# American Institute of Mathematical Sciences

November  2014, 8(4): 1033-1051. doi: 10.3934/ipi.2014.8.1033

## Reconstruction of complex-valued tensors in the Maxwell system from knowledge of internal magnetic fields

 1 Department of Applied Physics and Applied Mathematics, Columbia University, New York, NY 10027 2 Department of Applied Physics and Applied Mathematics, Columbia University, 200 S. W. Mudd Building, MC 4701, 500 W. 120th Street, New York, NY 10027

Received  August 2013 Revised  May 2014 Published  November 2014

This paper concerns the reconstruction of a complex-valued anisotropic tensor $\gamma = \sigma + \iota\omega\varepsilon$ from knowledge of several internal magnetic fields $H$, where $H$ satisfies the anisotropic Maxwell system on a bounded domain with prescribed boundary conditions. We show that $\gamma$ can be uniquely reconstructed with a loss of two derivatives from errors in the acquisition of $H$. A minimum number of $6$ such functionals is sufficient to obtain a local reconstruction of $\gamma$ in dimension three provided that the electric field satisfies appropriate boundary conditions. When $\gamma$ is close to a scalar tensor, such boundary conditions are shown to exist using the notion of complex geometric optics (CGO) solutions. For arbitrary symmetric tensors $\gamma$, a Runge approximation property is used instead to obtain partial results. This problem finds applications in the medical imaging modalities Current Density Imaging and Magnetic Resonance Electrical Impedance Tomography.
Citation: Chenxi Guo, Guillaume Bal. Reconstruction of complex-valued tensors in the Maxwell system from knowledge of internal magnetic fields. Inverse Problems and Imaging, 2014, 8 (4) : 1033-1051. doi: 10.3934/ipi.2014.8.1033
##### References:
 [1] G. Alessandrini, An identification problem for an elliptic equation in two variables, Ann. Mat. Pura Appl., 145 (1986), 265-295. doi: 10.1007/BF01790543. [2] H. Ammari, E. Bonnetier, Y. Capdeboscq, M. Tanter and M. Fink, Electrical Impedance Tomography by elastic deformation, SIAM J. Appl. Math., 68 (2008), 1557-1573. doi: 10.1137/070686408. [3] G. Bal, Inside Out, Cambridge University Press, 2012, ch. Hybrid inverse problems and internal functionals. [4] G. Bal, C. Guo and F. Monard, Linearized internal functionals for anisotropic conductivities, Inv. Probl. and Imaging, 8 (2014), 1-22. doi: 10.3934/ipi.2014.8.1. [5] _______, Inverse anisotropic conductivity from internal current densities, Inverse Problems, 30 (2014), 025001. [6] _______, Imaging of anisotropic conductivities from current densities in two dimensions, submitted, (2014). arXiv:1403.4964. [7] G. Bal and J. C. Schotland, Inverse scattering and acousto-optic imaging, Phys. Rev. Letters, 104 (2010), p. 043902. doi: 10.1103/PhysRevLett.104.043902. [8] G. Bal and G. Uhlmann, Inverse diffusion theory of photoacoustics, Inverse Problems, 26 (2010), 085010, 20pp. doi: 10.1088/0266-5611/26/8/085010. [9] _______, Reconstruction of coefficients in scalar second-order elliptic equations from knowledge of their solutions, Communications on Pure and Applied Mathematics, 66 (2013), 1629-1652. [10] A. Calderón, Uniqueness in the cauchy problem for partial differential equations, Amer.J.Math., 80 (1958), 16-36. doi: 10.2307/2372819. [11] P. Caro, P. Ola and M. Salo, Inverse boundary value problem for Maxwell equations with local data, Comm.PDE., 34 (2009), 1425-1464. doi: 10.1080/03605300903296272. [12] J. Chen and Y. Yang, Inverse problem of electro-seismic conversion, Inverse Problems, 29 (2013), 115006, 15pp. doi: 10.1088/0266-5611/29/11/115006. [13] D. Colton and R. Kress, Inverse Acoustic and Electromagnetic Scattering Theory, Applied Mathematical Sciences, Vol. 93. Springer, 2012. doi: 10.1007/978-1-4614-4942-3. [14] D. Colton and L. Päivärinta, The uniqueness of a solution to an inverse scattering problem for electromagnetic waves, Arch.Rational Mech.Anal., 119 (1992), 59-70. doi: 10.1007/BF00376010. [15] M. Eller and M.Yamamoto, A Carleman inequality for the stationary anisotropic Maxwell system, J.Math.Pures Appl., 86 (2006), 449-462. doi: 10.1016/j.matpur.2006.10.004. [16] L. Hormander, The Analysis of Linear Partial Differential Operators:Pseudo-Differential Operators I-IV, Springer Verlag, 1983. [17] R. Horn and C. Johnson, Matrix Analysis, Cambridge University Press, 1990. [18] Y. Ider and L. Muftuler, Measurement of AC magnetic field distribution using magnetic resonance imaging, IEEE Transactions on Medical Imaging, 16 (1997), 617-622. doi: 10.1109/42.640752. [19] Y. Ider and Özlem Birgül, Use of the magnetic field generated by the internal distribution of injected currents for electrical impedance tomography (MR-EIT), Elektrik, 6 (1998), 215-225. [20] C. E. Kenig, M. Salo and G. Uhlmann, Inverse problems for the anisotropic Maxwell equations, Duke Math.J., 157 (2011), 369-419. doi: 10.1215/00127094-1272903. [21] R. Kohn and M. Vogelius, Determining conductivity by boundary measurements, Commun. Pure Appl. Math., 37 (1984), 289-298. doi: 10.1002/cpa.3160370302. [22] P. Kuchment and L. Kunyansky, 2D and 3D reconstructions in acousto-electric tomography, Inverse Problems, 27 (2011), 055013, 21pp. doi: 10.1088/0266-5611/27/5/055013. [23] P. Kuchment and D. Steinhauer, Stabilizing inverse problems by internal data, Inverse Problems, 28 (2012), 084007, 20pp. doi: 10.1088/0266-5611/28/8/084007. [24] O. Kwon, E. Woo, J. Yoon and J. Seo, Magnetic resonance electrical impedance tomography (MREIT): Simulation study of J-substitution algorithm, IEEE Trans. Biomed. Eng., 49 (2002), 160-167. [25] P. Lax, A stability theorem for solutions of abstract differential equations, and tis application to the study of the local behavior of solutions to elliptic equations, Comm.Pure Applied Math., 9 (1956), 747-766. doi: 10.1002/cpa.3160090407. [26] F. Monard and G. Bal, Inverse anisotropic conductivity from power densities in dimension $n \ge 3$, Comm. Partial Differential Equations, 38 (2013), 1183-1207. doi: 10.1080/03605302.2013.787089. [27] A. Nachman, A. Tamasan and A. Timonov, Recovering the conductivity from a single measurement of interior data, Inverse Problems, 25 (2009), 035014, 16pp. doi: 10.1088/0266-5611/25/3/035014. [28] ________, Reconstruction of planar conductivities in subdomains from incomplete data, SIAM J. Appl. Math., 70 (2010), 3342-3362. [29] G. Nakamura, G. Uhlmann and J.N. Wang, Oscillating-decaying solutions, Runge approximation property for the anisotropic elasticity system and their applications to inverse problems, J.Math.Pures Appl., 84 (2005), 21-54. doi: 10.1016/j.matpur.2004.09.002. [30] L. Nirenberg, Lectures on Linear Partial Differential Equations, Amer. Math. Soc., Providencem R.I., 1973. [31] P. Ola, L. Päivärinta and E. Somersalo, An inverse boundary value problem in electromagnetics, Duke Math. J., 70 (1993), 617-653. doi: 10.1215/S0012-7094-93-07014-7. [32] P. Ola and E. Somersalo, Electromagnetic inverse problems and generalized Sommerfeld potentials, SIAM J.Appl.Math., 56 (1996), 1129-1145. doi: 10.1137/S0036139995283948. [33] J. K. Seo, D.-H. Kim, J. Lee, O. I. Kwon, S. Z. K. Sajib and E. J. Woo, Electrical tissue property imaging using MRI at dc and larmor frequency, Inverse Problems, 28 (2012), 084002, 26pp. doi: 10.1088/0266-5611/28/8/084002. [34] E. Somersalo, D. Isaacson and M. Cheney, A linearized inverse boundary value problem for Maxwell's equations, J.Comp.Appl.Math, 42 (1992), 123-136. doi: 10.1016/0377-0427(92)90167-V. [35] Z. Sun and G. Uhlmann, An inverse boundary value problem for Maxwell's equations, Arch.Rational Mech.Anal., 119 (1992), 71-93. doi: 10.1007/BF00376011. [36] J. Sylvester and G. Uhlmann, A global uniqueness theorem for an inverse boundary value problem, Ann. of Math., 125 (1987), 153-169. doi: 10.2307/1971291. [37] G. Uhlmann, Calderón's problem and electrical impedance tomography, Inverse Problems, 25 (2009), 123011. doi: 10.1088/0266-5611/25/12/123011. [38] C. Weber, Regularity theorems for Maxwell's equations, Mathematical Methods in the Applied Sciences, 3 (1981), 523-536. doi: 10.1002/mma.1670030137.

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##### References:
 [1] G. Alessandrini, An identification problem for an elliptic equation in two variables, Ann. Mat. Pura Appl., 145 (1986), 265-295. doi: 10.1007/BF01790543. [2] H. Ammari, E. Bonnetier, Y. Capdeboscq, M. Tanter and M. Fink, Electrical Impedance Tomography by elastic deformation, SIAM J. Appl. Math., 68 (2008), 1557-1573. doi: 10.1137/070686408. [3] G. Bal, Inside Out, Cambridge University Press, 2012, ch. Hybrid inverse problems and internal functionals. [4] G. Bal, C. Guo and F. Monard, Linearized internal functionals for anisotropic conductivities, Inv. Probl. and Imaging, 8 (2014), 1-22. doi: 10.3934/ipi.2014.8.1. [5] _______, Inverse anisotropic conductivity from internal current densities, Inverse Problems, 30 (2014), 025001. [6] _______, Imaging of anisotropic conductivities from current densities in two dimensions, submitted, (2014). arXiv:1403.4964. [7] G. Bal and J. C. Schotland, Inverse scattering and acousto-optic imaging, Phys. Rev. Letters, 104 (2010), p. 043902. doi: 10.1103/PhysRevLett.104.043902. [8] G. Bal and G. Uhlmann, Inverse diffusion theory of photoacoustics, Inverse Problems, 26 (2010), 085010, 20pp. doi: 10.1088/0266-5611/26/8/085010. [9] _______, Reconstruction of coefficients in scalar second-order elliptic equations from knowledge of their solutions, Communications on Pure and Applied Mathematics, 66 (2013), 1629-1652. [10] A. Calderón, Uniqueness in the cauchy problem for partial differential equations, Amer.J.Math., 80 (1958), 16-36. doi: 10.2307/2372819. [11] P. Caro, P. Ola and M. Salo, Inverse boundary value problem for Maxwell equations with local data, Comm.PDE., 34 (2009), 1425-1464. doi: 10.1080/03605300903296272. [12] J. Chen and Y. Yang, Inverse problem of electro-seismic conversion, Inverse Problems, 29 (2013), 115006, 15pp. doi: 10.1088/0266-5611/29/11/115006. [13] D. Colton and R. Kress, Inverse Acoustic and Electromagnetic Scattering Theory, Applied Mathematical Sciences, Vol. 93. Springer, 2012. doi: 10.1007/978-1-4614-4942-3. [14] D. Colton and L. Päivärinta, The uniqueness of a solution to an inverse scattering problem for electromagnetic waves, Arch.Rational Mech.Anal., 119 (1992), 59-70. doi: 10.1007/BF00376010. [15] M. Eller and M.Yamamoto, A Carleman inequality for the stationary anisotropic Maxwell system, J.Math.Pures Appl., 86 (2006), 449-462. doi: 10.1016/j.matpur.2006.10.004. [16] L. Hormander, The Analysis of Linear Partial Differential Operators:Pseudo-Differential Operators I-IV, Springer Verlag, 1983. [17] R. Horn and C. Johnson, Matrix Analysis, Cambridge University Press, 1990. [18] Y. Ider and L. Muftuler, Measurement of AC magnetic field distribution using magnetic resonance imaging, IEEE Transactions on Medical Imaging, 16 (1997), 617-622. doi: 10.1109/42.640752. [19] Y. Ider and Özlem Birgül, Use of the magnetic field generated by the internal distribution of injected currents for electrical impedance tomography (MR-EIT), Elektrik, 6 (1998), 215-225. [20] C. E. Kenig, M. Salo and G. Uhlmann, Inverse problems for the anisotropic Maxwell equations, Duke Math.J., 157 (2011), 369-419. doi: 10.1215/00127094-1272903. [21] R. Kohn and M. Vogelius, Determining conductivity by boundary measurements, Commun. Pure Appl. Math., 37 (1984), 289-298. doi: 10.1002/cpa.3160370302. [22] P. Kuchment and L. Kunyansky, 2D and 3D reconstructions in acousto-electric tomography, Inverse Problems, 27 (2011), 055013, 21pp. doi: 10.1088/0266-5611/27/5/055013. [23] P. Kuchment and D. Steinhauer, Stabilizing inverse problems by internal data, Inverse Problems, 28 (2012), 084007, 20pp. doi: 10.1088/0266-5611/28/8/084007. [24] O. Kwon, E. Woo, J. Yoon and J. Seo, Magnetic resonance electrical impedance tomography (MREIT): Simulation study of J-substitution algorithm, IEEE Trans. Biomed. Eng., 49 (2002), 160-167. [25] P. Lax, A stability theorem for solutions of abstract differential equations, and tis application to the study of the local behavior of solutions to elliptic equations, Comm.Pure Applied Math., 9 (1956), 747-766. doi: 10.1002/cpa.3160090407. [26] F. Monard and G. Bal, Inverse anisotropic conductivity from power densities in dimension $n \ge 3$, Comm. Partial Differential Equations, 38 (2013), 1183-1207. doi: 10.1080/03605302.2013.787089. [27] A. Nachman, A. Tamasan and A. Timonov, Recovering the conductivity from a single measurement of interior data, Inverse Problems, 25 (2009), 035014, 16pp. doi: 10.1088/0266-5611/25/3/035014. [28] ________, Reconstruction of planar conductivities in subdomains from incomplete data, SIAM J. Appl. Math., 70 (2010), 3342-3362. [29] G. Nakamura, G. Uhlmann and J.N. Wang, Oscillating-decaying solutions, Runge approximation property for the anisotropic elasticity system and their applications to inverse problems, J.Math.Pures Appl., 84 (2005), 21-54. doi: 10.1016/j.matpur.2004.09.002. [30] L. Nirenberg, Lectures on Linear Partial Differential Equations, Amer. Math. Soc., Providencem R.I., 1973. [31] P. Ola, L. Päivärinta and E. Somersalo, An inverse boundary value problem in electromagnetics, Duke Math. J., 70 (1993), 617-653. doi: 10.1215/S0012-7094-93-07014-7. [32] P. Ola and E. Somersalo, Electromagnetic inverse problems and generalized Sommerfeld potentials, SIAM J.Appl.Math., 56 (1996), 1129-1145. doi: 10.1137/S0036139995283948. [33] J. K. Seo, D.-H. Kim, J. Lee, O. I. Kwon, S. Z. K. Sajib and E. J. Woo, Electrical tissue property imaging using MRI at dc and larmor frequency, Inverse Problems, 28 (2012), 084002, 26pp. doi: 10.1088/0266-5611/28/8/084002. [34] E. Somersalo, D. Isaacson and M. Cheney, A linearized inverse boundary value problem for Maxwell's equations, J.Comp.Appl.Math, 42 (1992), 123-136. doi: 10.1016/0377-0427(92)90167-V. [35] Z. Sun and G. Uhlmann, An inverse boundary value problem for Maxwell's equations, Arch.Rational Mech.Anal., 119 (1992), 71-93. doi: 10.1007/BF00376011. [36] J. Sylvester and G. Uhlmann, A global uniqueness theorem for an inverse boundary value problem, Ann. of Math., 125 (1987), 153-169. doi: 10.2307/1971291. [37] G. Uhlmann, Calderón's problem and electrical impedance tomography, Inverse Problems, 25 (2009), 123011. doi: 10.1088/0266-5611/25/12/123011. [38] C. Weber, Regularity theorems for Maxwell's equations, Mathematical Methods in the Applied Sciences, 3 (1981), 523-536. doi: 10.1002/mma.1670030137.
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