doi: 10.3934/ipi.2020056

Numerical resolution by the quasi-reversibility method of a data completion problem for Maxwell's equations

1. 

LAMFA CNRS UMR 7352, Université de Picardie Jules Verne, 33 rue Saint-Leu, 80039 Amiens cedex 1

2. 

LMR CNRS UMR 9008, Université de Reims Champagne-Ardenne, Moulin de la Housse, 51687 Reims cedex 2

* Corresponding author: Jérémy Heleine, jeremy.heleine@u-picardie.fr

Received  February 2020 Revised  July 2020 Published  August 2020

This paper concerns the numerical resolution of a data completion problem for the time-harmonic Maxwell equations in the electric field. The aim is to recover the missing data on the inaccessible part of the boundary of a bounded domain from measured data on the accessible part. The non-iterative quasi-reversibility method is studied and different mixed variational formulations are proposed. Well-posedness, convergence and regularity results are proved. Discretization is performed by means of edge finite elements. Various two- and three-dimensional numerical simulations attest the efficiency of the method, in particular for noisy data.

Citation: Marion Darbas, Jérémy Heleine, Stephanie Lohrengel. Numerical resolution by the quasi-reversibility method of a data completion problem for Maxwell's equations. Inverse Problems & Imaging, doi: 10.3934/ipi.2020056
References:
[1]

G. Alessandrini, L. Rondi, E. Rosset and S. Vessella, The stability for the Cauchy problem for elliptic equations, Inverse problems, 25 (2009), 123004. doi: 10.1088/0266-5611/25/12/123004.  Google Scholar

[2]

S. AndrieuxT. N. Baranger and A. Ben Abda, Solving Cauchy problems by minimizing an energy-like functional, Inverse Problems, 22 (2006), 115-133.  doi: 10.1088/0266-5611/22/1/007.  Google Scholar

[3]

M. AzaïezF. Ben Belgacem and H. El Fekih, On Cauchy's problem: Ⅱ. Completion, regularization and approximation, Inverse Problems, 22 (2006), 1307-1336.  doi: 10.1088/0266-5611/22/4/012.  Google Scholar

[4]

M. AzaïezF. Ben BelgacemT. D. Du and F. Jelassi, A finite element model for the data completion problem: Analysis and assessment, Inverse Probl. Sci. Eng., 19.8 (2011), 1063-1086.  doi: 10.1080/17415977.2011.587515.  Google Scholar

[5]

S. BauerD. Pauly and M. Schomburg, The Maxwell compactness property in bounded weak Lipschitz domains with mixed boundary conditions, SIAM J. Math. Anal., 48 (2016), 2912-2943.  doi: 10.1137/16M1065951.  Google Scholar

[6]

F. S. V. Bazán and J. B. Francisco, An improved fixed-point algorithm for determining a Tikhonov regularization parameter, Inverse Problems, 25 (2009), 045007. doi: 10.1088/0266-5611/25/4/045007.  Google Scholar

[7]

F. Ben Belgacem, Why is the Cauchy problem severely ill-posed?, Inverse Problems, 23 (2007), 823-836.  doi: 10.1088/0266-5611/23/2/020.  Google Scholar

[8]

F. Ben Belgacem and H. El Fekih, On Cauchy's problem: Ⅰ. A variational Steklov-Poincaré theory, Inverse Problems, 21 (2005), 1915-1936.  doi: 10.1088/0266-5611/21/6/008.  Google Scholar

[9]

L. Bourgeois, A mixed formulation of quasi-reversibility to solve the Cauchy problem for Laplace's equation, Inverse Problems, 21 (2005), 1087-1104.  doi: 10.1088/0266-5611/21/3/018.  Google Scholar

[10]

L. Bourgeois and L. Chesnel, On quasi-reversibility solutions to the Cauchy problem for the Laplace equation: Regularity and error estimates, ESAIM Math. Model. Numer. Anal., 54 (2020), 493-529.  doi: 10.1051/m2an/2019073.  Google Scholar

[11]

L. Bourgeois and J. Dardé, A duality-based method of quasi-reversibility to solve the Cauchy problem in the presence of noisy data, Inverse Problems, 26.9 (2010), 095016. doi: 10.1088/0266-5611/26/9/095016.  Google Scholar

[12]

L. Bourgeois and J. Dardé, A quasi-reversibility approach to solve the inverse obstacle problem, Inverse Probl. Imaging, 4.3 (2010), 351-377.  doi: 10.3934/ipi.2010.4.351.  Google Scholar

[13]

L. Bourgeois and A. Recoquillay, A mixed formulation of the Tikhonov regularization and its application to inverse PDE problems, Math. Modell. Numer. Anal., 52.1 (2018), 123-145.  doi: 10.1051/m2an/2018008.  Google Scholar

[14]

B. M. BrownM. Marletta and J. M. Reyes, Uniqueness for an inverse problem in electromagnetism with partial data, J. Differential Equations, 260 (2016), 6525-6547.  doi: 10.1016/j.jde.2016.01.002.  Google Scholar

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A. Buffa and P. Ciarlet Jr., On traces for functional spaces related to Maxwell's equations: Ⅰ. An integration by parts formula in Lipschitz polyhedra, Math. Meth. Appl. Sci., 24 (2001), 9-30.  doi: 10.1002/1099-1476(20010110)24:1<9::AID-MMA191>3.0.CO;2-2.  Google Scholar

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A. Buffa and P. Ciarlet Jr., On traces for functional spaces related to Maxwell's equations: Ⅱ. Hodge decompositions on the boundary of Lipschitz polyhedra and applications, Math. Meth. Appl. Sci., 24 (2001), 31-48.  doi: 10.1002/1099-1476(20010110)24:1<31::AID-MMA193>3.0.CO;2-X.  Google Scholar

[17]

F. Cakoni and R. Kress, Integral equations for inverse problems in corrosion detection from partial Cauchy data, Inverse Probl. Imaging, 1.2 (2007), 299-245.  doi: 10.3934/ipi.2007.1.229.  Google Scholar

[18]

P. Caro, On an inverse problem in electromagnetism with local data: stability and uniqueness, Inverse Probl. Imaging, 5 (2011), 297-322.  doi: 10.3934/ipi.2011.5.297.  Google Scholar

[19]

P. CaroA. García and J. M. Reyes, Stability of the Calderón problem for less regular conductivities, J. Differential Equations, 254.2 (2013), 469-492.  doi: 10.1016/j.jde.2012.08.018.  Google Scholar

[20]

P. Caro and T. Zhou, Global uniqueness for an IBVP for the time-harmonic Maxwell equations, Analysis & PDE, 7.2 (2014), 375-405.  doi: 10.2140/apde.2014.7.375.  Google Scholar

[21]

J. L. CastellanosS. Gómez and V. Guerra, The triangle method for finding the corner of the L-curve, Appl. Numer. Math., 43.4 (2002), 359-373.  doi: 10.1016/S0168-9274(01)00179-9.  Google Scholar

[22]

R. Chapko and B. T. Johansson, An alternating boundary integral based method for a Cauchy problem for the Laplace equation in semi-infinite regions, Inverse Probl. Imaging, 2.3 (2008), 317-333.  doi: 10.3934/ipi.2008.2.317.  Google Scholar

[23]

R. Chapko and B. T. Johansson, On the numerical solution of a Cauchy problem for the Laplace equation via a direct integral equation approach, Inverse Probl. Imaging, 6.1 (2012), 25-38.  doi: 10.3934/ipi.2012.6.25.  Google Scholar

[24]

A. CimetièreF. DelvareM. Jaoua and F. Pons, Solution of the Cauchy problem using iterated Tikhonov regularization, Inverse Problems, 17 (2001), 553-570.  doi: 10.1088/0266-5611/17/3/313.  Google Scholar

[25]

M. Clerc, J. Leblond, J.-P. Marmorat and T. Papadopoulo, Source localization in EEG using rational approximation on plane sections, Inverse Problems, 28 (2012), 055018. doi: 10.1088/0266-5611/28/5/055018.  Google Scholar

[26]

M. Clerc, J. Leblond, J.-P. Marmorat and C. Papageorgakis, Uniqueness result for an inverse conductivity recovery problem with application to EEG, Rendiconti dell'Istituto di Matematica dell'Università di Trieste. An International Journal of Mathematics, 48 (2016). doi: 10.13137/2464-8728/13165.  Google Scholar

[27]

D. ColtonM. Piana and R. Potthast, A simple method using Morozov's discrepancy principle for solving inverse scattering problems, Inverse Problems, 13 (1999), 1477-1493.  doi: 10.1088/0266-5611/13/6/005.  Google Scholar

[28]

M. Costabel, A remark on the regularity of solutions of Mawxell's equations on Lipschitz domains, Math. Meth. Appl. Sci., 12 (1990), 365-368.  doi: 10.1002/mma.1670120406.  Google Scholar

[29]

M. DarbasJ. Lohrengel and et S. Heleine, Sensitivity analysis for 3D Maxwell's equations and its use in the resolution of an inverse medium problem at fixed frequency, Inverse Probl. Sci. Eng., 28.4 (2020), 459-496.  doi: 10.1080/17415977.2019.1588896.  Google Scholar

[30]

J. Dardé, Iterated quasi-reversibility method applied to elliptic and parabolic data completion problems, Inverse Probl. Imaging, 10.2, (2016), 379â€"407. doi: 10.3934/ipi.2016005.  Google Scholar

[31]

C. Greif and D. Schötzau, Preconditioners for the discretized time-harmonic Maxwell equations in mixed form, Numer. Linear Algebra Appl., 14 (2007), 281-297.  doi: 10.1002/nla.515.  Google Scholar

[32]

P. Grisvard, Singularities in Boundary Value Problems, RMA 22, Masson, Springer-Verlag, 1992.  Google Scholar

[33]

J. Hadamard, Lectures on Cauchy's Problem in Linear Partial Differential Equation, New York, Dover, 1953.  Google Scholar

[34]

M. Hanke, Limitations of the L-curve method in ill-posed problems, BIT Numer. Math., 36 (1996), 287-301.  doi: 10.1007/BF01731984.  Google Scholar

[35]

F. Hecht, New Development in FreeFem++, J. Numer. Math., 20.3-4 (2012), 251-265.  doi: 10.1515/jnum-2012-0013.  Google Scholar

[36]

J. Heleine, Identification de Paramètres Électromagnétiques par Imagerie Micro-ondes, Ph.D thesis, Université de Picardie-Jules Verne in Amiens, France, 2019. http://theses.fr/2019AMIE0013/document Google Scholar

[37]

R. Hiptmair and J. Xu, Nodal auxiliary space preconditioning in $H(\mathbf{ curl})$ and $H(\text{ div})$ spaces, SIAM J. Numer. Anal., 45.6 (2007), 2483-2509.  doi: 10.1137/060660588.  Google Scholar

[38]

V. A. KozlovV. G. Mazya and A. V. Fomin, An iterative method for solving the Cauchy problem for elliptic equations, Comput. Math. Math. Phys., 31 (1991), 45-52.   Google Scholar

[39]

M. V. Klibanov and F. Santosa, A computational quasi-reversibility method for Cauchy problems for Laplace's equation, SIAM J. Appl. Math., 51 (1991), 1653-1675.  doi: 10.1137/0151085.  Google Scholar

[40]

R. Lattès and J.-L. Lions, Méthode de Quasi-réversibilité et Applications, Dunod, Paris, 1967.  Google Scholar

[41]

L. Ling and T. Takeuchi, Boundary control for inverse Cauchy problems of the Laplace equations, Comp. Model. Eng. Sci., 29 (2008), 45-54.   Google Scholar

[42]

H. McCann, G. Pisano and L. Beltrachini, Variation in reported human head tissue electrical conductivity values, Brain Topography, (2019). doi: 10.1007/s10548-019-00710-2.  Google Scholar

[43]

V. Melicher and M. Slodička, Determination of missing boundary data for a steady-state Maxwell problem, Inverse Problems, 22 (2006), 297-310.  doi: 10.1088/0266-5611/22/1/016.  Google Scholar

[44] P. Monk, Finite Element Methods for Maxwell's Equations, Oxford University Press, 2003.  doi: 10.1093/acprof:oso/9780198508885.001.0001.  Google Scholar
[45]

J.-C. Nédélec, Mixed finite elements in $ {\mathbb{R}}^3$, Numerische Mathematik, 35 (1980), 315-341.  doi: 10.1007/BF01396415.  Google Scholar

[46]

P. Ola, L. Päivärinta and E. Somersalo, Inverse problems for time harmonic electrodynamics, Inside Out: Inverse Problems and Applications, Publications of the Research Institute for Mathematical Sciences, 47, Cambridge University Press, Cambridge, 2003, 169-191.  Google Scholar

[47]

M. Slodička and V. Melicher, An iterative algorithm for a Cauchy problem in eddy-current modelling, Appl. Math. Comput., 207 (2010), 237-246.  doi: 10.1016/j.amc.2010.05.054.  Google Scholar

[48]

P.-H. Tournier, et al., Microwave imaging of cerebrovascular accidents by using highperformance computing, Parallel Computing, 85 (2019), 88-97. doi: 10.1016/j.parco.2019.02.004.  Google Scholar

[49]

P.-H. Tournier, et al., Numerical modeling and high speed parallel computing: New perspectives for tomographic microwave imaging for brain stroke detection and monitoring, IEEE Antennas and Propagation Magazine, 59.5 (2017), 98-110. doi: 10.1109/MAP.2017.2731199.  Google Scholar

[50]

C. R. Vogel, Computational methods for inverse problems, SIAM Frontiers in Applied Mathematics, Philadelphia, 2002. doi: 10.1137/1.9780898717570.  Google Scholar

show all references

References:
[1]

G. Alessandrini, L. Rondi, E. Rosset and S. Vessella, The stability for the Cauchy problem for elliptic equations, Inverse problems, 25 (2009), 123004. doi: 10.1088/0266-5611/25/12/123004.  Google Scholar

[2]

S. AndrieuxT. N. Baranger and A. Ben Abda, Solving Cauchy problems by minimizing an energy-like functional, Inverse Problems, 22 (2006), 115-133.  doi: 10.1088/0266-5611/22/1/007.  Google Scholar

[3]

M. AzaïezF. Ben Belgacem and H. El Fekih, On Cauchy's problem: Ⅱ. Completion, regularization and approximation, Inverse Problems, 22 (2006), 1307-1336.  doi: 10.1088/0266-5611/22/4/012.  Google Scholar

[4]

M. AzaïezF. Ben BelgacemT. D. Du and F. Jelassi, A finite element model for the data completion problem: Analysis and assessment, Inverse Probl. Sci. Eng., 19.8 (2011), 1063-1086.  doi: 10.1080/17415977.2011.587515.  Google Scholar

[5]

S. BauerD. Pauly and M. Schomburg, The Maxwell compactness property in bounded weak Lipschitz domains with mixed boundary conditions, SIAM J. Math. Anal., 48 (2016), 2912-2943.  doi: 10.1137/16M1065951.  Google Scholar

[6]

F. S. V. Bazán and J. B. Francisco, An improved fixed-point algorithm for determining a Tikhonov regularization parameter, Inverse Problems, 25 (2009), 045007. doi: 10.1088/0266-5611/25/4/045007.  Google Scholar

[7]

F. Ben Belgacem, Why is the Cauchy problem severely ill-posed?, Inverse Problems, 23 (2007), 823-836.  doi: 10.1088/0266-5611/23/2/020.  Google Scholar

[8]

F. Ben Belgacem and H. El Fekih, On Cauchy's problem: Ⅰ. A variational Steklov-Poincaré theory, Inverse Problems, 21 (2005), 1915-1936.  doi: 10.1088/0266-5611/21/6/008.  Google Scholar

[9]

L. Bourgeois, A mixed formulation of quasi-reversibility to solve the Cauchy problem for Laplace's equation, Inverse Problems, 21 (2005), 1087-1104.  doi: 10.1088/0266-5611/21/3/018.  Google Scholar

[10]

L. Bourgeois and L. Chesnel, On quasi-reversibility solutions to the Cauchy problem for the Laplace equation: Regularity and error estimates, ESAIM Math. Model. Numer. Anal., 54 (2020), 493-529.  doi: 10.1051/m2an/2019073.  Google Scholar

[11]

L. Bourgeois and J. Dardé, A duality-based method of quasi-reversibility to solve the Cauchy problem in the presence of noisy data, Inverse Problems, 26.9 (2010), 095016. doi: 10.1088/0266-5611/26/9/095016.  Google Scholar

[12]

L. Bourgeois and J. Dardé, A quasi-reversibility approach to solve the inverse obstacle problem, Inverse Probl. Imaging, 4.3 (2010), 351-377.  doi: 10.3934/ipi.2010.4.351.  Google Scholar

[13]

L. Bourgeois and A. Recoquillay, A mixed formulation of the Tikhonov regularization and its application to inverse PDE problems, Math. Modell. Numer. Anal., 52.1 (2018), 123-145.  doi: 10.1051/m2an/2018008.  Google Scholar

[14]

B. M. BrownM. Marletta and J. M. Reyes, Uniqueness for an inverse problem in electromagnetism with partial data, J. Differential Equations, 260 (2016), 6525-6547.  doi: 10.1016/j.jde.2016.01.002.  Google Scholar

[15]

A. Buffa and P. Ciarlet Jr., On traces for functional spaces related to Maxwell's equations: Ⅰ. An integration by parts formula in Lipschitz polyhedra, Math. Meth. Appl. Sci., 24 (2001), 9-30.  doi: 10.1002/1099-1476(20010110)24:1<9::AID-MMA191>3.0.CO;2-2.  Google Scholar

[16]

A. Buffa and P. Ciarlet Jr., On traces for functional spaces related to Maxwell's equations: Ⅱ. Hodge decompositions on the boundary of Lipschitz polyhedra and applications, Math. Meth. Appl. Sci., 24 (2001), 31-48.  doi: 10.1002/1099-1476(20010110)24:1<31::AID-MMA193>3.0.CO;2-X.  Google Scholar

[17]

F. Cakoni and R. Kress, Integral equations for inverse problems in corrosion detection from partial Cauchy data, Inverse Probl. Imaging, 1.2 (2007), 299-245.  doi: 10.3934/ipi.2007.1.229.  Google Scholar

[18]

P. Caro, On an inverse problem in electromagnetism with local data: stability and uniqueness, Inverse Probl. Imaging, 5 (2011), 297-322.  doi: 10.3934/ipi.2011.5.297.  Google Scholar

[19]

P. CaroA. García and J. M. Reyes, Stability of the Calderón problem for less regular conductivities, J. Differential Equations, 254.2 (2013), 469-492.  doi: 10.1016/j.jde.2012.08.018.  Google Scholar

[20]

P. Caro and T. Zhou, Global uniqueness for an IBVP for the time-harmonic Maxwell equations, Analysis & PDE, 7.2 (2014), 375-405.  doi: 10.2140/apde.2014.7.375.  Google Scholar

[21]

J. L. CastellanosS. Gómez and V. Guerra, The triangle method for finding the corner of the L-curve, Appl. Numer. Math., 43.4 (2002), 359-373.  doi: 10.1016/S0168-9274(01)00179-9.  Google Scholar

[22]

R. Chapko and B. T. Johansson, An alternating boundary integral based method for a Cauchy problem for the Laplace equation in semi-infinite regions, Inverse Probl. Imaging, 2.3 (2008), 317-333.  doi: 10.3934/ipi.2008.2.317.  Google Scholar

[23]

R. Chapko and B. T. Johansson, On the numerical solution of a Cauchy problem for the Laplace equation via a direct integral equation approach, Inverse Probl. Imaging, 6.1 (2012), 25-38.  doi: 10.3934/ipi.2012.6.25.  Google Scholar

[24]

A. CimetièreF. DelvareM. Jaoua and F. Pons, Solution of the Cauchy problem using iterated Tikhonov regularization, Inverse Problems, 17 (2001), 553-570.  doi: 10.1088/0266-5611/17/3/313.  Google Scholar

[25]

M. Clerc, J. Leblond, J.-P. Marmorat and T. Papadopoulo, Source localization in EEG using rational approximation on plane sections, Inverse Problems, 28 (2012), 055018. doi: 10.1088/0266-5611/28/5/055018.  Google Scholar

[26]

M. Clerc, J. Leblond, J.-P. Marmorat and C. Papageorgakis, Uniqueness result for an inverse conductivity recovery problem with application to EEG, Rendiconti dell'Istituto di Matematica dell'Università di Trieste. An International Journal of Mathematics, 48 (2016). doi: 10.13137/2464-8728/13165.  Google Scholar

[27]

D. ColtonM. Piana and R. Potthast, A simple method using Morozov's discrepancy principle for solving inverse scattering problems, Inverse Problems, 13 (1999), 1477-1493.  doi: 10.1088/0266-5611/13/6/005.  Google Scholar

[28]

M. Costabel, A remark on the regularity of solutions of Mawxell's equations on Lipschitz domains, Math. Meth. Appl. Sci., 12 (1990), 365-368.  doi: 10.1002/mma.1670120406.  Google Scholar

[29]

M. DarbasJ. Lohrengel and et S. Heleine, Sensitivity analysis for 3D Maxwell's equations and its use in the resolution of an inverse medium problem at fixed frequency, Inverse Probl. Sci. Eng., 28.4 (2020), 459-496.  doi: 10.1080/17415977.2019.1588896.  Google Scholar

[30]

J. Dardé, Iterated quasi-reversibility method applied to elliptic and parabolic data completion problems, Inverse Probl. Imaging, 10.2, (2016), 379â€"407. doi: 10.3934/ipi.2016005.  Google Scholar

[31]

C. Greif and D. Schötzau, Preconditioners for the discretized time-harmonic Maxwell equations in mixed form, Numer. Linear Algebra Appl., 14 (2007), 281-297.  doi: 10.1002/nla.515.  Google Scholar

[32]

P. Grisvard, Singularities in Boundary Value Problems, RMA 22, Masson, Springer-Verlag, 1992.  Google Scholar

[33]

J. Hadamard, Lectures on Cauchy's Problem in Linear Partial Differential Equation, New York, Dover, 1953.  Google Scholar

[34]

M. Hanke, Limitations of the L-curve method in ill-posed problems, BIT Numer. Math., 36 (1996), 287-301.  doi: 10.1007/BF01731984.  Google Scholar

[35]

F. Hecht, New Development in FreeFem++, J. Numer. Math., 20.3-4 (2012), 251-265.  doi: 10.1515/jnum-2012-0013.  Google Scholar

[36]

J. Heleine, Identification de Paramètres Électromagnétiques par Imagerie Micro-ondes, Ph.D thesis, Université de Picardie-Jules Verne in Amiens, France, 2019. http://theses.fr/2019AMIE0013/document Google Scholar

[37]

R. Hiptmair and J. Xu, Nodal auxiliary space preconditioning in $H(\mathbf{ curl})$ and $H(\text{ div})$ spaces, SIAM J. Numer. Anal., 45.6 (2007), 2483-2509.  doi: 10.1137/060660588.  Google Scholar

[38]

V. A. KozlovV. G. Mazya and A. V. Fomin, An iterative method for solving the Cauchy problem for elliptic equations, Comput. Math. Math. Phys., 31 (1991), 45-52.   Google Scholar

[39]

M. V. Klibanov and F. Santosa, A computational quasi-reversibility method for Cauchy problems for Laplace's equation, SIAM J. Appl. Math., 51 (1991), 1653-1675.  doi: 10.1137/0151085.  Google Scholar

[40]

R. Lattès and J.-L. Lions, Méthode de Quasi-réversibilité et Applications, Dunod, Paris, 1967.  Google Scholar

[41]

L. Ling and T. Takeuchi, Boundary control for inverse Cauchy problems of the Laplace equations, Comp. Model. Eng. Sci., 29 (2008), 45-54.   Google Scholar

[42]

H. McCann, G. Pisano and L. Beltrachini, Variation in reported human head tissue electrical conductivity values, Brain Topography, (2019). doi: 10.1007/s10548-019-00710-2.  Google Scholar

[43]

V. Melicher and M. Slodička, Determination of missing boundary data for a steady-state Maxwell problem, Inverse Problems, 22 (2006), 297-310.  doi: 10.1088/0266-5611/22/1/016.  Google Scholar

[44] P. Monk, Finite Element Methods for Maxwell's Equations, Oxford University Press, 2003.  doi: 10.1093/acprof:oso/9780198508885.001.0001.  Google Scholar
[45]

J.-C. Nédélec, Mixed finite elements in $ {\mathbb{R}}^3$, Numerische Mathematik, 35 (1980), 315-341.  doi: 10.1007/BF01396415.  Google Scholar

[46]

P. Ola, L. Päivärinta and E. Somersalo, Inverse problems for time harmonic electrodynamics, Inside Out: Inverse Problems and Applications, Publications of the Research Institute for Mathematical Sciences, 47, Cambridge University Press, Cambridge, 2003, 169-191.  Google Scholar

[47]

M. Slodička and V. Melicher, An iterative algorithm for a Cauchy problem in eddy-current modelling, Appl. Math. Comput., 207 (2010), 237-246.  doi: 10.1016/j.amc.2010.05.054.  Google Scholar

[48]

P.-H. Tournier, et al., Microwave imaging of cerebrovascular accidents by using highperformance computing, Parallel Computing, 85 (2019), 88-97. doi: 10.1016/j.parco.2019.02.004.  Google Scholar

[49]

P.-H. Tournier, et al., Numerical modeling and high speed parallel computing: New perspectives for tomographic microwave imaging for brain stroke detection and monitoring, IEEE Antennas and Propagation Magazine, 59.5 (2017), 98-110. doi: 10.1109/MAP.2017.2731199.  Google Scholar

[50]

C. R. Vogel, Computational methods for inverse problems, SIAM Frontiers in Applied Mathematics, Philadelphia, 2002. doi: 10.1137/1.9780898717570.  Google Scholar

Figure 1.  Choice of the accessible part $ \Gamma_0 $ (grey line). Left: configuration G34. Right: configuration GE37
Figure 2.  Unit disc. QR method. Relative error $ \frac{\| {{\bm{E}}} - {{\bm{E}}}_\delta\|_{0, \Omega}}{\| {{\bm{E}}}\|_{0, \Omega}} $ with respect to the regularization parameter $ \delta $. Left: configuration G34. Right: configuration GE37
Figure 3.  Unit disc. QR method. Modulus of the error $ | {{\bm{E}}} - {{\bm{E}}}_{\delta}| $. Left: configuration G34. Right: configuration GE37
Figure 4.  Choice of the accessible part $ \Gamma_0 $ (grey line). Left: configuration GExt ($ \Gamma_0 $ covers 57% of the ring boundary). Middle: configuration G34 (43%). Right: configuration GE37 (47%)
Figure 5.  Ring. QR method. Relative error $ \frac{\| {{\bm{E}}} - {{\bm{E}}}_\delta\|_{0, \Omega}}{\| {{\bm{E}}}\|_{0, \Omega}} $ with respect to the regularization parameter $ \delta $. Left: configuration GExt. Middle: configuration G34. Right: configuration GE37
Figure 6.  Ring. QR method. Modulus of the error $ | {{\bm{E}}} - {{\bm{E}}}_\delta| $. Left: configuration GExt. Middle: configuration G34. Right: configuration GE37
Figure 7.  Ring. QR method. Extension/restriction method. Relative error $ \frac{\| {{\bm{E}}} - {{\bm{E}}}_\delta\|_{0, \Omega}}{\| {{\bm{E}}}\|_{0, \Omega}} $ with respect to the regularization parameter $ \delta $. Left: configuration GExt. Middle: configuration G34. Right: configuration GE37
Figure 8.  Ring. QR method. Extension/restriction method. Modulus of the error $ | {{\bm{E}}} - {{\bm{E}}}_\delta| $. Left: configuration GExt. Middle: configuration G34. Right: configuration GE37
Figure 9.  Unit disc. 5% noisy data. RR-QR method. Relative error $ \frac{\| {{\bm{E}}} - {{\bm{E}}}_\beta\|_{0, \Omega}}{\| {{\bm{E}}}\|_{0, \Omega}} $ with respect to the regularization parameter $ \delta $ at fixed $ \eta $ and $ \nu = \delta $. Left: configuration G34. Right: configuration GE37
Figure 10.  Unit disc. 5% noisy data. RR-QR method. Modulus of the error $ | {{\bm{E}}} - {{\bm{E}}}_\beta| $. Left: configuration G34. Right: configuration GE37
Figure 11.  Ring. 5% noisy data. RR-QR method with extension/restriction. Relative error for $ {{\bm{E}}} $ in $ L^2(\Omega) $-norm with respect to $ \delta $ for $ \nu = \delta $. $ \eta $ automatically fixed from noise level
Figure 12.  Ring. 5% noisy data. RR-QR method with extension/restriction. Modulus of the error in the approximation of $ {{\bm{E}}} $ for optimal $ \delta $ and $ \nu = \delta $. $ \eta $ automatically fixed from noise level. Left: configuration GExt. Middle: configuration G34. Right: configuration GE37
Figure 13.  Accessible part $ \Gamma_0 $ in 3D configurations. Set of 128 electrodes
Figure 14.  Relative error $ \frac{\| {{\bm{E}}} - {{\bm{E}}}_\delta\|_{0, \Omega}}{\| {{\bm{E}}}\|_{0, \Omega}} $ with respect to the regularization parameter $ \delta $. Left: Unit ball of $ {\mathbb{R}}^3 $. Right: 3D ring of internal radius 0.7
Figure 15.  Error in the three-dimensional ring. Left: view of the external boundary. Right: cut showing the internal boundary
Figure 16.  L-curve corresponding to the RR-QR method with extension/restriction in the ring. 5% noisy data. Left: configuration G37 (2D). Right: 3D ring
Table 1.  Unit disc. QR method. Errors for $ \delta = 9.103\text{e}-7 $
Configuration $ \frac{\| {{\bm{E}}} - {{\bm{E}}}_\delta\|_{0, \Omega}}{\| {{\bm{E}}}\|_{0, \Omega}} $ $ \frac{\|( {{\bm{E}}} - {{\bm{E}}}_\delta) \times {{\bm{n}}}\|_{0, \Gamma_1}}{\| {{\bm{E}}} \times {{\bm{n}}}\|_{0, \Gamma_1}} $ $ \| {{\bm{F}}}_\delta\|_{0, \Omega} $
G34 1.1244e-02 1.7302e-01 1.8374e-04
GE37 7.6288e-04 1.5185e-02 1.8224e-04
Configuration $ \frac{\| {{\bm{E}}} - {{\bm{E}}}_\delta\|_{0, \Omega}}{\| {{\bm{E}}}\|_{0, \Omega}} $ $ \frac{\|( {{\bm{E}}} - {{\bm{E}}}_\delta) \times {{\bm{n}}}\|_{0, \Gamma_1}}{\| {{\bm{E}}} \times {{\bm{n}}}\|_{0, \Gamma_1}} $ $ \| {{\bm{F}}}_\delta\|_{0, \Omega} $
G34 1.1244e-02 1.7302e-01 1.8374e-04
GE37 7.6288e-04 1.5185e-02 1.8224e-04
Table 2.  Ring. QR method. Errors
Configuration $ \frac{\| {{\bm{E}}} - {{\bm{E}}}_\delta\|_{0, \Omega}}{\| {{\bm{E}}}\|_{0, \Omega}} $ $ \frac{\|( {{\bm{E}}} - {{\bm{E}}}_\delta) \times {{\bm{n}}}\|_{0, \Gamma_1}}{\| {{\bm{E}}} \times {{\bm{n}}}\|_{0, \Gamma_1}} $ $ \frac{\|( {{\bm{E}}} - {{\bm{E}}}_\delta) \times {{\bm{n}}}\|_{0, \Gamma_\text{i}}}{\| {{\bm{E}}} \times {{\bm{n}}}\|_{0, \Gamma_\text{i}}} $ $ \| {{\bm{F}}}_\delta\|_{0, \Omega} $
GExt 3.9423e-04 6.6661e-04 6.6661e-04 1.7964e-04
G34 3.5093e-01 6.9598e-01 4.8011e-01 8.1894e-05
GE37 9.9784e-03 3.9068e-02 3.8737e-02 8.4337e-05
Configuration $ \frac{\| {{\bm{E}}} - {{\bm{E}}}_\delta\|_{0, \Omega}}{\| {{\bm{E}}}\|_{0, \Omega}} $ $ \frac{\|( {{\bm{E}}} - {{\bm{E}}}_\delta) \times {{\bm{n}}}\|_{0, \Gamma_1}}{\| {{\bm{E}}} \times {{\bm{n}}}\|_{0, \Gamma_1}} $ $ \frac{\|( {{\bm{E}}} - {{\bm{E}}}_\delta) \times {{\bm{n}}}\|_{0, \Gamma_\text{i}}}{\| {{\bm{E}}} \times {{\bm{n}}}\|_{0, \Gamma_\text{i}}} $ $ \| {{\bm{F}}}_\delta\|_{0, \Omega} $
GExt 3.9423e-04 6.6661e-04 6.6661e-04 1.7964e-04
G34 3.5093e-01 6.9598e-01 4.8011e-01 8.1894e-05
GE37 9.9784e-03 3.9068e-02 3.8737e-02 8.4337e-05
Table 3.  Ring. QR method. Extension/restriction method. Errors
Configuration $ \frac{\| {{\bm{E}}} - {{\bm{E}}}_\delta\|_{0, \Omega}}{\| {{\bm{E}}}\|_{0, \Omega}} $ $ \frac{\|( {{\bm{E}}} - {{\bm{E}}}_\delta) \times {{\bm{n}}}\|_{0, \Gamma_1}}{\| {{\bm{E}}} \times {{\bm{n}}}\|_{0, \Gamma_1}} $ $ \frac{\|( {{\bm{E}}} - {{\bm{E}}}_\delta) \times {{\bm{n}}}\|_{0, \Gamma_\text{i}}}{\| {{\bm{E}}} \times {{\bm{n}}}\|_{0, \Gamma_\text{i}}} $ $ \| {{\bm{F}}}_\delta\|_{0, \Omega} $
GExt 3.7745e-04 1.5665e-04 1.5665e-04 3.1125e-04
G34 3.4345e-02 1.2558e-01 8.1477e-03 2.6564e-04
GE37 1.7728e-03 1.1968e-02 2.9520e-04 9.0131e-05
Configuration $ \frac{\| {{\bm{E}}} - {{\bm{E}}}_\delta\|_{0, \Omega}}{\| {{\bm{E}}}\|_{0, \Omega}} $ $ \frac{\|( {{\bm{E}}} - {{\bm{E}}}_\delta) \times {{\bm{n}}}\|_{0, \Gamma_1}}{\| {{\bm{E}}} \times {{\bm{n}}}\|_{0, \Gamma_1}} $ $ \frac{\|( {{\bm{E}}} - {{\bm{E}}}_\delta) \times {{\bm{n}}}\|_{0, \Gamma_\text{i}}}{\| {{\bm{E}}} \times {{\bm{n}}}\|_{0, \Gamma_\text{i}}} $ $ \| {{\bm{F}}}_\delta\|_{0, \Omega} $
GExt 3.7745e-04 1.5665e-04 1.5665e-04 3.1125e-04
G34 3.4345e-02 1.2558e-01 8.1477e-03 2.6564e-04
GE37 1.7728e-03 1.1968e-02 2.9520e-04 9.0131e-05
Table 4.  Unit disc. 5% noisy data. RR-QR method. Errors in the approximation of $ {{\bm{E}}} $ for $ \nu = \delta $ at optimal $ \delta $ and automatically fixed $ \eta $
Configuration $ \frac{\| {{\bm{E}}} - {{\bm{E}}}_\delta\|_{0, \Omega}}{\| {{\bm{E}}}\|_{0, \Omega}} $ $ \frac{\|( {{\bm{E}}} - {{\bm{E}}}_\delta) \times {{\bm{n}}}\|_{0, \Gamma_0}}{\| {{\bm{E}}} \times {{\bm{n}}}\|_{0, \Gamma_0}} $ $ \frac{\|( {{\bm{E}}} - {{\bm{E}}}_\delta) \times {{\bm{n}}}\|_{0, \Gamma_1}}{\| {{\bm{E}}} \times {{\bm{n}}}\|_{0, \Gamma_1}} $ $ \| {{\bm{F}}}_\delta\|_{0, \Omega} $
G34 3.4698e-02 2.9145e-02 3.5686e-01 7.0667e-03
GE37 1.8550e-02 3.2848e-02 1.7154e-01 1.1857e-02
Configuration $ \frac{\| {{\bm{E}}} - {{\bm{E}}}_\delta\|_{0, \Omega}}{\| {{\bm{E}}}\|_{0, \Omega}} $ $ \frac{\|( {{\bm{E}}} - {{\bm{E}}}_\delta) \times {{\bm{n}}}\|_{0, \Gamma_0}}{\| {{\bm{E}}} \times {{\bm{n}}}\|_{0, \Gamma_0}} $ $ \frac{\|( {{\bm{E}}} - {{\bm{E}}}_\delta) \times {{\bm{n}}}\|_{0, \Gamma_1}}{\| {{\bm{E}}} \times {{\bm{n}}}\|_{0, \Gamma_1}} $ $ \| {{\bm{F}}}_\delta\|_{0, \Omega} $
G34 3.4698e-02 2.9145e-02 3.5686e-01 7.0667e-03
GE37 1.8550e-02 3.2848e-02 1.7154e-01 1.1857e-02
Table 5.  Ring. 5% noisy data. RR-QR method with extension/restriction. Errors in the approximation of $ {{\bm{E}}} $ for optimal $ \delta $ and $ \nu = \delta $. $ \eta $ automatically fixed from noise level. Errors on parts of the boundary refer to the $ L^2 $-norm of the tangential trace
Configuration $ \frac{\| {{\bm{E}}} - {{\bm{E}}}_\delta\|_{0, \Omega}}{\| {{\bm{E}}}\|_{0, \Omega}} $ $ \Gamma_0 $ $ \Gamma_1 $ $ \Gamma_\text{i} $ $ \| {{\bm{F}}}_\delta\|_{0, \Omega} $
GExt 9.2775e-03 3.3871e-02 4.2425e-03 4.2425e-03 1.0602e-02
G34 7.9236e-02 3.0600e-02 1.5695e-01 4.2697e-02 8.7901e-03
GE37 2.6712e-02 3.3503e-02 7.7043e-02 1.6996e-02 1.3596e-02
Configuration $ \frac{\| {{\bm{E}}} - {{\bm{E}}}_\delta\|_{0, \Omega}}{\| {{\bm{E}}}\|_{0, \Omega}} $ $ \Gamma_0 $ $ \Gamma_1 $ $ \Gamma_\text{i} $ $ \| {{\bm{F}}}_\delta\|_{0, \Omega} $
GExt 9.2775e-03 3.3871e-02 4.2425e-03 4.2425e-03 1.0602e-02
G34 7.9236e-02 3.0600e-02 1.5695e-01 4.2697e-02 8.7901e-03
GE37 2.6712e-02 3.3503e-02 7.7043e-02 1.6996e-02 1.3596e-02
Table 6.  Errors in the unit ball of $ {\mathbb{R}}^3 $ and in a three dimensional ring. Errors on parts of the boundary refer to the $ L^2 $-norm of the tangential trace
Domain $ \frac{\| {{\bm{E}}} - {{\bm{E}}}_\delta\|_{0, \Omega}}{\| {{\bm{E}}}\|_{0, \Omega}} $ $ \Gamma_0 $ $ \Gamma_1 $ $ \Gamma_\text{i} $ $ \| {{\bm{F}}}_\delta\|_{0, \Omega} $
Ball 1.0834e-01 2.3864e-02 4.0949e-01 $ \cdot $ 2.9336e-03
Ring 1.2414e-01 1.1567e-02 2.7969e-01 5.6528e-02 1.6497e-03
Domain $ \frac{\| {{\bm{E}}} - {{\bm{E}}}_\delta\|_{0, \Omega}}{\| {{\bm{E}}}\|_{0, \Omega}} $ $ \Gamma_0 $ $ \Gamma_1 $ $ \Gamma_\text{i} $ $ \| {{\bm{F}}}_\delta\|_{0, \Omega} $
Ball 1.0834e-01 2.3864e-02 4.0949e-01 $ \cdot $ 2.9336e-03
Ring 1.2414e-01 1.1567e-02 2.7969e-01 5.6528e-02 1.6497e-03
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