# 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 & 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.  doi: 10.1007/BF01790543.  Google Scholar [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.  doi: 10.1137/070686408.  Google Scholar [3] G. Bal, Inside Out,, Cambridge University Press, (2012).   Google Scholar [4] G. Bal, C. Guo and F. Monard, Linearized internal functionals for anisotropic conductivities,, Inv. Probl. and Imaging, 8 (2014), 1.  doi: 10.3934/ipi.2014.8.1.  Google Scholar [5] _______, Inverse anisotropic conductivity from internal current densities,, Inverse Problems, 30 (2014).   Google Scholar [6] _______, Imaging of anisotropic conductivities from current densities in two dimensions, submitted, (2014)., , ().   Google Scholar [7] G. Bal and J. C. Schotland, Inverse scattering and acousto-optic imaging,, Phys. Rev. Letters, 104 (2010).  doi: 10.1103/PhysRevLett.104.043902.  Google Scholar [8] G. Bal and G. Uhlmann, Inverse diffusion theory of photoacoustics,, Inverse Problems, 26 (2010).  doi: 10.1088/0266-5611/26/8/085010.  Google Scholar [9] _______, Reconstruction of coefficients in scalar second-order elliptic equations from knowledge of their solutions,, Communications on Pure and Applied Mathematics, 66 (2013), 1629.   Google Scholar [10] A. Calderón, Uniqueness in the cauchy problem for partial differential equations,, Amer.J.Math., 80 (1958), 16.  doi: 10.2307/2372819.  Google Scholar [11] P. Caro, P. Ola and M. Salo, Inverse boundary value problem for Maxwell equations with local data,, Comm.PDE., 34 (2009), 1425.  doi: 10.1080/03605300903296272.  Google Scholar [12] J. Chen and Y. Yang, Inverse problem of electro-seismic conversion,, Inverse Problems, 29 (2013).  doi: 10.1088/0266-5611/29/11/115006.  Google Scholar [13] D. Colton and R. Kress, Inverse Acoustic and Electromagnetic Scattering Theory,, Applied Mathematical Sciences, (2012).  doi: 10.1007/978-1-4614-4942-3.  Google Scholar [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.  doi: 10.1007/BF00376010.  Google Scholar [15] M. Eller and M.Yamamoto, A Carleman inequality for the stationary anisotropic Maxwell system,, J.Math.Pures Appl., 86 (2006), 449.  doi: 10.1016/j.matpur.2006.10.004.  Google Scholar [16] L. Hormander, The Analysis of Linear Partial Differential Operators:Pseudo-Differential Operators I-IV,, Springer Verlag, (1983).   Google Scholar [17] R. Horn and C. Johnson, Matrix Analysis,, Cambridge University Press, (1990).   Google Scholar [18] Y. Ider and L. Muftuler, Measurement of AC magnetic field distribution using magnetic resonance imaging,, IEEE Transactions on Medical Imaging, 16 (1997), 617.  doi: 10.1109/42.640752.  Google Scholar [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.   Google Scholar [20] C. E. Kenig, M. Salo and G. Uhlmann, Inverse problems for the anisotropic Maxwell equations,, Duke Math.J., 157 (2011), 369.  doi: 10.1215/00127094-1272903.  Google Scholar [21] R. Kohn and M. Vogelius, Determining conductivity by boundary measurements,, Commun. Pure Appl. Math., 37 (1984), 289.  doi: 10.1002/cpa.3160370302.  Google Scholar [22] P. Kuchment and L. Kunyansky, 2D and 3D reconstructions in acousto-electric tomography,, Inverse Problems, 27 (2011).  doi: 10.1088/0266-5611/27/5/055013.  Google Scholar [23] P. Kuchment and D. Steinhauer, Stabilizing inverse problems by internal data,, Inverse Problems, 28 (2012).  doi: 10.1088/0266-5611/28/8/084007.  Google Scholar [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.   Google Scholar [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.  doi: 10.1002/cpa.3160090407.  Google Scholar [26] F. Monard and G. Bal, Inverse anisotropic conductivity from power densities in dimension $n \ge 3$,, Comm. Partial Differential Equations, 38 (2013), 1183.  doi: 10.1080/03605302.2013.787089.  Google Scholar [27] A. Nachman, A. Tamasan and A. Timonov, Recovering the conductivity from a single measurement of interior data,, Inverse Problems, 25 (2009).  doi: 10.1088/0266-5611/25/3/035014.  Google Scholar [28] ________, Reconstruction of planar conductivities in subdomains from incomplete data,, SIAM J. Appl. Math., 70 (2010), 3342.   Google Scholar [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.  doi: 10.1016/j.matpur.2004.09.002.  Google Scholar [30] L. Nirenberg, Lectures on Linear Partial Differential Equations,, Amer. Math. Soc., (1973).   Google Scholar [31] P. Ola, L. Päivärinta and E. Somersalo, An inverse boundary value problem in electromagnetics,, Duke Math. J., 70 (1993), 617.  doi: 10.1215/S0012-7094-93-07014-7.  Google Scholar [32] P. Ola and E. Somersalo, Electromagnetic inverse problems and generalized Sommerfeld potentials,, SIAM J.Appl.Math., 56 (1996), 1129.  doi: 10.1137/S0036139995283948.  Google Scholar [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).  doi: 10.1088/0266-5611/28/8/084002.  Google Scholar [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.  doi: 10.1016/0377-0427(92)90167-V.  Google Scholar [35] Z. Sun and G. Uhlmann, An inverse boundary value problem for Maxwell's equations,, Arch.Rational Mech.Anal., 119 (1992), 71.  doi: 10.1007/BF00376011.  Google Scholar [36] J. Sylvester and G. Uhlmann, A global uniqueness theorem for an inverse boundary value problem,, Ann. of Math., 125 (1987), 153.  doi: 10.2307/1971291.  Google Scholar [37] G. Uhlmann, Calderón's problem and electrical impedance tomography,, Inverse Problems, 25 (2009).  doi: 10.1088/0266-5611/25/12/123011.  Google Scholar [38] C. Weber, Regularity theorems for Maxwell's equations,, Mathematical Methods in the Applied Sciences, 3 (1981), 523.  doi: 10.1002/mma.1670030137.  Google Scholar

show all references

##### References:
 [1] G. Alessandrini, An identification problem for an elliptic equation in two variables,, Ann. Mat. Pura Appl., 145 (1986), 265.  doi: 10.1007/BF01790543.  Google Scholar [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.  doi: 10.1137/070686408.  Google Scholar [3] G. Bal, Inside Out,, Cambridge University Press, (2012).   Google Scholar [4] G. Bal, C. Guo and F. Monard, Linearized internal functionals for anisotropic conductivities,, Inv. Probl. and Imaging, 8 (2014), 1.  doi: 10.3934/ipi.2014.8.1.  Google Scholar [5] _______, Inverse anisotropic conductivity from internal current densities,, Inverse Problems, 30 (2014).   Google Scholar [6] _______, Imaging of anisotropic conductivities from current densities in two dimensions, submitted, (2014)., , ().   Google Scholar [7] G. Bal and J. C. Schotland, Inverse scattering and acousto-optic imaging,, Phys. Rev. Letters, 104 (2010).  doi: 10.1103/PhysRevLett.104.043902.  Google Scholar [8] G. Bal and G. Uhlmann, Inverse diffusion theory of photoacoustics,, Inverse Problems, 26 (2010).  doi: 10.1088/0266-5611/26/8/085010.  Google Scholar [9] _______, Reconstruction of coefficients in scalar second-order elliptic equations from knowledge of their solutions,, Communications on Pure and Applied Mathematics, 66 (2013), 1629.   Google Scholar [10] A. Calderón, Uniqueness in the cauchy problem for partial differential equations,, Amer.J.Math., 80 (1958), 16.  doi: 10.2307/2372819.  Google Scholar [11] P. Caro, P. Ola and M. Salo, Inverse boundary value problem for Maxwell equations with local data,, Comm.PDE., 34 (2009), 1425.  doi: 10.1080/03605300903296272.  Google Scholar [12] J. Chen and Y. Yang, Inverse problem of electro-seismic conversion,, Inverse Problems, 29 (2013).  doi: 10.1088/0266-5611/29/11/115006.  Google Scholar [13] D. Colton and R. Kress, Inverse Acoustic and Electromagnetic Scattering Theory,, Applied Mathematical Sciences, (2012).  doi: 10.1007/978-1-4614-4942-3.  Google Scholar [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.  doi: 10.1007/BF00376010.  Google Scholar [15] M. Eller and M.Yamamoto, A Carleman inequality for the stationary anisotropic Maxwell system,, J.Math.Pures Appl., 86 (2006), 449.  doi: 10.1016/j.matpur.2006.10.004.  Google Scholar [16] L. Hormander, The Analysis of Linear Partial Differential Operators:Pseudo-Differential Operators I-IV,, Springer Verlag, (1983).   Google Scholar [17] R. Horn and C. Johnson, Matrix Analysis,, Cambridge University Press, (1990).   Google Scholar [18] Y. Ider and L. Muftuler, Measurement of AC magnetic field distribution using magnetic resonance imaging,, IEEE Transactions on Medical Imaging, 16 (1997), 617.  doi: 10.1109/42.640752.  Google Scholar [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.   Google Scholar [20] C. E. Kenig, M. Salo and G. Uhlmann, Inverse problems for the anisotropic Maxwell equations,, Duke Math.J., 157 (2011), 369.  doi: 10.1215/00127094-1272903.  Google Scholar [21] R. Kohn and M. Vogelius, Determining conductivity by boundary measurements,, Commun. Pure Appl. Math., 37 (1984), 289.  doi: 10.1002/cpa.3160370302.  Google Scholar [22] P. Kuchment and L. Kunyansky, 2D and 3D reconstructions in acousto-electric tomography,, Inverse Problems, 27 (2011).  doi: 10.1088/0266-5611/27/5/055013.  Google Scholar [23] P. Kuchment and D. Steinhauer, Stabilizing inverse problems by internal data,, Inverse Problems, 28 (2012).  doi: 10.1088/0266-5611/28/8/084007.  Google Scholar [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.   Google Scholar [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.  doi: 10.1002/cpa.3160090407.  Google Scholar [26] F. Monard and G. Bal, Inverse anisotropic conductivity from power densities in dimension $n \ge 3$,, Comm. Partial Differential Equations, 38 (2013), 1183.  doi: 10.1080/03605302.2013.787089.  Google Scholar [27] A. Nachman, A. Tamasan and A. Timonov, Recovering the conductivity from a single measurement of interior data,, Inverse Problems, 25 (2009).  doi: 10.1088/0266-5611/25/3/035014.  Google Scholar [28] ________, Reconstruction of planar conductivities in subdomains from incomplete data,, SIAM J. Appl. Math., 70 (2010), 3342.   Google Scholar [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.  doi: 10.1016/j.matpur.2004.09.002.  Google Scholar [30] L. Nirenberg, Lectures on Linear Partial Differential Equations,, Amer. Math. Soc., (1973).   Google Scholar [31] P. Ola, L. Päivärinta and E. Somersalo, An inverse boundary value problem in electromagnetics,, Duke Math. J., 70 (1993), 617.  doi: 10.1215/S0012-7094-93-07014-7.  Google Scholar [32] P. Ola and E. Somersalo, Electromagnetic inverse problems and generalized Sommerfeld potentials,, SIAM J.Appl.Math., 56 (1996), 1129.  doi: 10.1137/S0036139995283948.  Google Scholar [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).  doi: 10.1088/0266-5611/28/8/084002.  Google Scholar [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.  doi: 10.1016/0377-0427(92)90167-V.  Google Scholar [35] Z. Sun and G. Uhlmann, An inverse boundary value problem for Maxwell's equations,, Arch.Rational Mech.Anal., 119 (1992), 71.  doi: 10.1007/BF00376011.  Google Scholar [36] J. Sylvester and G. Uhlmann, A global uniqueness theorem for an inverse boundary value problem,, Ann. of Math., 125 (1987), 153.  doi: 10.2307/1971291.  Google Scholar [37] G. Uhlmann, Calderón's problem and electrical impedance tomography,, Inverse Problems, 25 (2009).  doi: 10.1088/0266-5611/25/12/123011.  Google Scholar [38] C. Weber, Regularity theorems for Maxwell's equations,, Mathematical Methods in the Applied Sciences, 3 (1981), 523.  doi: 10.1002/mma.1670030137.  Google Scholar
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