November  2021, 20(11): 4025-4041. doi: 10.3934/cpaa.2021142

Stability of current density impedance imaging II

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* Corresponding author

Received  January 2021 Revised  June 2021 Published  November 2021 Early access  August 2021

This paper is a continuation of the authors earlier work on stability of Current Density Impedance Imaging (CDII) [R. Lopez, A. Moradifam, Stability of Current Density Impedance Imaging, SIAM J. Math. Anal. (2020).] We show that CDII is stable with respect to errors in both measurement of the magnitude of the current density vector field in the interior and the measurement of the voltage potential on the boundary. This completes the authors study of the stability of Current Density Independence Imaging which was previously shown only by numerical simulations.

Citation: Amir Moradifam, Robert Lopez. Stability of current density impedance imaging II. Communications on Pure and Applied Analysis, 2021, 20 (11) : 4025-4041. doi: 10.3934/cpaa.2021142
References:
[1]

Giovanni Alberti, A Lusin type theorem for gradients, J. Funct. Anal., 100 (1991), 110-118.  doi: 10.1016/0022-1236(91)90104-D.

[2]

Gabriele Anzellotti, Pairings between measures and bounded functions and compensated compactness, Ann. Mat. Pura Appl., 135 (1983), 293-318.  doi: 10.1007/BF01781073.

[3]

L. Borcea, Electrical impedance tomography, Inverse Probl., 18 (2002), R99-R136. doi: 10.1088/0266-5611/18/6/201.

[4]

M. Cheney and D. Isaacso, An overview of inversion algorithms for impedance imaging, Contemp. Math., 122 (1991), 29-39.  doi: 10.1090/conm/122/1135853.

[5]

M. CheneyD. Isaacson and and J. C. Newell, Electrical impedance tomography, SIAM Rev., 41 (1999), 85-101.  doi: 10.1137/S0036144598333613.

[6]

M. Dos Santos, Characteristic functions on the boundary of a planar domain need not be traces of least gradient functions, Confluentes Math., 9 (2017), 65-93.  doi: 10.5802/cml.36.

[7]

W. GórnyP. Rybka and A. Sabra, Special cases of the planar least gradient problem, Nonlinear Anal., 151 (2017), 66-95.  doi: 10.1016/j.na.2016.11.020.

[8]

A. GreenleafY. KurylevM. Lassas and G. Uhlmann, Invisibility and inverse problems, Bull. Amer. Math. Soc., 46 (2009), 55-97.  doi: 10.1090/S0273-0979-08-01232-9.

[9]

E. Giusti, Minimal Surfaces and Functions of Bounded Variations, Birkhäuser, Boston, 1984. doi: 10.1007/978-1-4684-9486-0.

[10]

N. HoellA. Moradifam and A. Nachman, Current density impedance imaging of an anisotropic conductivity in a known conformal class, SIAM J. Math. Anal., 46 (2014), 1820-1842.  doi: 10.1137/130911524.

[11]

D. Isaacson and M. Cheney, Effects of measurement precision and finite numbers of electrodes on linear impedance imaging algorithms,, SIAM J. Appl. Math., 51 (1991), 1705-1731.  doi: 10.1137/0151087.

[12]

R. L. JerrardA. Moradifam and A. I. Nachman., Existence and uniqueness of minimizers of general least gradient problems, J. Reine Angew. Math., 734 (2018), 71-97.  doi: 10.1515/crelle-2014-0151.

[13]

M. L. JoyG. C. Scott and M. Henkelman, In vivo detection of applied electric currents by magnetic resonance imaging, Magn. Reson. Imaging, 7 (1989), 89-94. 

[14]

M. J. Joy, A. I. Nachman, K. F. Hasanov, R. S. Yoon and A. W. Ma, A New Approach to Current Density Impedance Imaging, Kyoto, Japan, 2004.

[15]

R. Lopez and A. Moradifam, Stability of current density impedance imaging, SIAM J. Math. Anal., 52 (2020), 4506-4523.  doi: 10.1137/19M126520X.

[16]

N. Mandache, Exponential instability in an inverse problem for the Schrodinger equation, Inverse Probl., 17 (2001), 1435-1444.  doi: 10.1088/0266-5611/17/5/313.

[17]

C. Montalto and A. Tamasan, Stability in conductivity imaging from partial measurements of one interior current, Inverse Probl. Imag., 11 (2017), 339-353.  doi: 10.3934/ipi.2017016.

[18]

C. Montalto and P. Stefanov, Stability of coupled-physics inverse problems with one internal measurement, Inverse Probl., 29 (2013), 125004, 13 pp. doi: 10.1088/0266-5611/29/12/125004.

[19]

A. Moradifam, Least gradient problems with Neumann boundary condition, J. Differ. Equ., 263 (2017), 7900-7918.  doi: 10.1016/j.jde.2017.08.031.

[20]

A. Moradifam, Existence and structure of minimizers of least gradient problems, Indiana Uni. Math J., 67 (2018), 1025-1037.  doi: 10.1512/iumj.2018.67.7360.

[21]

A. MoradifamA. Nachman and A. Tamasan, Conductivity imaging from one interior measurement in the presence of perfectly conducting and insulating inclusions, SIAM J. Math. Anal., 44 (2012), 3969-3990.  doi: 10.1137/120866701.

[22]

A. Moradifam, A. Nachman and A. Timonov, A convergent algorithm for the hybrid problem of reconstructing conductivity from minimal interior data, Inverse Probl., 28 (2012), 084003, 23pp. doi: 10.1088/0266-5611/28/8/084003.

[23]

A. Moradifam, A. Nachman and A. Tamasan, Uniqueness of minimizers of weighted least gradient problems arising in hybrid inverse problems, Calc. Var. Partial Differ. Equ., 57 (2018), 14 pp. doi: 10.1007/s00526-017-1274-x.

[24]

M. W. Hirsch, Differential Topology, Springer-Verlag, New York-Heidelberg, 1976.

[25]

A. NachmanA. Tamasan and A. Timonov, Conductivity imaging with a single measurement of boundary and interior data, Inverse Probl., 23 (2007), 2551-2563.  doi: 10.1088/0266-5611/23/6/017.

[26]

A. Nachman, A. Tamasan and A. Timonov, Recovering the conductivity from a single measurement of interior data, Inverse Probl., 25 (2009), 035014, 16pp. doi: 10.1088/0266-5611/25/3/035014.

[27]

A. NachmanA. Tamasan and A. Timonov, Reconstruction of Planar Conductivities in Subdomains from Incomplete Data, SIAM J. Appl. Math., 70 (2010), 3342-3362.  doi: 10.1137/10079241X.

[28]

A. Nachman, A. Tamasan and A. Timonov, Current density impedance imaging, in Tomography and Inverse Transport Theory, Contemp. Math., Amer. Math. Soc., Providence, RI, 2011. doi: 10.1090/conm/559/11076.

[29]

M. Z. Nashed and A. Tamasan, Structural stability in a minimization problem and applications to conductivity imaging, Inverse Probl. Imag., 5 (2011), 219-236.  doi: 10.3934/ipi.2011.5.219.

[30]

G. S. Spradlin and A. Tamasan, Not all traces on the circle come from functions of least gradient in the disk, Indiana Univ. Math. J., 63 (2014), 1819-1837.  doi: 10.1512/iumj.2014.63.5421.

show all references

References:
[1]

Giovanni Alberti, A Lusin type theorem for gradients, J. Funct. Anal., 100 (1991), 110-118.  doi: 10.1016/0022-1236(91)90104-D.

[2]

Gabriele Anzellotti, Pairings between measures and bounded functions and compensated compactness, Ann. Mat. Pura Appl., 135 (1983), 293-318.  doi: 10.1007/BF01781073.

[3]

L. Borcea, Electrical impedance tomography, Inverse Probl., 18 (2002), R99-R136. doi: 10.1088/0266-5611/18/6/201.

[4]

M. Cheney and D. Isaacso, An overview of inversion algorithms for impedance imaging, Contemp. Math., 122 (1991), 29-39.  doi: 10.1090/conm/122/1135853.

[5]

M. CheneyD. Isaacson and and J. C. Newell, Electrical impedance tomography, SIAM Rev., 41 (1999), 85-101.  doi: 10.1137/S0036144598333613.

[6]

M. Dos Santos, Characteristic functions on the boundary of a planar domain need not be traces of least gradient functions, Confluentes Math., 9 (2017), 65-93.  doi: 10.5802/cml.36.

[7]

W. GórnyP. Rybka and A. Sabra, Special cases of the planar least gradient problem, Nonlinear Anal., 151 (2017), 66-95.  doi: 10.1016/j.na.2016.11.020.

[8]

A. GreenleafY. KurylevM. Lassas and G. Uhlmann, Invisibility and inverse problems, Bull. Amer. Math. Soc., 46 (2009), 55-97.  doi: 10.1090/S0273-0979-08-01232-9.

[9]

E. Giusti, Minimal Surfaces and Functions of Bounded Variations, Birkhäuser, Boston, 1984. doi: 10.1007/978-1-4684-9486-0.

[10]

N. HoellA. Moradifam and A. Nachman, Current density impedance imaging of an anisotropic conductivity in a known conformal class, SIAM J. Math. Anal., 46 (2014), 1820-1842.  doi: 10.1137/130911524.

[11]

D. Isaacson and M. Cheney, Effects of measurement precision and finite numbers of electrodes on linear impedance imaging algorithms,, SIAM J. Appl. Math., 51 (1991), 1705-1731.  doi: 10.1137/0151087.

[12]

R. L. JerrardA. Moradifam and A. I. Nachman., Existence and uniqueness of minimizers of general least gradient problems, J. Reine Angew. Math., 734 (2018), 71-97.  doi: 10.1515/crelle-2014-0151.

[13]

M. L. JoyG. C. Scott and M. Henkelman, In vivo detection of applied electric currents by magnetic resonance imaging, Magn. Reson. Imaging, 7 (1989), 89-94. 

[14]

M. J. Joy, A. I. Nachman, K. F. Hasanov, R. S. Yoon and A. W. Ma, A New Approach to Current Density Impedance Imaging, Kyoto, Japan, 2004.

[15]

R. Lopez and A. Moradifam, Stability of current density impedance imaging, SIAM J. Math. Anal., 52 (2020), 4506-4523.  doi: 10.1137/19M126520X.

[16]

N. Mandache, Exponential instability in an inverse problem for the Schrodinger equation, Inverse Probl., 17 (2001), 1435-1444.  doi: 10.1088/0266-5611/17/5/313.

[17]

C. Montalto and A. Tamasan, Stability in conductivity imaging from partial measurements of one interior current, Inverse Probl. Imag., 11 (2017), 339-353.  doi: 10.3934/ipi.2017016.

[18]

C. Montalto and P. Stefanov, Stability of coupled-physics inverse problems with one internal measurement, Inverse Probl., 29 (2013), 125004, 13 pp. doi: 10.1088/0266-5611/29/12/125004.

[19]

A. Moradifam, Least gradient problems with Neumann boundary condition, J. Differ. Equ., 263 (2017), 7900-7918.  doi: 10.1016/j.jde.2017.08.031.

[20]

A. Moradifam, Existence and structure of minimizers of least gradient problems, Indiana Uni. Math J., 67 (2018), 1025-1037.  doi: 10.1512/iumj.2018.67.7360.

[21]

A. MoradifamA. Nachman and A. Tamasan, Conductivity imaging from one interior measurement in the presence of perfectly conducting and insulating inclusions, SIAM J. Math. Anal., 44 (2012), 3969-3990.  doi: 10.1137/120866701.

[22]

A. Moradifam, A. Nachman and A. Timonov, A convergent algorithm for the hybrid problem of reconstructing conductivity from minimal interior data, Inverse Probl., 28 (2012), 084003, 23pp. doi: 10.1088/0266-5611/28/8/084003.

[23]

A. Moradifam, A. Nachman and A. Tamasan, Uniqueness of minimizers of weighted least gradient problems arising in hybrid inverse problems, Calc. Var. Partial Differ. Equ., 57 (2018), 14 pp. doi: 10.1007/s00526-017-1274-x.

[24]

M. W. Hirsch, Differential Topology, Springer-Verlag, New York-Heidelberg, 1976.

[25]

A. NachmanA. Tamasan and A. Timonov, Conductivity imaging with a single measurement of boundary and interior data, Inverse Probl., 23 (2007), 2551-2563.  doi: 10.1088/0266-5611/23/6/017.

[26]

A. Nachman, A. Tamasan and A. Timonov, Recovering the conductivity from a single measurement of interior data, Inverse Probl., 25 (2009), 035014, 16pp. doi: 10.1088/0266-5611/25/3/035014.

[27]

A. NachmanA. Tamasan and A. Timonov, Reconstruction of Planar Conductivities in Subdomains from Incomplete Data, SIAM J. Appl. Math., 70 (2010), 3342-3362.  doi: 10.1137/10079241X.

[28]

A. Nachman, A. Tamasan and A. Timonov, Current density impedance imaging, in Tomography and Inverse Transport Theory, Contemp. Math., Amer. Math. Soc., Providence, RI, 2011. doi: 10.1090/conm/559/11076.

[29]

M. Z. Nashed and A. Tamasan, Structural stability in a minimization problem and applications to conductivity imaging, Inverse Probl. Imag., 5 (2011), 219-236.  doi: 10.3934/ipi.2011.5.219.

[30]

G. S. Spradlin and A. Tamasan, Not all traces on the circle come from functions of least gradient in the disk, Indiana Univ. Math. J., 63 (2014), 1819-1837.  doi: 10.1512/iumj.2014.63.5421.

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