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February  2022, 16(1): 229-249. doi: 10.3934/ipi.2021048

Refined stability estimates in electrical impedance tomography with multi-layer structure

1. 

School of Mathematical Sciences, Laboratory of Mathematics and Complex Systems, Ministry of Education, Beijing Normal University, Beijing 100875, China

2. 

Institute of Applied Mathematical Sciences, National Taiwan University, Taipei 106, Taiwan

3. 

School of Mathematical Sciences, Beijing Normal University, Beijing 100875, China

* Corresponding author: Jenn-Nan Wang

Received  January 2021 Revised  May 2021 Published  February 2022 Early access  July 2021

Fund Project: H.G. Li was partially supported by BJNSF (1202013) and NSFC (11631002, 11971061). J.N. Wang was supported in part by MOST 108-2115-M-002-002-MY3 and 109-2115-M-002-001-MY3

In this paper we study the inverse problem of determining an electrical inclusion in a multi-layer composite from boundary measurements in 2D. We assume the conductivities in different layers are different and derive a stability estimate for the linearized map with explicit formulae on the conductivity and the thickness of each layer. Intuitively, if an inclusion is surrounded by a highly conductive layer, then, in view of "the principle of the least work", the current will take a path in the highly conductive layer and disregard the existence of the inclusion. Consequently, a worse stability of identifying the hidden inclusion is expected in this case. Our estimates indeed show that the ill-posedness of the problem increases as long as the conductivity of some layer becomes large. This work is an extension of the previous result by Nagayasu-Uhlmann-Wang[15], where a depth-dependent estimate is derived when an inclusion is deeply hidden in a conductor. Estimates in this work also show the influence of the depth of the inclusion.

Citation: Haigang Li, Jenn-Nan Wang, Ling Wang. Refined stability estimates in electrical impedance tomography with multi-layer structure. Inverse Problems and Imaging, 2022, 16 (1) : 229-249. doi: 10.3934/ipi.2021048
References:
[1]

G. Alessandrini, Stable determination of conductivity by boundary measurements, Appl. Anal., 27 (1988), 153-172.  doi: 10.1080/00036818808839730.

[2]

G. Alessandrini, Singular solutions of elliptic equations and the determination of conductivity by boundary measurements, J. Diff. Equations, 84 (1990), 252-272.  doi: 10.1016/0022-0396(90)90078-4.

[3]

G. Alessandrini and M. D. Cristo, Stable determination of an inclusion by boundary measurements, SIAM J. Math. Anal., 37 (2005), 200-217.  doi: 10.1137/S003614100444191X.

[4]

G. Alessandrini and R. Gaburro, Determining conductivity with special anisotropy by boundary measurements, SIAM J. Math. Anal., 33 (2001), 153-171.  doi: 10.1137/S0036141000369563.

[5]

G. Alessandrini and R. Gaburro, The local Calderón problem and the determination at the boundary of the conductivity, Comm. Partial Differential Equations, 34 (2009), 918-936.  doi: 10.1080/03605300903017397.

[6]

G. Alessandrini and A. Scapin, Depth dependent resolution in electrical impedance tomography, J. Inverse Ill-Posed Probl., 25 (2017), 391-402.  doi: 10.1515/jiip-2017-0029.

[7]

A. Calderón, On an inverse boundary value problem, Seminar on Numerical Analysis and Its Applications to Continuum Physics, Sociedade Brasileira de Matemática, Río de Janeiro, Brazil, 1980, 65–73.

[8]

H. Garde and N. Hyvonen, Optimal depth-dependent distinguishability bounds for electrical impedance tomography in arbitrary dimension, SIAM J. Appl. Math., 80 (2020), 20-43.  doi: 10.1137/19M1258761.

[9]

H. Garde and K. Knudsen, Distinguishability revisited: Depth dependent bounds on reconstruction quality in electrical impedance tomography, SIAM J. Appl. Math., 77 (2017), 697-720.  doi: 10.1137/16M1072991.

[10]

D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, 2$^{nd}$ edition, Springer-Verlag, Berlin Heidelberg, 2001.

[11]

V. Isakov, On uniqueness of recovery of a discontinuous conductivity coefficient, Comm. Pure Appl. Math., 41 (1988), 865-877.  doi: 10.1002/cpa.3160410702.

[12]

J. Jossinet, The impedivity of freshly excised human breast tissue, Physiological Measurement, 19 (1998), 61–75. doi: 10.1088/0967-3334/19/1/006.

[13]

A. Lorenzi and C. D. Pagani, On the stability of the surface separating two homogeneous media with different thermal conductivities, Acta Math. Sci., 7 (1987), 411-429.  doi: 10.1016/S0252-9602(18)30464-8.

[14]

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

[15]

S. Nagayasu, G. Uhlmann and J.-N. Wang, A depth-dependent stability estimate in electrical impedance tomography, Inverse Problems, 25 (2009), 075001. doi: 10.1088/0266-5611/25/7/075001.

[16]

V. P. Palamodov, Gabor analysis of the continuum model for impedance tomography, Ark. Mat., 40 (2002), 169-187.  doi: 10.1007/BF02384508.

[17]

J. Sylvester and G. Uhlmann, Inverse boundary value problems at the boundary: Continuous dependence, Comm. Pure Appl. Math., 41 (1988), 197-219.  doi: 10.1002/cpa.3160410205.

[18]

G. Uhlmann and J. -N. Wang, Reconstructing discontinuities using complex geometrical optics solutions, SIAM J. Appl. Math., 68 (2008), 1026-1044.  doi: 10.1137/060676350.

[19]

G. UhlmannJ. -N. Wang and C. -T. Wu, Reconstruction of inclusions in an elastic body, J. Math. Pures Appl., 91 (2009), 569-582.  doi: 10.1016/j.matpur.2009.01.006.

show all references

References:
[1]

G. Alessandrini, Stable determination of conductivity by boundary measurements, Appl. Anal., 27 (1988), 153-172.  doi: 10.1080/00036818808839730.

[2]

G. Alessandrini, Singular solutions of elliptic equations and the determination of conductivity by boundary measurements, J. Diff. Equations, 84 (1990), 252-272.  doi: 10.1016/0022-0396(90)90078-4.

[3]

G. Alessandrini and M. D. Cristo, Stable determination of an inclusion by boundary measurements, SIAM J. Math. Anal., 37 (2005), 200-217.  doi: 10.1137/S003614100444191X.

[4]

G. Alessandrini and R. Gaburro, Determining conductivity with special anisotropy by boundary measurements, SIAM J. Math. Anal., 33 (2001), 153-171.  doi: 10.1137/S0036141000369563.

[5]

G. Alessandrini and R. Gaburro, The local Calderón problem and the determination at the boundary of the conductivity, Comm. Partial Differential Equations, 34 (2009), 918-936.  doi: 10.1080/03605300903017397.

[6]

G. Alessandrini and A. Scapin, Depth dependent resolution in electrical impedance tomography, J. Inverse Ill-Posed Probl., 25 (2017), 391-402.  doi: 10.1515/jiip-2017-0029.

[7]

A. Calderón, On an inverse boundary value problem, Seminar on Numerical Analysis and Its Applications to Continuum Physics, Sociedade Brasileira de Matemática, Río de Janeiro, Brazil, 1980, 65–73.

[8]

H. Garde and N. Hyvonen, Optimal depth-dependent distinguishability bounds for electrical impedance tomography in arbitrary dimension, SIAM J. Appl. Math., 80 (2020), 20-43.  doi: 10.1137/19M1258761.

[9]

H. Garde and K. Knudsen, Distinguishability revisited: Depth dependent bounds on reconstruction quality in electrical impedance tomography, SIAM J. Appl. Math., 77 (2017), 697-720.  doi: 10.1137/16M1072991.

[10]

D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, 2$^{nd}$ edition, Springer-Verlag, Berlin Heidelberg, 2001.

[11]

V. Isakov, On uniqueness of recovery of a discontinuous conductivity coefficient, Comm. Pure Appl. Math., 41 (1988), 865-877.  doi: 10.1002/cpa.3160410702.

[12]

J. Jossinet, The impedivity of freshly excised human breast tissue, Physiological Measurement, 19 (1998), 61–75. doi: 10.1088/0967-3334/19/1/006.

[13]

A. Lorenzi and C. D. Pagani, On the stability of the surface separating two homogeneous media with different thermal conductivities, Acta Math. Sci., 7 (1987), 411-429.  doi: 10.1016/S0252-9602(18)30464-8.

[14]

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

[15]

S. Nagayasu, G. Uhlmann and J.-N. Wang, A depth-dependent stability estimate in electrical impedance tomography, Inverse Problems, 25 (2009), 075001. doi: 10.1088/0266-5611/25/7/075001.

[16]

V. P. Palamodov, Gabor analysis of the continuum model for impedance tomography, Ark. Mat., 40 (2002), 169-187.  doi: 10.1007/BF02384508.

[17]

J. Sylvester and G. Uhlmann, Inverse boundary value problems at the boundary: Continuous dependence, Comm. Pure Appl. Math., 41 (1988), 197-219.  doi: 10.1002/cpa.3160410205.

[18]

G. Uhlmann and J. -N. Wang, Reconstructing discontinuities using complex geometrical optics solutions, SIAM J. Appl. Math., 68 (2008), 1026-1044.  doi: 10.1137/060676350.

[19]

G. UhlmannJ. -N. Wang and C. -T. Wu, Reconstruction of inclusions in an elastic body, J. Math. Pures Appl., 91 (2009), 569-582.  doi: 10.1016/j.matpur.2009.01.006.

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