Fourier method for identifying electromagnetic sources with multi-frequency far-field data. (English) Zbl 1415.35292
Summary: We consider the inverse problem of determining an unknown vectorial source current distribution associated with the homogeneous Maxwell system. We propose a novel non-iterative reconstruction method for solving the aforementioned inverse problem from multi-frequency far-field measurements. The method is based on recovering the Fourier coefficients of the unknown source. A key ingredient of the method is to establish the relationship between the Fourier coefficients and the multi-frequency far-field data. Uniqueness and stability results are established for the proposed reconstruction method. Numerical experiments are presented to illustrate the effectiveness and efficiency of the method.
MSC:
| 35R30 | Inverse problems for PDEs |
| 35P25 | Scattering theory for PDEs |
| 78A46 | Inverse problems (including inverse scattering) in optics and electromagnetic theory |
| 35Q60 | PDEs in connection with optics and electromagnetic theory |
References:
| [1] | Anastasio, M.; Zhang, J.; Modgil, D.; La Rivi, P., Application of inverse source concepts to photoacoustic tomography, Inverse Problems, 23, 6, 21-35 (2007) · Zbl 1125.92033 |
| [2] | Clason, C.; Klibanov, M., The quasi-reversibility method for thermoacoustic tomography in a heterogeneous medium, SIAM J. Sci. Comput., 30, 1, 1-23 (2007) · Zbl 1159.65346 |
| [3] | Klibanov, M., Thermoacoustic tomography with an arbitrary elliptic operator, Inverse Problems, 29, 2, Article 025014 pp. (2013) · Zbl 1302.35376 |
| [4] | Liu, H.; Uhlmann, G., Determining both sound speed and internal source in thermo- and photo-acoustic tomography, Inverse Problems, 31, 10, Article 105005 pp. (2015) · Zbl 1328.35316 |
| [5] | Ammari, H.; Bao, G.; Fleming, J., An inverse source problem for Maxwell’s equations in magnetoencephalography, SIAM J. Appl. Math., 62, 4, 1369-1382 (2002) · Zbl 1003.35123 |
| [6] | Arridge, S., Optical tomography in medical imaging, Inverse Problems, 15, 2, R41-R93 (1999) · Zbl 0926.35155 |
| [7] | Fokas, A.; Kurylev, Y.; Marinakis, V., The unique determination of neuronal currents in the brain via magnetoencephalography, Inverse Problems, 20, 4, 1067-1082 (2004) · Zbl 1075.92012 |
| [8] | El Badia, A.; Ha-Duong, T., On an inverse source problem for the heat equation. Application to a pollution detection problem, J. Inverse Ill-Posed Probl., 10, 6, 585-599 (2002) · Zbl 1028.35164 |
| [9] | Bao, G.; Li, P.; Zhao, Y., Stability in the inverse source problem for elastic and electromagnetic waves with multi-frequencies (2017) |
| [10] | Isakov, V., Inverse Source Problems, Mathematical Surveys and Monographs (1990), American Mathematical Society: American Mathematical Society Providence · Zbl 0721.31002 |
| [11] | Alzaalig, A.; Hu, G.; Liu, X.; Sun, J., Fast acoustic source imaging using multi-frequency sparse data (2017) |
| [12] | El Badia, A.; Nara, T., Inverse dipole source problem for time-harmonic Maxwell equations: algebraic algorithm and Hölder stability, Inverse Problems, 29, 1, Article 015007 pp. (2013) · Zbl 1266.78004 |
| [13] | He, S.; Romanov, V., Identification of dipole sources in a bounded domain for Maxwell’s equations, Wave Motion, 28, 1, 25-40 (1998) · Zbl 1074.78506 |
| [14] | Eller, M.; Valdivia, N., Acoustic source identification using multiple frequency information, Inverse Problems, 25, 11, Article 115005 pp. (2009) · Zbl 1181.35328 |
| [15] | Valdivia, N., Electromagnetic source identification using multiple frequency information, Inverse Problems, 28, 11, Article 115002 pp. (2012) · Zbl 1259.78014 |
| [16] | Bleistein, N.; Cohen, J., Nonuniqueness in the inverse source problem in acoustics and electromagnetics, J. Math. Phys., 18, 2, 194-201 (1977) · Zbl 0379.76076 |
| [17] | Marengo, E.; Devaney, A., Nonradiating sources with connections to the adjoint problem, Phys. Rev. E., 70, Article 037601 pp. (2004) |
| [18] | Albanese, R.; Monk, P., The inverse source problem for Maxwell’s equations, Inverse Problems, 22, 3, 1023-1035 (2006) · Zbl 1099.35161 |
| [19] | Wang, G.; Ma, F.; Guo, Y.; Li, J., Solving the multi-frequency electromagnetic inverse source problem by the Fourier method, J. Differential Equations, 265, 1, 417-443 (2018) · Zbl 1388.78005 |
| [20] | Wang, X.; Guo, Y.; Zhang, D.; Liu, H., Fourier method for recovering acoustic sources from multi-frequency far-field data, Inverse Problems, 33, 3, Article 035001 pp. (2017) · Zbl 1401.35352 |
| [21] | Zhang, D.; Guo, Y., Fourier method for solving the multi-frequency inverse source problem for the Helmholtz equation, Inverse Problems, 31, 3, Article 035007 pp. (2015) · Zbl 1325.35288 |
| [22] | Tai, C., Dyadic Green Functions in Electromagnetic Theory, 48-50 (1994), IEEE: IEEE New York · Zbl 0913.73002 |
| [23] | Colton, D.; Kress, R., Inverse Acoustic and Electromagnetic Scattering Theory (2013), Springer-Verlag: Springer-Verlag Berlin · Zbl 1266.35121 |
| [24] | Lindell, I. V., TE/TM decomposition of electromagnetic sources, IEEE Trans. Antennas and Propagation, 36, 10, 1382-1388 (1988) · Zbl 1058.78506 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
