# American Institute of Mathematical Sciences

February  2015, 9(1): 1-25. doi: 10.3934/ipi.2015.9.1

## Optimization approach for the simultaneous reconstruction of the dielectric permittivity and magnetic permeability functions from limited observations

 1 Department of Mathematical Sciences, Chalmers University of Technology and Gothenburg University, SE-412 96 Gothenburg, Sweden 2 Institut de Mathématiques de Marseille, Aix-Marseille Université, 13453 Marseille, France 3 Aix-Marseille Université, 13453 Marseille, France

Received  April 2014 Revised  September 2014 Published  January 2015

We consider the inverse problem of the simultaneous reconstruction of the dielectric permittivity and magnetic permeability functions of the Maxwell's system in 3D with limited boundary observations of the electric field. The theoretical stability for the problem is provided by the Carleman estimates. For the numerical computations the problem is formulated as an optimization problem and hybrid finite element/difference method is used to solve the parameter identification problem.
Citation: Larisa Beilina, Michel Cristofol, Kati Niinimäki. Optimization approach for the simultaneous reconstruction of the dielectric permittivity and magnetic permeability functions from limited observations. Inverse Problems and Imaging, 2015, 9 (1) : 1-25. doi: 10.3934/ipi.2015.9.1
##### References:
 [1] F. Assous, P. Degond, E. Heintze and P. Raviart, On a finite-element method for solving the three-dimensional Maxwell equations, Journal of Computational Physics, 109 (1993), 222-237. doi: 10.1006/jcph.1993.1214. [2] A. Bakushinsky, M. Y. Kokurin and A. Smirnova, Iterative Methods for Ill-posed Problems, Inverse and Ill-Posed Problems Series 54, De Gruyter, 2011. [3] L. Beilina, Energy estimates and numerical verification of the stabilized Domain Decomposition Finite Element/Finite Difference approach for time-dependent Maxwell's system, Central European Journal of Mathematics, 11 (2013), 702-733. doi: 10.2478/s11533-013-0202-3. [4] L. Beilina, Adaptive Finite Element Method for a coefficient inverse problem for the Maxwell's system, Applicable Analysis, 90 (2011), 1461-1479. doi: 10.1080/00036811.2010.502116. [5] L. Beilina and M. Grote, Adaptive Hybrid Finite Element/Difference Method for Maxwell's equations, TWMS Journal of Pure and Applied Mathematics, 1 (2010), 176-197. [6] L. Beilina and C. Johnson, A posteriori error estimation in computational inverse scattering, Mathematical Models in Applied Sciences, 15 (2005), 23-35. doi: 10.1142/S0218202505003885. [7] L. Beilina, N. T. Thành, M. V. Klibanov and J. Bondestam-Malmberg, Reconstruction of shapes and refractive indices from blind backscattering experimental data using the adaptivity, Inverse Problems, 30 (2014), 105007, 28pp. [8] L. Beilina, N. T. Thành, M. V. Klibanov and M. A. Fiddy, Reconstruction from blind experimental data for an inverse problem for a hyperbolic equation, Inverse Problems, 30 (2014), 025002, 24pp. doi: 10.1088/0266-5611/30/2/025002. [9] M. I. Belishev and V. M. Isakov, On the uniqueness of the recovery of parameters of the Maxwell system from dynamical boundary data, Journal of Mathematical Sciences, 122 (2004), 3459-3469. doi: 10.1023/B:JOTH.0000034024.38243.02. [10] M. Bellassoued, M. Cristofol and E. Soccorsi, Inverse boundary value problem for the dynamical heterogeneous Maxwell's system, Inverse Problems, 28 (2012), 095009, 18pp. doi: 10.1088/0266-5611/28/9/095009. [11] S. C. Brenner and L. R. Scott, The Mathematical Theory of Finite Element Methods, Springer-Verlag, Berlin, 1994. doi: 10.1007/978-1-4757-4338-8. [12] P. Caro, Stable determination of the electromagnetic coefficients by boundary measurements, Inverse Problems, 26 (2010), 105014, 25pp. doi: 10.1088/0266-5611/26/10/105014. [13] P. Caro, P. Ola and M. Salo, Inverse boundary value problem for Maxwell equations with local data, Communications in Partial Differential Equations, 34 (2009), 1425-1464. doi: 10.1080/03605300903296272. [14] J. M. Catalá-Civera, A. J. Canós-Marin, L. Sempere and E. de los Reyes, Microwaveresonator for the non-invasive evaluation of degradation processes in liquid composites, IEEE Electronic Letters, 37 (2001), 99-100. [15] H. S. Cho, S. Kim, J. Kim and J. Jung, Determination of allowable defect size in multimode polymer waveguides fabricated by printed circuit board compatible processes, Journal of Micromechanics and Microengineering, 20 (2010), 035035. doi: 10.1088/0960-1317/20/3/035035. [16] P. Ciarlet, Jr. and J. Zou, Fully discrete finite element approaches for time-dependent Maxwell's equations, Numerische Mathematik, 82 (1999), 193-219. doi: 10.1007/s002110050417. [17] P. Ciarlet, H. Wu and J. Zou, Edge element methods for Maxwell's equations with strong convergence for Gauss' laws, SIAM Journal on Numerical Analysis, 52 (2014), 779-807. doi: 10.1137/120899856. [18] G. C. Cohen, Higher Order Numerical Methods for Transient Wave Equations, Springer-Verlag, 2002. doi: 10.1007/978-3-662-04823-8. [19] R. Courant, K. Friedrichs and H. Lewy, On the partial differential equations of mathematical physics, IBM Journal of Research and Development, 11 (1967), 215-234. doi: 10.1147/rd.112.0215. [20] D. W. Einters, J. D. Shea, P. Komas, B. D. Van Veen and S. C. Hagness, Three-dimensional microwave breast imaging: Dispersive dielectric properties estimation using patient-specific basis functions, IEEE Transactions on Medical Imaging, 28 (2009), 969-981. [21] M. Eller, V. Isakov, G. Nakamura and D. Tataru, Uniqueness and stability in the Cauchy problem for Maxwell and elasticity systems, in Nonlinear Partial Differential Equations and their Applications, Collège de France Seminar, 31 (2002), 329-349. doi: 10.1016/S0168-2024(02)80016-9. [22] A. Elmkies and P. Joly, Finite elements and mass lumping for Maxwell's equations: the 2D case, Numerical Analysis, 324 (1997), 1287-1293. doi: 10.1016/S0764-4442(99)80415-7. [23] H. W. Engl, M. Hanke and A. Neubauer, Regularization of Inverse Problems, Boston: Kluwer Academic Publishers, 2000. [24] B. Engquist and A. Majda, Absorbing boundary conditions for the numerical simulation of waves, Mathematics of Computation, 31 (1977), 629-651. doi: 10.1090/S0025-5718-1977-0436612-4. [25] H. Feng, D. Jiang and J. Zou, Simultaneous identification of electric permittivity and magnetic permeability, Inverse Problems, 26 (2010), 095009, 20pp. doi: 10.1088/0266-5611/26/9/095009. [26] M. J. Grote, A. Schneebeli and D. Schötzau, Interior penalty discontinuous Galerkin method for Maxwell's equations: Energy norm error estimates, Journal of Computational and Applied Mathematics, 204 (2007), 375-386. doi: 10.1016/j.cam.2006.01.044. [27] M. V. de Hoop, L. Qiu and O. Scherzer, Local analysis of inverse problems: Höder stability and iterative reconstruction, Inverse Problems, 28 (2012), 045001, 16pp. doi: 10.1088/0266-5611/28/4/045001. [28] K. Ito, B. Jin and T. Takeuchi, Multi-parameter Tikhonov regularization, Methods and Applications of Analysis, 18 (2011), 31-46. doi: 10.4310/MAA.2011.v18.n1.a2. [29] P. Joly, Variational methods for time-dependent wave propagation problems, Lecture Notes in Computational Science and Engineering, Springer, 31 (2003), 201-264. doi: 10.1007/978-3-642-55483-4_6. [30] M. V. Klibanov, Uniqueness of the solution of two inverse problems for a Maxwellian system, Computational Mathematics and Mathematical Physics, 26 (1986), 1063-1071. [31] Y. Kurylev, M. Lassas and E. Somersalo, Maxwell's equations with a polarization independent wave velocity: Direct and inverse problems, Journal de Mathematiques Pures et Appliquees, 86 (2006), 237-270. doi: 10.1016/j.matpur.2006.01.008. [32] A. V. Kuzhuget, L. Beilina, M. V. Klibanov, A. Sullivan, L. Nguyen and M. A. Fiddy, Blind experimental data collected in the field and an approximately globally convergent inverse algorithm, Inverse Problems, 28 (2012), 095007. doi: 10.1088/0266-5611/28/9/095007. [33] R. L. Lee and N. K. Madsen, A mixed finite element formulation for Maxwell's equations in the time domain, Journal of Computational Physics, 88 (1990), 284-304. doi: 10.1016/0021-9991(90)90181-Y. [34] S. Li, An inverse problem for Maxwell's equations in bi-isotropic media, SIAM Journal on Mathematical Analysis, 37 (2005), 1027-1043. doi: 10.1137/S003614100444366X. [35] S. Li and M. Yamomoto, Carleman estimate for Maxwell's Equations in anisotropic media and the observability inequality, Journal of Physics: Conference Series, 12 (2005), 110-115. doi: 10.1088/1742-6596/12/1/011. [36] 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. [37] P. B. Monk, Finite Element Methods for Maxwell's Equations, Oxford University Press, 2003. doi: 10.1093/acprof:oso/9780198508885.001.0001. [38] J. Monzó, A. Díaz, J. V. Balbastre, D. Sánchez-Hernández and E. de los Reyes, Selective Heating and Moisture Leveling in Microwave-assisted Trying of Laminal Materials: Explicit Model, 8th Int. Conf. on Microwave and High Frequency heating AMPERE, Bayreuth, Germany, 2001. [39] C. D. Munz, P. Omnes, R. Schneider, E. Sonnendrucker and U. Voss, Divergence correction techniques for Maxwell Solvers based on a hyperbolic model, Journal of Computational Physics, 161 (2000), 484-511. doi: 10.1006/jcph.2000.6507. [40] J. C. Nédélec, A new family of mixed finite elements in $\mathbbR^3$, NUMMA, 50 (1986), 57-81. doi: 10.1007/BF01389668. [41] P. Ola, L. Päivarinta and E. Somersalo, An inverse boundary value problem in electrodynamics, Duke Mathematical Journal, 70 (1993), 617-653. doi: 10.1215/S0012-7094-93-07014-7. [42] K. D. Paulsen and D. R. Lynch, Elimination of vector parasities in Finite Element Maxwell solutions, IEEE Transactions on Microwave Theory and Techniques, 39 (1991), 395-404. [43] PETSc, Portable, Extensible Toolkit for Scientific Computation, http://www.mcs.anl.gov/petsc/. [44] O. Pironneau, Optimal Shape Design for Elliptic Systems, Springer-Verlag, Berlin, 1984. doi: 10.1007/978-3-642-87722-3. [45] J. Pitarch, M. Contelles-Cervera, F. L. Peñaranda-Foix and J. M. Catalá-Civera, Determination of the permittivity and permeability for waveguides partially loaded with isotropic samples, Measurement Science and Technology, 17 (2006), 145-152. doi: 10.1088/0957-0233/17/1/024. [46] M. Salo, C. Kenig and G. Uhlmann, Inverse problems for the anisotropic Maxwell equations, Duke Matematical Journal, 157 (2011), 369-419. doi: 10.1215/00127094-1272903. [47] Y. G. Smirnov, Y. G Shestopalov and E. D. Derevyanchyk, Reconstruction of permittivity and permeability tensors of anisotropic materials in a rectangular waveguides from the reflection and transmission coefficients from different frequencies, Proceedings of Progress in Electromagnetic Research Symposium, Stockholm, Sweden, (2013), 290-295. [48] D. R. Smith, S. Schultz, P. Markos and C. M. Soukoulis, Determination of effective permittivity and permeability of metamaterials from reflection and transmission coefficients, Physical Review B, 65 (2002), p195104. doi: 10.1103/PhysRevB.65.195104. [49] N. T. Thành, L. Beilina, M. V. Klibanov and M. A. Fiddy, Reconstruction of the refractive index from experimental backscattering data using a globally convergent inverse method, SIAM J. Scientific Computing, 36 (2014), 273-293. [50] A. N. Tikhonov, A. V. Goncharsky, V. V. Stepanov and A. G. Yagola, Numerical Methods for the Solution of Ill-Posed Problems, Kluwer, London, 1995. doi: 10.1007/978-94-015-8480-7. [51] WavES, the software package, http://www.waves24.com. [52] X. Zhang, H. Tortel, S. Ruy and A. Litman, Microwave imaging of soil water diffusion using the linear sampling method, IEEE Geoscience and Remote Sensing Letters, 8 (2011), 421-425. doi: 10.1109/LGRS.2010.2082490.

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##### References:
 [1] F. Assous, P. Degond, E. Heintze and P. Raviart, On a finite-element method for solving the three-dimensional Maxwell equations, Journal of Computational Physics, 109 (1993), 222-237. doi: 10.1006/jcph.1993.1214. [2] A. Bakushinsky, M. Y. Kokurin and A. Smirnova, Iterative Methods for Ill-posed Problems, Inverse and Ill-Posed Problems Series 54, De Gruyter, 2011. [3] L. Beilina, Energy estimates and numerical verification of the stabilized Domain Decomposition Finite Element/Finite Difference approach for time-dependent Maxwell's system, Central European Journal of Mathematics, 11 (2013), 702-733. doi: 10.2478/s11533-013-0202-3. [4] L. Beilina, Adaptive Finite Element Method for a coefficient inverse problem for the Maxwell's system, Applicable Analysis, 90 (2011), 1461-1479. doi: 10.1080/00036811.2010.502116. [5] L. Beilina and M. Grote, Adaptive Hybrid Finite Element/Difference Method for Maxwell's equations, TWMS Journal of Pure and Applied Mathematics, 1 (2010), 176-197. [6] L. Beilina and C. Johnson, A posteriori error estimation in computational inverse scattering, Mathematical Models in Applied Sciences, 15 (2005), 23-35. doi: 10.1142/S0218202505003885. [7] L. Beilina, N. T. Thành, M. V. Klibanov and J. Bondestam-Malmberg, Reconstruction of shapes and refractive indices from blind backscattering experimental data using the adaptivity, Inverse Problems, 30 (2014), 105007, 28pp. [8] L. Beilina, N. T. Thành, M. V. Klibanov and M. A. Fiddy, Reconstruction from blind experimental data for an inverse problem for a hyperbolic equation, Inverse Problems, 30 (2014), 025002, 24pp. doi: 10.1088/0266-5611/30/2/025002. [9] M. I. Belishev and V. M. Isakov, On the uniqueness of the recovery of parameters of the Maxwell system from dynamical boundary data, Journal of Mathematical Sciences, 122 (2004), 3459-3469. doi: 10.1023/B:JOTH.0000034024.38243.02. [10] M. Bellassoued, M. Cristofol and E. Soccorsi, Inverse boundary value problem for the dynamical heterogeneous Maxwell's system, Inverse Problems, 28 (2012), 095009, 18pp. doi: 10.1088/0266-5611/28/9/095009. [11] S. C. Brenner and L. R. Scott, The Mathematical Theory of Finite Element Methods, Springer-Verlag, Berlin, 1994. doi: 10.1007/978-1-4757-4338-8. [12] P. Caro, Stable determination of the electromagnetic coefficients by boundary measurements, Inverse Problems, 26 (2010), 105014, 25pp. doi: 10.1088/0266-5611/26/10/105014. [13] P. Caro, P. Ola and M. Salo, Inverse boundary value problem for Maxwell equations with local data, Communications in Partial Differential Equations, 34 (2009), 1425-1464. doi: 10.1080/03605300903296272. [14] J. M. Catalá-Civera, A. J. Canós-Marin, L. Sempere and E. de los Reyes, Microwaveresonator for the non-invasive evaluation of degradation processes in liquid composites, IEEE Electronic Letters, 37 (2001), 99-100. [15] H. S. Cho, S. Kim, J. Kim and J. Jung, Determination of allowable defect size in multimode polymer waveguides fabricated by printed circuit board compatible processes, Journal of Micromechanics and Microengineering, 20 (2010), 035035. doi: 10.1088/0960-1317/20/3/035035. [16] P. Ciarlet, Jr. and J. Zou, Fully discrete finite element approaches for time-dependent Maxwell's equations, Numerische Mathematik, 82 (1999), 193-219. doi: 10.1007/s002110050417. [17] P. Ciarlet, H. Wu and J. Zou, Edge element methods for Maxwell's equations with strong convergence for Gauss' laws, SIAM Journal on Numerical Analysis, 52 (2014), 779-807. doi: 10.1137/120899856. [18] G. C. Cohen, Higher Order Numerical Methods for Transient Wave Equations, Springer-Verlag, 2002. doi: 10.1007/978-3-662-04823-8. [19] R. Courant, K. Friedrichs and H. Lewy, On the partial differential equations of mathematical physics, IBM Journal of Research and Development, 11 (1967), 215-234. doi: 10.1147/rd.112.0215. [20] D. W. Einters, J. D. Shea, P. Komas, B. D. Van Veen and S. C. Hagness, Three-dimensional microwave breast imaging: Dispersive dielectric properties estimation using patient-specific basis functions, IEEE Transactions on Medical Imaging, 28 (2009), 969-981. [21] M. Eller, V. Isakov, G. Nakamura and D. Tataru, Uniqueness and stability in the Cauchy problem for Maxwell and elasticity systems, in Nonlinear Partial Differential Equations and their Applications, Collège de France Seminar, 31 (2002), 329-349. doi: 10.1016/S0168-2024(02)80016-9. [22] A. Elmkies and P. Joly, Finite elements and mass lumping for Maxwell's equations: the 2D case, Numerical Analysis, 324 (1997), 1287-1293. doi: 10.1016/S0764-4442(99)80415-7. [23] H. W. Engl, M. Hanke and A. Neubauer, Regularization of Inverse Problems, Boston: Kluwer Academic Publishers, 2000. [24] B. Engquist and A. Majda, Absorbing boundary conditions for the numerical simulation of waves, Mathematics of Computation, 31 (1977), 629-651. doi: 10.1090/S0025-5718-1977-0436612-4. [25] H. Feng, D. Jiang and J. Zou, Simultaneous identification of electric permittivity and magnetic permeability, Inverse Problems, 26 (2010), 095009, 20pp. doi: 10.1088/0266-5611/26/9/095009. [26] M. J. Grote, A. Schneebeli and D. Schötzau, Interior penalty discontinuous Galerkin method for Maxwell's equations: Energy norm error estimates, Journal of Computational and Applied Mathematics, 204 (2007), 375-386. doi: 10.1016/j.cam.2006.01.044. [27] M. V. de Hoop, L. Qiu and O. Scherzer, Local analysis of inverse problems: Höder stability and iterative reconstruction, Inverse Problems, 28 (2012), 045001, 16pp. doi: 10.1088/0266-5611/28/4/045001. [28] K. Ito, B. Jin and T. Takeuchi, Multi-parameter Tikhonov regularization, Methods and Applications of Analysis, 18 (2011), 31-46. doi: 10.4310/MAA.2011.v18.n1.a2. [29] P. Joly, Variational methods for time-dependent wave propagation problems, Lecture Notes in Computational Science and Engineering, Springer, 31 (2003), 201-264. doi: 10.1007/978-3-642-55483-4_6. [30] M. V. Klibanov, Uniqueness of the solution of two inverse problems for a Maxwellian system, Computational Mathematics and Mathematical Physics, 26 (1986), 1063-1071. [31] Y. Kurylev, M. Lassas and E. Somersalo, Maxwell's equations with a polarization independent wave velocity: Direct and inverse problems, Journal de Mathematiques Pures et Appliquees, 86 (2006), 237-270. doi: 10.1016/j.matpur.2006.01.008. [32] A. V. Kuzhuget, L. Beilina, M. V. Klibanov, A. Sullivan, L. Nguyen and M. A. Fiddy, Blind experimental data collected in the field and an approximately globally convergent inverse algorithm, Inverse Problems, 28 (2012), 095007. doi: 10.1088/0266-5611/28/9/095007. [33] R. L. Lee and N. K. Madsen, A mixed finite element formulation for Maxwell's equations in the time domain, Journal of Computational Physics, 88 (1990), 284-304. doi: 10.1016/0021-9991(90)90181-Y. [34] S. Li, An inverse problem for Maxwell's equations in bi-isotropic media, SIAM Journal on Mathematical Analysis, 37 (2005), 1027-1043. doi: 10.1137/S003614100444366X. [35] S. Li and M. Yamomoto, Carleman estimate for Maxwell's Equations in anisotropic media and the observability inequality, Journal of Physics: Conference Series, 12 (2005), 110-115. doi: 10.1088/1742-6596/12/1/011. [36] 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. [37] P. B. Monk, Finite Element Methods for Maxwell's Equations, Oxford University Press, 2003. doi: 10.1093/acprof:oso/9780198508885.001.0001. [38] J. Monzó, A. Díaz, J. V. Balbastre, D. Sánchez-Hernández and E. de los Reyes, Selective Heating and Moisture Leveling in Microwave-assisted Trying of Laminal Materials: Explicit Model, 8th Int. Conf. on Microwave and High Frequency heating AMPERE, Bayreuth, Germany, 2001. [39] C. D. Munz, P. Omnes, R. Schneider, E. Sonnendrucker and U. Voss, Divergence correction techniques for Maxwell Solvers based on a hyperbolic model, Journal of Computational Physics, 161 (2000), 484-511. doi: 10.1006/jcph.2000.6507. [40] J. C. Nédélec, A new family of mixed finite elements in $\mathbbR^3$, NUMMA, 50 (1986), 57-81. doi: 10.1007/BF01389668. [41] P. Ola, L. Päivarinta and E. Somersalo, An inverse boundary value problem in electrodynamics, Duke Mathematical Journal, 70 (1993), 617-653. doi: 10.1215/S0012-7094-93-07014-7. [42] K. D. Paulsen and D. R. Lynch, Elimination of vector parasities in Finite Element Maxwell solutions, IEEE Transactions on Microwave Theory and Techniques, 39 (1991), 395-404. [43] PETSc, Portable, Extensible Toolkit for Scientific Computation, http://www.mcs.anl.gov/petsc/. [44] O. Pironneau, Optimal Shape Design for Elliptic Systems, Springer-Verlag, Berlin, 1984. doi: 10.1007/978-3-642-87722-3. [45] J. Pitarch, M. Contelles-Cervera, F. L. Peñaranda-Foix and J. M. Catalá-Civera, Determination of the permittivity and permeability for waveguides partially loaded with isotropic samples, Measurement Science and Technology, 17 (2006), 145-152. doi: 10.1088/0957-0233/17/1/024. [46] M. Salo, C. Kenig and G. Uhlmann, Inverse problems for the anisotropic Maxwell equations, Duke Matematical Journal, 157 (2011), 369-419. doi: 10.1215/00127094-1272903. [47] Y. G. Smirnov, Y. G Shestopalov and E. D. Derevyanchyk, Reconstruction of permittivity and permeability tensors of anisotropic materials in a rectangular waveguides from the reflection and transmission coefficients from different frequencies, Proceedings of Progress in Electromagnetic Research Symposium, Stockholm, Sweden, (2013), 290-295. [48] D. R. Smith, S. Schultz, P. Markos and C. M. Soukoulis, Determination of effective permittivity and permeability of metamaterials from reflection and transmission coefficients, Physical Review B, 65 (2002), p195104. doi: 10.1103/PhysRevB.65.195104. [49] N. T. Thành, L. Beilina, M. V. Klibanov and M. A. Fiddy, Reconstruction of the refractive index from experimental backscattering data using a globally convergent inverse method, SIAM J. Scientific Computing, 36 (2014), 273-293. [50] A. N. Tikhonov, A. V. Goncharsky, V. V. Stepanov and A. G. Yagola, Numerical Methods for the Solution of Ill-Posed Problems, Kluwer, London, 1995. doi: 10.1007/978-94-015-8480-7. [51] WavES, the software package, http://www.waves24.com. [52] X. Zhang, H. Tortel, S. Ruy and A. Litman, Microwave imaging of soil water diffusion using the linear sampling method, IEEE Geoscience and Remote Sensing Letters, 8 (2011), 421-425. doi: 10.1109/LGRS.2010.2082490.
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