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Partial inversion of the 2D attenuated $ X $-ray transform with data on an arc
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 |
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[
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. 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. |
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. 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. |
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