Reconstruction of extended regions in EIT with a generalized Robin transmission condition

Govanni Granados; Isaac Harris; Heejin Lee

doi:10.3934/cac.2023017

Article Contents

2023, Volume 1, Issue 4: 347-368. Doi: 10.3934/cac.2023017

This issue Previous Article Efficient numerical method for shape optimization problem constrained by stochastic elliptic interface equation Next Article Operator-valued martingale transform on mixed martingale Hardy spaces

Reconstruction of extended regions in EIT with a generalized Robin transmission condition

1.
Department of Mathematics, Purdue University, West Lafayette, IN 47907, USA

^*Corresponding author: Isaac Harris
^*Corresponding author: Isaac Harris

Received: November 2022

Revised: November 2023

Early access: December 2023

Published: December 2023

The authors G. Granados, I. Harris and H. Lee are supported by NSF grant 2107891.

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Abstract

In this paper, we consider an inverse shape problem coming from electrical impedance tomography (EIT) with a generalized Robin transmission condition. We will derive an algorithm in order to detect whether two materials that should be in contact are separated or delaminated. More precisely, we assume that the undamaged material or background state is known and shares an interface or boundary with the damaged subregion. The Robin transmission condition on this boundary asymptotically models delamination. We assume that the Dirichlet-to-Neumann (DtN) operator is given from measuring the current on the surface of the material from an imposed voltage. We show that this mapping uniquely recovers the boundary parameters. Furthermore, using this electrostatic Cauchy data as physical measurements, we can determine if all of the coefficients from the Robin transmission condition are real-valued or complex-valued. We study these two cases separately and show that the regularized factorization method can be used to detect whether delamination has occurred and recover the damaged subregion. Numerical examples will be presented for both cases in two dimensions in the unit circle.

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Mathematics Subject Classification: Primary: 35J05, 35J25.

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Figure 1. Reconstruction of a circular region with $ \rho = 0.2 $ via the regularized factorization method. Boundary coefficients are $ \gamma = 2 - 0.5 \text{i} $ and $ \mu = 0.1 - \text{i} $. Contour plot of $ W(z) $ on the left and level curve when $ W(z) = 0.2 $ on the right

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Figure 2. Reconstruction of a circular region with $ \rho = 0.7 $ via the regularized factorization method. Boundary coefficients are $ \gamma = 2-3 \text{i} $ and $ \mu = 1 - 4 \text{i} $. Contour plot of $ W(z) $ on the left and level curve when $ W(z) = 0.2 $ on the right

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Figure 3. Reconstruction of a circular region with $ \rho = 0.25 $ via the regularized factorization method. Boundary coefficients are $ \gamma = 1.2 $ and $ \mu = 0.5 $. Contour plot of $ W(z) $ on the left and level curve when $ W(z) = 0.1 $ on the right

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Figure 4. Reconstruction of a circular region with $ \rho = 0.75 $ via the regularized factorization method. Boundary coefficients are $ \gamma = 0.6 $ and $ \mu = 1.6 $. Contour plot of $ W(z) $ on the left and level curve when $ W(z) = 0.07 $ on the right

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References

[1]	T. Arens, Why linear sampling method works, Inverse Problems, 20 (2004), 163-173. doi: 10.1088/0266-5611/20/1/010.
[2]	L. Audibert and H. Haddar, A generalized formulation of the linear sampling method with exact characterization of targets in terms of far field measurements, Inverse Problems, 30 (2014), 035011. doi: 10.1088/0266-5611/30/3/035011.
[3]	E. Blasten, X. Li, H. Liu and Y. Wang, On vanishing and localizing of transmission eigenfunctions near singular points: a numerical study, Inverse Problems, 33 (2017), 105001. doi: 10.1088/1361-6420/aa8826.
[4]	L. Borcea, Electrical impedance tomography, Inverse Problems, 18 (2002), R99-R136. doi: 10.1088/0266-5611/18/6/201.
[5]	L. Borcea, Addendum to: Electrical impedance tomography, Inverse Problems, 19 (2003), 997-998. doi: 10.1088/0266-5611/19/4/501.
[6]	F. Cakoni, I. de Teresa and P. Monk, Nondestructive testing of delaminated interfaces between two materials using electromagnetic interrogation, Inverse Problems, 34 (2018), 065005. doi: 10.1088/1361-6420/aabb1c.
[7]	F. Cakoni, Y. Hu and R. Kress, Simultaneous reconstruction of shape and generalized impedance functions in electrostatic imaging, Inverse Problems, 30 (2014), 105009. doi: 10.1088/0266-5611/30/10/105009.
[8]	F. Cakoni, H. Lee, P. Monk and Y. Zhang, A spectral target signature for thin surfaces with higher order jump conditions, Inverse Problems and Imaging, 16 (2022), 1473-1500. doi: 10.3934/ipi.2022020.
[9]	S. Chaabane, B. Charfi and H. Haddar, Reconstruction of discontinuous parameters in a second order impedance boundary operator, Inverse Problems, 32 (2016), 105004. doi: 10.1088/0266-5611/32/10/105004.
[10]	M. Chamaillard, N. Chaulet and H. Haddar, Analysis of the factorization method for a general class of boundary conditions, Journal of Inverse and Ill-posed Problems, 22 (2014), 643-670. doi: 10.1515/jip-2013-0013.
[11]	M. Cheney, D. Isaacson and J.-C. Newell, Electrical impedance tomography, SIAM Rev., 41 (1999), 85-101. doi: 10.1137/S0036144598333613.
[12]	Y. T. Chow, K. Ito, K. Liu and J. Zou, Direct sampling method for diffusive optical tomography, SIAM J. Sci. Comput., 37 (2015), A1658-A1684. doi: 10.1137/14097519X.
[13]	Y. T. Chow, K. Ito, K. Liu and J. Zou, Direct sampling method for electrical impedance tomography, Inverse Problems, 30 (2014), 095003. doi: 10.1088/0266-5611/30/9/095003.
[14]	D. Colton and A. Kirsch, A simple method for solving inverse scattering problems in the resonance region, Inverse Problems, 12 (1996), 383-393. doi: 10.1088/0266-5611/12/4/003.
[15]	Y. Deng and X. Liu, Electromagnetic imaging methods for nondestructive evaluation applications, Sensors, 11 (2011), 11774-11808. doi: 10.3390/s111211774.
[16]	H. Diao, H. Liu and L. Wang, Further results on generalized Holmgren's principle to the Lame operator and applications, J. Differential Equations, 309 (2022), 841-882. doi: 10.1016/j.jde.2021.11.039.
[17]	M. R. Embry, Factorization of operators on Banach space, Proc. Amer. Math. Soc., 38 (1973), 587-590. doi: 10.1090/S0002-9939-1973-0312287-8.
[18]	L. Evans, Partial Differential Equation, Second edition, Grad. Stud. Math., 19, American Mathematical Society, Providence, RI, 2010. doi: 10.1090/gsm/019.
[19]	A. Franchois and C. Pichot, Microwave imaging-complex permittivity reconstruction with a Levenberg-Marquardt method, IEEE Trans. Antennas Propag., 45 (1997), 203-215. doi: 10.1109/8.560338.
[20]	B. Gebauer and N. Hyvönen, Factorization method and irregular inclusions in electrical impedance tomography, Inverse Problems, 23 (2007), 2159-2170. doi: 10.1088/0266-5611/23/5/020.
[21]	G. Granados and I. Harris, Reconstruction of small and extended regions in EIT with a Robin transmission condition, Inverse Problems, 38 (2022), 105009. doi: 10.1088/1361-6420/ac8b2e.
[22]	M. Hanke and M. Brühl, Recent progress in electrical impedance tomography, Inverse Problems, 19 (2003), S65-S90. doi: 10.1088/0266-5611/19/6/055.
[23]	B. Harrach, Recent progress on the factorization method for electrical impedance tomography, Comput. Math. Methods Med., 2013 (2013), 425184. doi: 10.1155/2013/425184.
[24]	B. Harrach, Uniqueness, stability and global convergence for a discrete inverse elliptic Robin transmission problem, Numer. Math., 147 (2021), 29-70. doi: 10.1007/s00211-020-01162-8.
[25]	B. Harrach and H. Meftahi, Global uniqueness and Lipschitz-stability for the inverse Robin transmission problem, SIAM J. App. Math., 79 (2019), 525-550. doi: 10.1137/18M1205388.
[26]	I. Harris, Regularization of the Factorization Method applied to diffuse optical tomography, Inverse Problems, 37 (2021), 125010. doi: 10.1088/1361-6420/ac37f9.
[27]	I. Harris, Detecting inclusions with a generalized impedance condition from electrostatic data via sampling, Math. Methods Appl. Sci., 49 (2019), 6741-6756. doi: 10.1002/mma.5777.
[28]	I. Harris, Regularized factorization method for a perturbed positive compact operator applied to inverse scattering, Inverse Problems, 39 (2023), 115007. doi: 10.1088/1361-6420/acfd59.
[29]	H. Hedenmalm, On the uniqueness theorem of Holmgren, Math. Z., 281 (2015), 357-378. doi: 10.1007/s00209-015-1488-6.
[30]	S. Kharkovsky and R. Zoughi, Microwave and millimeter wave nondestructive testing and evaluation-overview and recent advances, IEEE Instrum. Meas. Mag., 10 (2007), 26-38. doi: 10.1109/MIM.2007.364985.
[31]	A. Kirsch A and N. Grinberg, The Factorization Method for Inverse Problems, Oxford Lecture Ser. Math. Appl., 36, Oxford University Press, Oxford, 2008.
[32]	A. Laurain and H. Meftahi, Shape and parameter reconstruction for the Robin transmission inverse problem, J. Inverse Ill-Posed Probl., 24 (2016), 643-662. doi: 10.1515/jiip-2015-0008.
[33]	X. Liu, S. Meng and B. Zhang, Modified sampling method with near field measurements, SIAM J. App. Math., 82 (2022), 244-266. doi: 10.1137/21M1432235.
[34]	W. McLean, Strongly Elliptic Systems and Boundary Integral Equations, Cambridge University Press, Cambridge, 2000.
[35]	J. Mueller and S. Siltanen, Linear and Nonlinear Inverse Problems with Practical Applications, Comput. Sci. Eng., 10, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2012. doi: 10.1137/1.9781611972344.
[36]	Y. Tamori, E. Suzuki and W. M. Deng, Epithelial tumors originate in tumor hotspots, a tissue-intrinsic microenvironment, PLoS Biol., 14 (2016), e1002537. doi: 10.1371/journal.pbio.1002537.