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doi: 10.3934/ipi.2021048
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## 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 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 & Imaging, 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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [10] D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, 2$^{nd}$ edition, Springer-Verlag, Berlin Heidelberg, 2001.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [16] V. P. Palamodov, Gabor analysis of the continuum model for impedance tomography, Ark. Mat., 40 (2002), 169-187.  doi: 10.1007/BF02384508.  Google Scholar [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.  Google Scholar [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.  Google Scholar [19] G. Uhlmann, J. -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.  Google Scholar

show all references

##### References:
 [1] G. Alessandrini, Stable determination of conductivity by boundary measurements, Appl. Anal., 27 (1988), 153-172.  doi: 10.1080/00036818808839730.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [10] D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, 2$^{nd}$ edition, Springer-Verlag, Berlin Heidelberg, 2001.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [16] V. P. Palamodov, Gabor analysis of the continuum model for impedance tomography, Ark. Mat., 40 (2002), 169-187.  doi: 10.1007/BF02384508.  Google Scholar [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.  Google Scholar [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.  Google Scholar [19] G. Uhlmann, J. -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.  Google Scholar
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