

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

[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.

[35]

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

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.

[2]

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

[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.

[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.

[5]

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

[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.

[7]

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

[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.

[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.

[10]

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

[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.

[12]

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

[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.

[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.

[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.

[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.

[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.

[18]

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

[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.

[20]

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

[21]

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

[22]

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

[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.

[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.

[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.

[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.

[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.

[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.

[29]

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

[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.

[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.

[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.

[33]

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

[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.

[35]

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

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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