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Inverse source problems in electrodynamics
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 |
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.
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. |

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