December  2020, 14(6): 1107-1133. 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  December 2020 Early access  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 and Imaging, 2020, 14 (6) : 1107-1133. 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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[32]

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

[33]

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

[34]

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

[35]

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

[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

[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.

[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. 

[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.

[40]

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

[41]

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

[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.

[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.

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

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

[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.

[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.

[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.

[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.

[50]

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

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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[32]

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

[33]

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

[34]

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

[35]

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

[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

[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.

[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. 

[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.

[40]

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

[41]

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

[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.

[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.

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

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

[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.

[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.

[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.

[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.

[50]

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

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{\| {{\boldsymbol{E}}} - {{\boldsymbol{E}}}_\delta\|_{0, \Omega}}{\| {{\boldsymbol{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 $ | {{\boldsymbol{E}}} - {{\boldsymbol{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{\| {{\boldsymbol{E}}} - {{\boldsymbol{E}}}_\delta\|_{0, \Omega}}{\| {{\boldsymbol{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 $ | {{\boldsymbol{E}}} - {{\boldsymbol{E}}}_\delta| $. Left: configuration GExt. Middle: configuration G34. Right: configuration GE37
Figure 7.  Ring. QR method. Extension/restriction method. Relative error $ \frac{\| {{\boldsymbol{E}}} - {{\boldsymbol{E}}}_\delta\|_{0, \Omega}}{\| {{\boldsymbol{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 $ | {{\boldsymbol{E}}} - {{\boldsymbol{E}}}_\delta| $. Left: configuration GExt. Middle: configuration G34. Right: configuration GE37
Figure 9.  Unit disc. 5% noisy data. RR-QR method. Relative error $ \frac{\| {{\boldsymbol{E}}} - {{\boldsymbol{E}}}_\beta\|_{0, \Omega}}{\| {{\boldsymbol{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 $ | {{\boldsymbol{E}}} - {{\boldsymbol{E}}}_\beta| $. Left: configuration G34. Right: configuration GE37
Figure 11.  Ring. 5% noisy data. RR-QR method with extension/restriction. Relative error for $ {{\boldsymbol{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 $ {{\boldsymbol{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{\| {{\boldsymbol{E}}} - {{\boldsymbol{E}}}_\delta\|_{0, \Omega}}{\| {{\boldsymbol{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{\| {{\boldsymbol{E}}} - {{\boldsymbol{E}}}_\delta\|_{0, \Omega}}{\| {{\boldsymbol{E}}}\|_{0, \Omega}} $ $ \frac{\|( {{\boldsymbol{E}}} - {{\boldsymbol{E}}}_\delta) \times {{\boldsymbol{n}}}\|_{0, \Gamma_1}}{\| {{\boldsymbol{E}}} \times {{\boldsymbol{n}}}\|_{0, \Gamma_1}} $ $ \| {{\boldsymbol{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{\| {{\boldsymbol{E}}} - {{\boldsymbol{E}}}_\delta\|_{0, \Omega}}{\| {{\boldsymbol{E}}}\|_{0, \Omega}} $ $ \frac{\|( {{\boldsymbol{E}}} - {{\boldsymbol{E}}}_\delta) \times {{\boldsymbol{n}}}\|_{0, \Gamma_1}}{\| {{\boldsymbol{E}}} \times {{\boldsymbol{n}}}\|_{0, \Gamma_1}} $ $ \| {{\boldsymbol{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{\| {{\boldsymbol{E}}} - {{\boldsymbol{E}}}_\delta\|_{0, \Omega}}{\| {{\boldsymbol{E}}}\|_{0, \Omega}} $ $ \frac{\|( {{\boldsymbol{E}}} - {{\boldsymbol{E}}}_\delta) \times {{\boldsymbol{n}}}\|_{0, \Gamma_1}}{\| {{\boldsymbol{E}}} \times {{\boldsymbol{n}}}\|_{0, \Gamma_1}} $ $ \frac{\|( {{\boldsymbol{E}}} - {{\boldsymbol{E}}}_\delta) \times {{\boldsymbol{n}}}\|_{0, \Gamma_\text{i}}}{\| {{\boldsymbol{E}}} \times {{\boldsymbol{n}}}\|_{0, \Gamma_\text{i}}} $ $ \| {{\boldsymbol{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{\| {{\boldsymbol{E}}} - {{\boldsymbol{E}}}_\delta\|_{0, \Omega}}{\| {{\boldsymbol{E}}}\|_{0, \Omega}} $ $ \frac{\|( {{\boldsymbol{E}}} - {{\boldsymbol{E}}}_\delta) \times {{\boldsymbol{n}}}\|_{0, \Gamma_1}}{\| {{\boldsymbol{E}}} \times {{\boldsymbol{n}}}\|_{0, \Gamma_1}} $ $ \frac{\|( {{\boldsymbol{E}}} - {{\boldsymbol{E}}}_\delta) \times {{\boldsymbol{n}}}\|_{0, \Gamma_\text{i}}}{\| {{\boldsymbol{E}}} \times {{\boldsymbol{n}}}\|_{0, \Gamma_\text{i}}} $ $ \| {{\boldsymbol{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{\| {{\boldsymbol{E}}} - {{\boldsymbol{E}}}_\delta\|_{0, \Omega}}{\| {{\boldsymbol{E}}}\|_{0, \Omega}} $ $ \frac{\|( {{\boldsymbol{E}}} - {{\boldsymbol{E}}}_\delta) \times {{\boldsymbol{n}}}\|_{0, \Gamma_1}}{\| {{\boldsymbol{E}}} \times {{\boldsymbol{n}}}\|_{0, \Gamma_1}} $ $ \frac{\|( {{\boldsymbol{E}}} - {{\boldsymbol{E}}}_\delta) \times {{\boldsymbol{n}}}\|_{0, \Gamma_\text{i}}}{\| {{\boldsymbol{E}}} \times {{\boldsymbol{n}}}\|_{0, \Gamma_\text{i}}} $ $ \| {{\boldsymbol{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{\| {{\boldsymbol{E}}} - {{\boldsymbol{E}}}_\delta\|_{0, \Omega}}{\| {{\boldsymbol{E}}}\|_{0, \Omega}} $ $ \frac{\|( {{\boldsymbol{E}}} - {{\boldsymbol{E}}}_\delta) \times {{\boldsymbol{n}}}\|_{0, \Gamma_1}}{\| {{\boldsymbol{E}}} \times {{\boldsymbol{n}}}\|_{0, \Gamma_1}} $ $ \frac{\|( {{\boldsymbol{E}}} - {{\boldsymbol{E}}}_\delta) \times {{\boldsymbol{n}}}\|_{0, \Gamma_\text{i}}}{\| {{\boldsymbol{E}}} \times {{\boldsymbol{n}}}\|_{0, \Gamma_\text{i}}} $ $ \| {{\boldsymbol{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 $ {{\boldsymbol{E}}} $ for $ \nu = \delta $ at optimal $ \delta $ and automatically fixed $ \eta $
Configuration $ \frac{\| {{\boldsymbol{E}}} - {{\boldsymbol{E}}}_\delta\|_{0, \Omega}}{\| {{\boldsymbol{E}}}\|_{0, \Omega}} $ $ \frac{\|( {{\boldsymbol{E}}} - {{\boldsymbol{E}}}_\delta) \times {{\boldsymbol{n}}}\|_{0, \Gamma_0}}{\| {{\boldsymbol{E}}} \times {{\boldsymbol{n}}}\|_{0, \Gamma_0}} $ $ \frac{\|( {{\boldsymbol{E}}} - {{\boldsymbol{E}}}_\delta) \times {{\boldsymbol{n}}}\|_{0, \Gamma_1}}{\| {{\boldsymbol{E}}} \times {{\boldsymbol{n}}}\|_{0, \Gamma_1}} $ $ \| {{\boldsymbol{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{\| {{\boldsymbol{E}}} - {{\boldsymbol{E}}}_\delta\|_{0, \Omega}}{\| {{\boldsymbol{E}}}\|_{0, \Omega}} $ $ \frac{\|( {{\boldsymbol{E}}} - {{\boldsymbol{E}}}_\delta) \times {{\boldsymbol{n}}}\|_{0, \Gamma_0}}{\| {{\boldsymbol{E}}} \times {{\boldsymbol{n}}}\|_{0, \Gamma_0}} $ $ \frac{\|( {{\boldsymbol{E}}} - {{\boldsymbol{E}}}_\delta) \times {{\boldsymbol{n}}}\|_{0, \Gamma_1}}{\| {{\boldsymbol{E}}} \times {{\boldsymbol{n}}}\|_{0, \Gamma_1}} $ $ \| {{\boldsymbol{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 $ {{\boldsymbol{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{\| {{\boldsymbol{E}}} - {{\boldsymbol{E}}}_\delta\|_{0, \Omega}}{\| {{\boldsymbol{E}}}\|_{0, \Omega}} $ $ \Gamma_0 $ $ \Gamma_1 $ $ \Gamma_\text{i} $ $ \| {{\boldsymbol{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{\| {{\boldsymbol{E}}} - {{\boldsymbol{E}}}_\delta\|_{0, \Omega}}{\| {{\boldsymbol{E}}}\|_{0, \Omega}} $ $ \Gamma_0 $ $ \Gamma_1 $ $ \Gamma_\text{i} $ $ \| {{\boldsymbol{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{\| {{\boldsymbol{E}}} - {{\boldsymbol{E}}}_\delta\|_{0, \Omega}}{\| {{\boldsymbol{E}}}\|_{0, \Omega}} $ $ \Gamma_0 $ $ \Gamma_1 $ $ \Gamma_\text{i} $ $ \| {{\boldsymbol{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{\| {{\boldsymbol{E}}} - {{\boldsymbol{E}}}_\delta\|_{0, \Omega}}{\| {{\boldsymbol{E}}}\|_{0, \Omega}} $ $ \Gamma_0 $ $ \Gamma_1 $ $ \Gamma_\text{i} $ $ \| {{\boldsymbol{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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