doi: 10.3934/ipi.2022032
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Monotonicity in inverse scattering for Maxwell's equations

Institute for Applied and Numerical Mathematics, Karlsruhe Institute of Technology, 76131 Karlsruhe, Germany

Received  October 2021 Revised  April 2022 Early access June 2022

Fund Project: This work was funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) – Project-ID 258734477 – SFB 1173

A recent area of research in inverse scattering theory has been the study of monotonicity relations for the eigenvalues of far field operators and their use in shape reconstruction for inverse scattering problems. We develop such monotonicity relations for an electromagnetic inverse scattering problem governed by Maxwell's equations, and we apply them to establish novel rigorous characterizations of the shape of scattering objects in terms of the corresponding far field operators. Along the way we establish the existence of electromagnetic fields that have arbitrarily large energy in some prescribed region, while at the same time having arbitrarily small energy in some other prescribed region. These localized vector wave functions not only play an important role in the proofs of the novel monotonicity based shape characterizations but they are also of independent interest. We conclude with some simple numerical demonstrations of our theoretical results.

Citation: Annalena Albicker, Roland Griesmaier. Monotonicity in inverse scattering for Maxwell's equations. Inverse Problems and Imaging, doi: 10.3934/ipi.2022032
References:
[1]

A. Albicker and R. Griesmaier, Monotonicity in inverse obstacle scattering on unbounded domains, Inverse Problems, 36 (2020), 085014.  doi: 10.1088/1361-6420/ab98a3.

[2]

K. Atkinson and W. Han, Spherical Harmonics and Approximations on the Unit Sphere: An Introduction, Lecture Notes in Mathematics, 2044, Springer, Heidelberg, 2012. doi: 10.1007/978-3-642-25983-8.

[3]

L. AudibertL. Chesnel and H. Haddar, Inside-outside duality with artificial backgrounds, Inverse Problems, 35 (2019), 104008.  doi: 10.1088/1361-6420/ab3244.

[4]

L. AudibertL. ChesnelH. Haddar and K. Napal, Qualitative indicator functions for imaging crack networks using acoustic waves, SIAM J. Sci. Comput., 43 (2021), B271-B297.  doi: 10.1137/20M134650X.

[5]

J. M. BallY. Capdeboscq and B. Tsering-Xiao, On uniqueness for time harmonic anisotropic Maxwell's equations with piecewise regular coefficients, Math. Models Methods Appl. Sci., 22 (2012), 1250036.  doi: 10.1142/S0218202512500364.

[6]

A. BarthB. HarrachN. Hyvönen and L. Mustonen, Detecting stochastic inclusions in electrical impedance tomography, Inverse Problems, 33 (2017), 115012.  doi: 10.1088/1361-6420/aa8f5c.

[7]

T. BranderB. HarrachM. Kar and M. Salo, Monotonicity and enclosure methods for the $p$-Laplace equation, SIAM J. Appl. Math., 78 (2018), 742-758.  doi: 10.1137/17M1128599.

[8]

F. Cakoni, D. Colton and H. Haddar, Inverse Scattering Theory and Transmission Eigenvalues, CBMS-NSF Regional Conference Series in Applied Mathematics, 88, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2016. doi: 10.1137/1.9781611974461.ch1.

[9]

F. Cakoni, D. Colton and P. Monk, The Linear Sampling Method in Inverse Electromagnetic Scattering, CBMS-NSF Regional Conference Series in Applied Mathematics, 80, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2011. doi: 10.1137/1.9780898719406.

[10]

V. CandianiJ. DardéH. Garde and N. Hyvönen, Monotonicity-based reconstruction of extreme inclusions in electrical impedance tomography, SIAM J. Math. Anal., 52 (2020), 6234-6259.  doi: 10.1137/19M1299219.

[11]

D. Colton and R. Kress, Inverse Acoustic and Electromagnetic Scattering Theory, Applied Mathematical Sciences, 93, Springer, Cham, Fourth edition, 2019. doi: 10.1007/978-3-030-30351-8.

[12]

A. Corbo EspositoL. FaellaG. PiscitelliR. Prakash and A. Tamburrino, Monotonicity principle in tomography of nonlinear conducting materials, Inverse Problems, 37 (2021), 045012.  doi: 10.1088/1361-6420/abd29a.

[13]

T. Furuya, The factorization and monotonicity method for the defect in an open periodic waveguide, J. Inverse Ill-Posed Probl., 28 (2020), 783-796.  doi: 10.1515/jiip-2019-0088.

[14]

T. Furuya, Remarks on the factorization and monotonicity method for inverse acoustic scatterings, Inverse Problems, 37 (2021), 065006.  doi: 10.1088/1361-6420/abf75f.

[15]

H. Garde, Comparison of linear and non-linear monotonicity-based shape reconstruction using exact matrix characterizations, Inverse Probl. Sci. Eng., 26 (2018), 33-50.  doi: 10.1080/17415977.2017.1290088.

[16]

H. Garde and S. Staboulis, Convergence and regularization for monotonicity-based shape reconstruction in electrical impedance tomography, Numer. Math., 135 (2017), 1221-1251.  doi: 10.1007/s00211-016-0830-1.

[17]

H. Garde and S. Staboulis, The regularized monotonicity method: Detecting irregular indefinite inclusions, Inverse Probl. Imaging, 13 (2019), 93-116.  doi: 10.3934/ipi.2019006.

[18]

B. Gebauer, Localized potentials in electrical impedance tomography, Inverse Probl. Imaging, 2 (2008), 251-269.  doi: 10.3934/ipi.2008.2.251.

[19] G. H. Golub and C. F. Van Loan, Matrix Computations, third edition, Johns Hopkins Studies in the Mathematical Sciences, Johns Hopkins University Press, Baltimore, MD, 1996.  doi: 10.2307/3621013.
[20]

R. Griesmaier and B. Harrach, Monotonicity in inverse medium scattering on unbounded domains, SIAM J. Appl. Math., 78 (2018), 2533-2557.  doi: 10.1137/18M1171679.

[21]

R. Griesmaier and B. Harrach, Erratum: Monotonicity in inverse medium scattering on unbounded domains, SIAM J. Appl. Math., 81 (2021), 1332-1337.  doi: 10.1137/21M1399221.

[22]

R. Griesmaier and J. Sylvester, Uncertainty principles for inverse source problems for electromagnetic and elastic waves, Inverse Problems, 34 (2018), 065003.  doi: 10.1088/1361-6420/aab45c.

[23]

N. I. Grinberg, Obstacle visualization via the factorization method for the mixed boundary value problem, Inverse Problems, 18 (2002), 1687-1704.  doi: 10.1088/0266-5611/18/6/317.

[24]

N. I. Grinberg and A. Kirsch, The factorization method for obstacles with a priori separated sound-soft and sound-hard parts, Math. Comput. Simulation, 66 (2004), 267-279.  doi: 10.1016/j.matcom.2004.02.011.

[25]

P. Grisvard, Elliptic Problems in Nonsmooth Domains, Classics in Applied Mathematics, 69, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2011. doi: 10.1137/1.9781611972030.ch1.

[26]

H. Haddar and P. Monk, The linear sampling method for solving the electromagnetic inverse medium problem, Inverse Problems, 18 (2002), 891-906.  doi: 10.1088/0266-5611/18/3/323.

[27]

B. Harrach and Y.-H. Lin, Monotonicity-based inversion of the fractional Schrödinger equation I. Positive potentials, SIAM J. Math. Anal., 51 (2019), 3092-3111.  doi: 10.1137/18M1166298.

[28]

B. Harrach and Y.-H. Lin, Monotonicity-based inversion of the fractional Schödinger equation Ⅱ. General potentials and stability, SIAM J. Math. Anal., 52 (2020), 402-436.  doi: 10.1137/19M1251576.

[29]

B. HarrachY.-H. Lin and H. Liu, On localizing and concentrating electromagnetic fields, SIAM J. Appl. Math., 78 (2018), 2558-2574.  doi: 10.1137/18M1173605.

[30]

B. Harrach and M. N. Minh, Enhancing residual-based techniques with shape reconstruction features in electrical impedance tomography, Inverse Problems, 32 (2016), 125002.  doi: 10.1088/0266-5611/32/12/125002.

[31]

B. Harrach and M. N. Minh, Monotonicity-based regularization for phantom experiment data in electrical impedance tomography, In New Trends in Parameter Identification for Mathematical Models, Trends Math., Birkhäuser/Springer, Cham, 2018, 107–120. doi: 10.1007/978-3-319-70824-9_6.

[32]

B. HarrachV. Pohjola and M. Salo, Dimension bounds in monotonicity methods for the Helmholtz equation, SIAM J. Math. Anal., 51 (2019), 2995-3019.  doi: 10.1137/19M1240708.

[33]

B. HarrachV. Pohjola and M. Salo, Monotonicity and local uniqueness for the Helmholtz equation, Anal. PDE, 12 (2019), 1741-1771.  doi: 10.2140/apde.2019.12.1741.

[34]

B. Harrach and M. Ullrich, Monotonicity-based shape reconstruction in electrical impedance tomography, SIAM J. Math. Anal., 45 (2013), 3382-3403.  doi: 10.1137/120886984.

[35]

B. Harrach and M. Ullrich, Resolution guarantees in electrical impedance tomography, IEEE Transactions on Medical Imaging, 34 (2015), 1513-1521.  doi: 10.1109/TMI.2015.2404133.

[36]

M. Ikehata, Size estimation of inclusion, J. Inverse Ill-Posed Probl., 6 (1998), 127-140.  doi: 10.1515/jiip.1998.6.2.127.

[37]

H. KangJ. K. Seo and D. Sheen, The inverse conductivity problem with one measurement: Stability and estimation of size, SIAM J. Math. Anal., 28 (1997), 1389-1405.  doi: 10.1137/S0036141096299375.

[38]

A. Kirsch, The factorization method for Maxwell's equations, Inverse Problems, 20 (2004), S117-S134.  doi: 10.1088/0266-5611/20/6/S08.

[39]

A. Kirsch, An integral equation for Maxwell's equations in a layered medium with an application to the factorization method, J. Integral Equations Appl., 19 (2007), 333-358.  doi: 10.1216/jiea/1190905490.

[40] A. Kirsch and N. Grinberg, The Factorization Method for Inverse Problems, Oxford Lecture Series in Mathematics and its Applications, 36 Oxford University Press, Oxford, 2008.  doi: 10.1093/acprof:oso/9780199213535.001.0001.
[41]

A. Kirsch and F. Hettlich, The Mathematical Theory of Time-Harmonic Maxwell's Equations, Applied Mathematical Sciences, 190, Springer, Cham, 2015. doi: 10.1007/978-3-319-11086-8.

[42]

A. Kirsch and A. Lechleiter, The inside-outside duality for scattering problems by inhomogeneous media, Inverse Problems, 29 (2013), 104011.  doi: 10.1088/0266-5611/29/10/104011.

[43]

E. Lakshtanov and A. Lechleiter, Difference factorizations and monotonicity in inverse medium scattering for contrasts with fixed sign on the boundary, SIAM J. Math. Anal., 48 (2016), 3688-3707.  doi: 10.1137/16M1060819.

[44]

A. Lechleiter and M. Rennoch, Inside-outside duality and the determination of electromagnetic interior transmission eigenvalues, SIAM J. Math. Anal., 47 (2015), 684-705.  doi: 10.1137/14098538X.

[45] W. McLean, Strongly Elliptic Systems and Boundary Integral Equations, Cambridge University Press, Cambridge, 2000.  doi: 10.1017/S0013091501244435.
[46] P. Monk, Finite Element Methods for Maxwell's Equations, Numerical Mathematics and Scientific Computation, Oxford University Press, New York, 2003.  doi: 10.1093/acprof:oso/9780198508885.001.0001.
[47]

S. Schmitt, The factorization method for EIT in the case of mixed inclusions, Inverse Problems, 25 (2009), 065012.  doi: 10.1088/0266-5611/25/6/065012.

[48]

W. Sickel, On the regularity of characteristic functions, In Anomalies in Partial Differential Equations, Springer INdAM Ser., 43 Springer, Cham, 2021, 395–441. doi: 10.1007/978-3-030-61346-4_18.

[49]

W. Śmigaj, T. Betcke, S. Arridge, J. Phillips and M. Schweiger, Solving boundary integral problems with BEM++, ACM Trans. Math. Software, 41 (2015), Art. 6, 40 pp. doi: 10.1145/2590830.

[50]

Z. Su, S. Ventre, L. Udpa and A. Tamburrino, Monotonicity based on imaging method for time-domain eddy current problems, Inverse Problems, 33 (2017), 125007, 23 pp. doi: 10.1088/1361-6420/aa909a.

[51]

A. TamburrinoG. Piscitelli and Z. Zhou, The monotonicity principle for magnetic induction tomography, Inverse Problems, 37 (2021), 095003.  doi: 10.1088/1361-6420/ac156c.

[52]

A. Tamburrino and G. Rubinacci, A new non-iterative inversion method for electrical resistance tomography, Inverse Problems, 18 (2002), 1809-1829.  doi: 10.1088/0266-5611/18/6/323.

[53]

M. Tucsnak and G. Weiss, Observation and Control for Operator Semigroups, Birkhäuser Advanced Texts: Basler Lehrbücher, Birkhäuser Verlag, Basel, 2009. doi: 10.1007/978-3-7643-8994-9.

[54]

C. Weber, Regularity theorems for Maxwell's equations, Math. Methods Appl. Sci., 3 (1981), 523-536.  doi: 10.1002/mma.1670030137.

show all references

References:
[1]

A. Albicker and R. Griesmaier, Monotonicity in inverse obstacle scattering on unbounded domains, Inverse Problems, 36 (2020), 085014.  doi: 10.1088/1361-6420/ab98a3.

[2]

K. Atkinson and W. Han, Spherical Harmonics and Approximations on the Unit Sphere: An Introduction, Lecture Notes in Mathematics, 2044, Springer, Heidelberg, 2012. doi: 10.1007/978-3-642-25983-8.

[3]

L. AudibertL. Chesnel and H. Haddar, Inside-outside duality with artificial backgrounds, Inverse Problems, 35 (2019), 104008.  doi: 10.1088/1361-6420/ab3244.

[4]

L. AudibertL. ChesnelH. Haddar and K. Napal, Qualitative indicator functions for imaging crack networks using acoustic waves, SIAM J. Sci. Comput., 43 (2021), B271-B297.  doi: 10.1137/20M134650X.

[5]

J. M. BallY. Capdeboscq and B. Tsering-Xiao, On uniqueness for time harmonic anisotropic Maxwell's equations with piecewise regular coefficients, Math. Models Methods Appl. Sci., 22 (2012), 1250036.  doi: 10.1142/S0218202512500364.

[6]

A. BarthB. HarrachN. Hyvönen and L. Mustonen, Detecting stochastic inclusions in electrical impedance tomography, Inverse Problems, 33 (2017), 115012.  doi: 10.1088/1361-6420/aa8f5c.

[7]

T. BranderB. HarrachM. Kar and M. Salo, Monotonicity and enclosure methods for the $p$-Laplace equation, SIAM J. Appl. Math., 78 (2018), 742-758.  doi: 10.1137/17M1128599.

[8]

F. Cakoni, D. Colton and H. Haddar, Inverse Scattering Theory and Transmission Eigenvalues, CBMS-NSF Regional Conference Series in Applied Mathematics, 88, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2016. doi: 10.1137/1.9781611974461.ch1.

[9]

F. Cakoni, D. Colton and P. Monk, The Linear Sampling Method in Inverse Electromagnetic Scattering, CBMS-NSF Regional Conference Series in Applied Mathematics, 80, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2011. doi: 10.1137/1.9780898719406.

[10]

V. CandianiJ. DardéH. Garde and N. Hyvönen, Monotonicity-based reconstruction of extreme inclusions in electrical impedance tomography, SIAM J. Math. Anal., 52 (2020), 6234-6259.  doi: 10.1137/19M1299219.

[11]

D. Colton and R. Kress, Inverse Acoustic and Electromagnetic Scattering Theory, Applied Mathematical Sciences, 93, Springer, Cham, Fourth edition, 2019. doi: 10.1007/978-3-030-30351-8.

[12]

A. Corbo EspositoL. FaellaG. PiscitelliR. Prakash and A. Tamburrino, Monotonicity principle in tomography of nonlinear conducting materials, Inverse Problems, 37 (2021), 045012.  doi: 10.1088/1361-6420/abd29a.

[13]

T. Furuya, The factorization and monotonicity method for the defect in an open periodic waveguide, J. Inverse Ill-Posed Probl., 28 (2020), 783-796.  doi: 10.1515/jiip-2019-0088.

[14]

T. Furuya, Remarks on the factorization and monotonicity method for inverse acoustic scatterings, Inverse Problems, 37 (2021), 065006.  doi: 10.1088/1361-6420/abf75f.

[15]

H. Garde, Comparison of linear and non-linear monotonicity-based shape reconstruction using exact matrix characterizations, Inverse Probl. Sci. Eng., 26 (2018), 33-50.  doi: 10.1080/17415977.2017.1290088.

[16]

H. Garde and S. Staboulis, Convergence and regularization for monotonicity-based shape reconstruction in electrical impedance tomography, Numer. Math., 135 (2017), 1221-1251.  doi: 10.1007/s00211-016-0830-1.

[17]

H. Garde and S. Staboulis, The regularized monotonicity method: Detecting irregular indefinite inclusions, Inverse Probl. Imaging, 13 (2019), 93-116.  doi: 10.3934/ipi.2019006.

[18]

B. Gebauer, Localized potentials in electrical impedance tomography, Inverse Probl. Imaging, 2 (2008), 251-269.  doi: 10.3934/ipi.2008.2.251.

[19] G. H. Golub and C. F. Van Loan, Matrix Computations, third edition, Johns Hopkins Studies in the Mathematical Sciences, Johns Hopkins University Press, Baltimore, MD, 1996.  doi: 10.2307/3621013.
[20]

R. Griesmaier and B. Harrach, Monotonicity in inverse medium scattering on unbounded domains, SIAM J. Appl. Math., 78 (2018), 2533-2557.  doi: 10.1137/18M1171679.

[21]

R. Griesmaier and B. Harrach, Erratum: Monotonicity in inverse medium scattering on unbounded domains, SIAM J. Appl. Math., 81 (2021), 1332-1337.  doi: 10.1137/21M1399221.

[22]

R. Griesmaier and J. Sylvester, Uncertainty principles for inverse source problems for electromagnetic and elastic waves, Inverse Problems, 34 (2018), 065003.  doi: 10.1088/1361-6420/aab45c.

[23]

N. I. Grinberg, Obstacle visualization via the factorization method for the mixed boundary value problem, Inverse Problems, 18 (2002), 1687-1704.  doi: 10.1088/0266-5611/18/6/317.

[24]

N. I. Grinberg and A. Kirsch, The factorization method for obstacles with a priori separated sound-soft and sound-hard parts, Math. Comput. Simulation, 66 (2004), 267-279.  doi: 10.1016/j.matcom.2004.02.011.

[25]

P. Grisvard, Elliptic Problems in Nonsmooth Domains, Classics in Applied Mathematics, 69, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2011. doi: 10.1137/1.9781611972030.ch1.

[26]

H. Haddar and P. Monk, The linear sampling method for solving the electromagnetic inverse medium problem, Inverse Problems, 18 (2002), 891-906.  doi: 10.1088/0266-5611/18/3/323.

[27]

B. Harrach and Y.-H. Lin, Monotonicity-based inversion of the fractional Schrödinger equation I. Positive potentials, SIAM J. Math. Anal., 51 (2019), 3092-3111.  doi: 10.1137/18M1166298.

[28]

B. Harrach and Y.-H. Lin, Monotonicity-based inversion of the fractional Schödinger equation Ⅱ. General potentials and stability, SIAM J. Math. Anal., 52 (2020), 402-436.  doi: 10.1137/19M1251576.

[29]

B. HarrachY.-H. Lin and H. Liu, On localizing and concentrating electromagnetic fields, SIAM J. Appl. Math., 78 (2018), 2558-2574.  doi: 10.1137/18M1173605.

[30]

B. Harrach and M. N. Minh, Enhancing residual-based techniques with shape reconstruction features in electrical impedance tomography, Inverse Problems, 32 (2016), 125002.  doi: 10.1088/0266-5611/32/12/125002.

[31]

B. Harrach and M. N. Minh, Monotonicity-based regularization for phantom experiment data in electrical impedance tomography, In New Trends in Parameter Identification for Mathematical Models, Trends Math., Birkhäuser/Springer, Cham, 2018, 107–120. doi: 10.1007/978-3-319-70824-9_6.

[32]

B. HarrachV. Pohjola and M. Salo, Dimension bounds in monotonicity methods for the Helmholtz equation, SIAM J. Math. Anal., 51 (2019), 2995-3019.  doi: 10.1137/19M1240708.

[33]

B. HarrachV. Pohjola and M. Salo, Monotonicity and local uniqueness for the Helmholtz equation, Anal. PDE, 12 (2019), 1741-1771.  doi: 10.2140/apde.2019.12.1741.

[34]

B. Harrach and M. Ullrich, Monotonicity-based shape reconstruction in electrical impedance tomography, SIAM J. Math. Anal., 45 (2013), 3382-3403.  doi: 10.1137/120886984.

[35]

B. Harrach and M. Ullrich, Resolution guarantees in electrical impedance tomography, IEEE Transactions on Medical Imaging, 34 (2015), 1513-1521.  doi: 10.1109/TMI.2015.2404133.

[36]

M. Ikehata, Size estimation of inclusion, J. Inverse Ill-Posed Probl., 6 (1998), 127-140.  doi: 10.1515/jiip.1998.6.2.127.

[37]

H. KangJ. K. Seo and D. Sheen, The inverse conductivity problem with one measurement: Stability and estimation of size, SIAM J. Math. Anal., 28 (1997), 1389-1405.  doi: 10.1137/S0036141096299375.

[38]

A. Kirsch, The factorization method for Maxwell's equations, Inverse Problems, 20 (2004), S117-S134.  doi: 10.1088/0266-5611/20/6/S08.

[39]

A. Kirsch, An integral equation for Maxwell's equations in a layered medium with an application to the factorization method, J. Integral Equations Appl., 19 (2007), 333-358.  doi: 10.1216/jiea/1190905490.

[40] A. Kirsch and N. Grinberg, The Factorization Method for Inverse Problems, Oxford Lecture Series in Mathematics and its Applications, 36 Oxford University Press, Oxford, 2008.  doi: 10.1093/acprof:oso/9780199213535.001.0001.
[41]

A. Kirsch and F. Hettlich, The Mathematical Theory of Time-Harmonic Maxwell's Equations, Applied Mathematical Sciences, 190, Springer, Cham, 2015. doi: 10.1007/978-3-319-11086-8.

[42]

A. Kirsch and A. Lechleiter, The inside-outside duality for scattering problems by inhomogeneous media, Inverse Problems, 29 (2013), 104011.  doi: 10.1088/0266-5611/29/10/104011.

[43]

E. Lakshtanov and A. Lechleiter, Difference factorizations and monotonicity in inverse medium scattering for contrasts with fixed sign on the boundary, SIAM J. Math. Anal., 48 (2016), 3688-3707.  doi: 10.1137/16M1060819.

[44]

A. Lechleiter and M. Rennoch, Inside-outside duality and the determination of electromagnetic interior transmission eigenvalues, SIAM J. Math. Anal., 47 (2015), 684-705.  doi: 10.1137/14098538X.

[45] W. McLean, Strongly Elliptic Systems and Boundary Integral Equations, Cambridge University Press, Cambridge, 2000.  doi: 10.1017/S0013091501244435.
[46] P. Monk, Finite Element Methods for Maxwell's Equations, Numerical Mathematics and Scientific Computation, Oxford University Press, New York, 2003.  doi: 10.1093/acprof:oso/9780198508885.001.0001.
[47]

S. Schmitt, The factorization method for EIT in the case of mixed inclusions, Inverse Problems, 25 (2009), 065012.  doi: 10.1088/0266-5611/25/6/065012.

[48]

W. Sickel, On the regularity of characteristic functions, In Anomalies in Partial Differential Equations, Springer INdAM Ser., 43 Springer, Cham, 2021, 395–441. doi: 10.1007/978-3-030-61346-4_18.

[49]

W. Śmigaj, T. Betcke, S. Arridge, J. Phillips and M. Schweiger, Solving boundary integral problems with BEM++, ACM Trans. Math. Software, 41 (2015), Art. 6, 40 pp. doi: 10.1145/2590830.

[50]

Z. Su, S. Ventre, L. Udpa and A. Tamburrino, Monotonicity based on imaging method for time-domain eddy current problems, Inverse Problems, 33 (2017), 125007, 23 pp. doi: 10.1088/1361-6420/aa909a.

[51]

A. TamburrinoG. Piscitelli and Z. Zhou, The monotonicity principle for magnetic induction tomography, Inverse Problems, 37 (2021), 095003.  doi: 10.1088/1361-6420/ac156c.

[52]

A. Tamburrino and G. Rubinacci, A new non-iterative inversion method for electrical resistance tomography, Inverse Problems, 18 (2002), 1809-1829.  doi: 10.1088/0266-5611/18/6/323.

[53]

M. Tucsnak and G. Weiss, Observation and Control for Operator Semigroups, Birkhäuser Advanced Texts: Basler Lehrbücher, Birkhäuser Verlag, Basel, 2009. doi: 10.1007/978-3-7643-8994-9.

[54]

C. Weber, Regularity theorems for Maxwell's equations, Math. Methods Appl. Sci., 3 (1981), 523-536.  doi: 10.1002/mma.1670030137.

Figure 1.  Number of negative eigenvalues (left) and number of positive eigenvalues (right) $ {\rm{Re}}\bigl(\lambda_n(r_D)\bigr) $ (dotted), $ -\alpha \mu_n(r_B) $ (dashed), and $ {\rm{Re}}\bigl(\lambda_n(r_D)\bigr)-\alpha\mu_n(r_B) $ (solid) within the range $ n = 0, \ldots, 1000 $ as function of $ r_B $
Figure 2.  Visualization of exact shape of scattering object in Example 9.1 (left), visualization of isosurface $ I_{20} = 2 $ of indicator function from (65) using simulated far field data without additional noise (center), and visualization of isosurface $ I_{20} = 11 $ using simulated far field data with $ 0.1 \% $ noise (right)
Figure 3.  Visualization of the indicator function $ I_\alpha $ in the $ {\boldsymbol x}_1, {\boldsymbol x}_2 $-plane for $ \alpha \in \{0.01, 0.1, 0.5, 1, 10, 20\} $ using simulated far field data without additional noise. The dashed lines show the exact boundaries of the cross-section of the scatterer
Figure 4.  Visualization of the exact shape of the mixed scattering object in Example 9.2 (left), and of the isosurfaces $ I_{0.5}^+ = 7 $ (center) and $ I_{-1}^- = 1 $ (right)
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