# American Institute of Mathematical Sciences

October  2021, 15(5): 1171-1198. doi: 10.3934/ipi.2021033

## Boundary determination of electromagnetic and Lamé parameters with corrupted data

 1 Basque Center for Applied Mathematics, Bilbao, Spain 2 School of Mathematics, University of Minnesota, USA 3 Department of Applied Mathematics, National Chiao Tung University, Taiwan 4 Department of Mathematics, Northeastern University, USA

* Corresponding author: Ting Zhou

Received  September 2020 Revised  January 2021 Published  October 2021 Early access  May 2021

We study boundary determination for an inverse problem associated to the time-harmonic Maxwell equations and another associated to the isotropic elasticity system. We identify the electromagnetic parameters and the Lamé moduli for these two systems from the corresponding boundary measurements. In a first step we reconstruct Lipschitz magnetic permeability, electric permittivity and conductivity on the surface from the ideal boundary measurements. Then, we study inverse problems for Maxwell equations and the isotropic elasticity system assuming that the data contains measurement errors. For both systems, we provide explicit formulas to reconstruct the parameters on the boundary as well as its rate of convergence formula.

Citation: Pedro Caro, Ru-Yu Lai, Yi-Hsuan Lin, Ting Zhou. Boundary determination of electromagnetic and Lamé parameters with corrupted data. Inverse Problems & Imaging, 2021, 15 (5) : 1171-1198. doi: 10.3934/ipi.2021033
##### References:
 [1] G. Alessandrini, Singular solutions of elliptic equations and the determination of conductivity by boundary measurements,, J. Differ. Equations, 84 (1990), 252-272.  doi: 10.1016/0022-0396(90)90078-4.  Google Scholar [2] K. Astala and L. Päiv$\ddot{\mathrm{r}}$inta, Calderón's inverse conductivity problem in the plane, Annals of Mathematics, 163 (2006), 265-299.  doi: 10.4007/annals.2006.163.265.  Google Scholar [3] R.-M. Brown, Recovering the conductivity at the boundary from the Dirichlet to Neumann map: A pointwise result, Journal of Inverse and III-Posed Problems, 9 (2001), 567-574.  doi: 10.1515/jiip.2001.9.6.567.  Google Scholar [4] A. García and G. Zhang, Appendix of "Reconstruction from boundary measurements for less regular conductivities", Inverse Problems, 32 (2016), 115015, 22 pp. doi: 10.1088/0266-5611/32/11/115015.  Google Scholar [5] R.-M. Brown and M. Salo, Identifiability at the boundary for first-order terms, Applicable Analysis, 85 (2006), 735-749.  doi: 10.1080/00036810600603377.  Google Scholar [6] A. Buffa, M. Costabel and D. Sheen, On traces for H(curl; $\Omega$) in Lipschitz domains, J. Math. Anal. Appl, 276 (2002), 845-867.  doi: 10.1016/S0022-247X(02)00455-9.  Google Scholar [7] P. Caro and A. Garcia, The Calderón problem with corrupted data, Inverse Problems, 33 (2017), 085001, 17 pp. doi: 10.1088/1361-6420/aa7425.  Google Scholar [8] P. Caro and C. J. Meroño, The observational limit of wave packets with noisy measurements, SIAM Journal on Mathematical Analysis, 52 (2020), 5196-5212.  doi: 10.1137/20M1324946.  Google Scholar [9] Pedro Caro and K. M. Rogers, Global uniqueness for the Calderón problem with Lipschitz conductivities, Forum of Mathematics, Pi, 4 (2016), 28 pp. doi: 10.1017/fmp.2015.9.  Google Scholar [10] P. Caro and T. Zhou, Global uniqueness for an IBVP for the time-harmonic Maxwell equations, Anal. PDE, 7 (2014), 375-405.  doi: 10.2140/apde.2014.7.375.  Google Scholar [11] B. Haberman and D. Tataru, Uniqueness in Calderón's problem with Lipschitz conductivities, Duke Math. J., 162 (2013), 497-516.  doi: 10.1215/00127094-2019591.  Google Scholar [12] H. Kekkonen, M. Lassas and S. Siltanen, Corrigendum: Analysis of regularized inversion of data corrupted by white Gaussian noise (2014 inverse problems 30 045009), Inverse Problems, 32 (2016), 099501. Google Scholar [13] M. Ikehata, A regularized extraction formula in the enclosure method, Inverse Problems, 18 (2002), 435-440.  doi: 10.1088/0266-5611/18/2/309.  Google Scholar [14] O. Y. Imanuvilov and M. Yamamoto, Global uniqueness in inverse boundary value problems for the Navier–Stokes equations and Lamé system in two dimensions, Inverse Problems, 31 (2015), 035004, 46 pp. doi: 10.1088/0266-5611/31/3/035004.  Google Scholar [15] M. S. Joshi and S. R. McDowall, Total determination of material parameters from electromagnetic boundary information, Pacific Journal of Mathematics, 193 (2000), 107-129.  doi: 10.2140/pjm.2000.193.107.  Google Scholar [16] H. Kekkonen, M. Lassas and S. Siltanen, Analysis of regularized inversion of data corrupted by white gaussian noise, Inverse Problems, 30 (2014), 045009, 18 pp. doi: 10.1088/0266-5611/30/4/045009.  Google Scholar [17] J. L. Mueller, K. Knudsen, M. Lassas and S. Siltanen, Regularized D-bar method for the inverse conductivity problem, Inverse Problems Imaging, 3 (2009), 599-624.  doi: 10.3934/ipi.2009.3.599.  Google Scholar [18] R. Kohn and M. Vogelius, Determining conductivity by boundary measurements, Communications on Pure and Applied Mathematics, 37 (1984), 289-298.  doi: 10.1002/cpa.3160370302.  Google Scholar [19] Y.-H. Lin and G. Nakamura, Boundary determination of the Lamé moduli for the isotropic elasticity system, Inverse Problems, 33 (2017), 125004, 23 pp. doi: 10.1088/1361-6420/aa942d.  Google Scholar [20] S. R. McDowall, Boundary determination of material parameters from electromagnetic boundary information, Inverse Problems, 13 (1997), 153-163.  doi: 10.1088/0266-5611/13/1/012.  Google Scholar [21] A. I. Nachman, Global uniqueness for a two-dimensional inverse boundary value problem, Annals of Mathematics, 143 (1996), 71-96.  doi: 10.2307/2118653.  Google Scholar [22] 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.  Google Scholar [23] P. Ola and E. Somersalo, Electromagnetic inverse problems and generalized sommerfeld potentials, SIAM Journal on Applied Mathematics, 56 (1996), 1129-1145.  doi: 10.1137/S0036139995283948.  Google Scholar [24] M. Pichler, An inverse problem for Maxwell equations with Lipschitz parameters, Inverse Problems, 34 (2018), 025006, 21 pp. doi: 10.1088/1361-6420/aaa352.  Google Scholar [25] M. Salo and L. Tzou, Carleman estimates and inverse problems for Dirac operators, Mathematische Annalen, 344 (2009), 161-184.  doi: 10.1007/s00208-008-0301-9.  Google Scholar [26] J. Sylvester and G. Uhlmann, A global uniqueness theorem for an inverse boundary value problem, Ann. Math., 125 (1987), 153-169.  doi: 10.2307/1971291.  Google Scholar [27] J. Sylvester and G. Uhlmann, Inverse boundary value problems at the boundary – continuous dependence, Communications on Pure and Applied Mathematics, 41 (1988), 197-219.  doi: 10.1002/cpa.3160410205.  Google Scholar [28] K. Tanuma, Stroh formalism and Rayleigh waves, J. Elasticity, 89 (2007), vi+159 pp. doi: 10.1007/s10659-007-9117-1.  Google Scholar [29] L. Tartar, On the characterization of traces of a Sobolev space used for Maxwell equation, in Proceedings of a Meeting Held in Bordeaux, in Honour of Michel Artola, (1997). Google Scholar

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##### References:
 [1] G. Alessandrini, Singular solutions of elliptic equations and the determination of conductivity by boundary measurements,, J. Differ. Equations, 84 (1990), 252-272.  doi: 10.1016/0022-0396(90)90078-4.  Google Scholar [2] K. Astala and L. Päiv$\ddot{\mathrm{r}}$inta, Calderón's inverse conductivity problem in the plane, Annals of Mathematics, 163 (2006), 265-299.  doi: 10.4007/annals.2006.163.265.  Google Scholar [3] R.-M. Brown, Recovering the conductivity at the boundary from the Dirichlet to Neumann map: A pointwise result, Journal of Inverse and III-Posed Problems, 9 (2001), 567-574.  doi: 10.1515/jiip.2001.9.6.567.  Google Scholar [4] A. García and G. Zhang, Appendix of "Reconstruction from boundary measurements for less regular conductivities", Inverse Problems, 32 (2016), 115015, 22 pp. doi: 10.1088/0266-5611/32/11/115015.  Google Scholar [5] R.-M. Brown and M. Salo, Identifiability at the boundary for first-order terms, Applicable Analysis, 85 (2006), 735-749.  doi: 10.1080/00036810600603377.  Google Scholar [6] A. Buffa, M. Costabel and D. Sheen, On traces for H(curl; $\Omega$) in Lipschitz domains, J. Math. Anal. Appl, 276 (2002), 845-867.  doi: 10.1016/S0022-247X(02)00455-9.  Google Scholar [7] P. Caro and A. Garcia, The Calderón problem with corrupted data, Inverse Problems, 33 (2017), 085001, 17 pp. doi: 10.1088/1361-6420/aa7425.  Google Scholar [8] P. Caro and C. J. Meroño, The observational limit of wave packets with noisy measurements, SIAM Journal on Mathematical Analysis, 52 (2020), 5196-5212.  doi: 10.1137/20M1324946.  Google Scholar [9] Pedro Caro and K. M. Rogers, Global uniqueness for the Calderón problem with Lipschitz conductivities, Forum of Mathematics, Pi, 4 (2016), 28 pp. doi: 10.1017/fmp.2015.9.  Google Scholar [10] P. Caro and T. Zhou, Global uniqueness for an IBVP for the time-harmonic Maxwell equations, Anal. PDE, 7 (2014), 375-405.  doi: 10.2140/apde.2014.7.375.  Google Scholar [11] B. Haberman and D. Tataru, Uniqueness in Calderón's problem with Lipschitz conductivities, Duke Math. J., 162 (2013), 497-516.  doi: 10.1215/00127094-2019591.  Google Scholar [12] H. Kekkonen, M. Lassas and S. Siltanen, Corrigendum: Analysis of regularized inversion of data corrupted by white Gaussian noise (2014 inverse problems 30 045009), Inverse Problems, 32 (2016), 099501. Google Scholar [13] M. Ikehata, A regularized extraction formula in the enclosure method, Inverse Problems, 18 (2002), 435-440.  doi: 10.1088/0266-5611/18/2/309.  Google Scholar [14] O. Y. Imanuvilov and M. Yamamoto, Global uniqueness in inverse boundary value problems for the Navier–Stokes equations and Lamé system in two dimensions, Inverse Problems, 31 (2015), 035004, 46 pp. doi: 10.1088/0266-5611/31/3/035004.  Google Scholar [15] M. S. Joshi and S. R. McDowall, Total determination of material parameters from electromagnetic boundary information, Pacific Journal of Mathematics, 193 (2000), 107-129.  doi: 10.2140/pjm.2000.193.107.  Google Scholar [16] H. Kekkonen, M. Lassas and S. Siltanen, Analysis of regularized inversion of data corrupted by white gaussian noise, Inverse Problems, 30 (2014), 045009, 18 pp. doi: 10.1088/0266-5611/30/4/045009.  Google Scholar [17] J. L. Mueller, K. Knudsen, M. Lassas and S. Siltanen, Regularized D-bar method for the inverse conductivity problem, Inverse Problems Imaging, 3 (2009), 599-624.  doi: 10.3934/ipi.2009.3.599.  Google Scholar [18] R. Kohn and M. Vogelius, Determining conductivity by boundary measurements, Communications on Pure and Applied Mathematics, 37 (1984), 289-298.  doi: 10.1002/cpa.3160370302.  Google Scholar [19] Y.-H. Lin and G. Nakamura, Boundary determination of the Lamé moduli for the isotropic elasticity system, Inverse Problems, 33 (2017), 125004, 23 pp. doi: 10.1088/1361-6420/aa942d.  Google Scholar [20] S. R. McDowall, Boundary determination of material parameters from electromagnetic boundary information, Inverse Problems, 13 (1997), 153-163.  doi: 10.1088/0266-5611/13/1/012.  Google Scholar [21] A. I. Nachman, Global uniqueness for a two-dimensional inverse boundary value problem, Annals of Mathematics, 143 (1996), 71-96.  doi: 10.2307/2118653.  Google Scholar [22] 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.  Google Scholar [23] P. Ola and E. Somersalo, Electromagnetic inverse problems and generalized sommerfeld potentials, SIAM Journal on Applied Mathematics, 56 (1996), 1129-1145.  doi: 10.1137/S0036139995283948.  Google Scholar [24] M. Pichler, An inverse problem for Maxwell equations with Lipschitz parameters, Inverse Problems, 34 (2018), 025006, 21 pp. doi: 10.1088/1361-6420/aaa352.  Google Scholar [25] M. Salo and L. Tzou, Carleman estimates and inverse problems for Dirac operators, Mathematische Annalen, 344 (2009), 161-184.  doi: 10.1007/s00208-008-0301-9.  Google Scholar [26] J. Sylvester and G. Uhlmann, A global uniqueness theorem for an inverse boundary value problem, Ann. Math., 125 (1987), 153-169.  doi: 10.2307/1971291.  Google Scholar [27] J. Sylvester and G. Uhlmann, Inverse boundary value problems at the boundary – continuous dependence, Communications on Pure and Applied Mathematics, 41 (1988), 197-219.  doi: 10.1002/cpa.3160410205.  Google Scholar [28] K. Tanuma, Stroh formalism and Rayleigh waves, J. Elasticity, 89 (2007), vi+159 pp. doi: 10.1007/s10659-007-9117-1.  Google Scholar [29] L. Tartar, On the characterization of traces of a Sobolev space used for Maxwell equation, in Proceedings of a Meeting Held in Bordeaux, in Honour of Michel Artola, (1997). Google Scholar
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