×

A level set method to reconstruct the discontinuity of the conductivity in EIT. (English) Zbl 1181.35274

Summary: In this paper, one level set method is applied to finding the interface of discontinuity of the conductivity in EIT (electrical impedance tomography) problem. By choosing one suitable velocity function, a level set reconstruction algorithm is proposed. The theoretical results for EIT problem and regularization are given. Finally the numerical examples demonstrate that the reconstruction algorithm is efficient and stable.

MSC:

35Q60 PDEs in connection with optics and electromagnetic theory
35J25 Boundary value problems for second-order elliptic equations
35A24 Methods of ordinary differential equations applied to PDEs
78A25 Electromagnetic theory (general)
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Gilbarg D, Trudinger N S. Elliptic Partial Differential Equations of Second Order. New York: Springer-Verlag, 1983 · Zbl 0562.35001
[2] Adams R A. Sobolev Spaces Pure and Applied Mathematics, 65. New York: Academic Press, 1975
[3] Cheney M, Issacson D, Newell J C. Electrical impedance tomogrpahy. SIAM Rev, 41: 85–101 (1999) · Zbl 0927.35130 · doi:10.1137/S0036144598333613
[4] Isaacson D, Cheney M. Current problems in impedance imaging. In: Colton D, Ewing R, Rundell W, eds. Inverse Problems in Partial Differential Equantisons. Philadelphia: SIAM, 1990, 141–149 · Zbl 0712.35108
[5] Parker R L. The inverse problem of resistivity sounding. Geophysics, 142: 2143–2158 (1984) · doi:10.1190/1.1441630
[6] Ramirez A, Daily W, Binley B, et al. Detection of leaks in underground storage tanks using electrical resistance methods. J Environ Eng Geophys, 1: 189–203 (1996) · doi:10.4133/JEEG1.3.189
[7] Ramirez A, Daily W, LaBreque D, et al. Monitoring an underground steam injection process using electrical resistance tomography. Water Resources Res, 29: 73–87 (1993) · doi:10.1029/92WR01608
[8] Calerón A P. On an inverse boundary problem. In: Seminar on Numerical Analysis and Its Applications to Continumm Physics (Soc Bransileria de Matèmatica, Rio de Janeiro), 1980, 65–73
[9] Woo H, Kim S, Seo J K, et al. A direct tracking method for a grounded conductor inside a pipeline from capacitance measurements. Inverse Problems, 22: 481–494 (2006) · Zbl 1094.35146 · doi:10.1088/0266-5611/22/2/006
[10] Brühl M, Hank M. Numerical implementation of two noniterative methods for locating inclusions by impedance tomography. Inverse Problems, 16: 1029–1042 (2000) · Zbl 0955.35076 · doi:10.1088/0266-5611/16/4/310
[11] Brühl M, Hank M. Recent progress in electrical impedance tomography. Inverse Problems, 19: 65–90 (2003)
[12] Brühl M, Hank M, Pidcock M. Crack detection using electrostatic measurements. Math Model Numer Anal, 35: 595–605 (2001) · Zbl 0985.35103 · doi:10.1051/m2an:2001128
[13] Muller J L, Sitanen S. Direct reconstruction of conductivities from boundary measurements. SIAM J Sci Comput, 24: 1232–1266 (2003) · Zbl 1031.78008 · doi:10.1137/S1064827501394568
[14] Siltanen S, Mueller J, Isaacson D. An implementation of the reconstruction algorithm of A. Nachman for 2D inverse conductivity problem. Inverse Problems, 16: 681–699 (2000) · Zbl 0962.35193 · doi:10.1088/0266-5611/16/3/310
[15] Dobson D C. Convergence of a reconstruction method for the inverse conductivity problem. SIAM J Appl Math, 8: 71–81 (1992) · Zbl 0747.35050
[16] Kohn R V, Vogelius M. Relaxation of a variational method for impedance computed tomography. Commun Pure Appl Math, XL: 745–777 (1987) · Zbl 0659.49009 · doi:10.1002/cpa.3160400605
[17] Kohn R V, Vogelius M. Determining conductivity by boundary measurements. Commun Pure Appl Math, 37: 113–123 (1984) · Zbl 0586.35089 · doi:10.1002/cpa.3160370302
[18] Liu J J, Cheng J, Nakamura G. Reconstruction and uniqueness of an inverse scattering problem with impedance boundary. Sci China Ser A-Math, 45(11): 1408–1419 (2002) · Zbl 1103.35368 · doi:10.1007/BF02880035
[19] Tong C L, Cheng J, Yamamoto M. Reconstruction of convection coefficients of an elliptic equation in the plane by Dirichlet to Neumann map. Sci China Ser A-Math, 48(1): 40–56 (2005) · Zbl 1126.35376 · doi:10.1360/03YS0171
[20] Feng L X, Ma F M. Uniqueness and local stability for the inverse scattering problem of determining the cavity. Sci China Ser A-Math, 48(8): 1113–1123 (2005) · Zbl 1120.78302 · doi:10.1360/022004-18
[21] Dong H P, Ma F M. Reconstruction of the shape of object with near field measurements in a half-plane. Sci China Ser A-Math, 51(6): 1059–1070 (2008) · Zbl 1154.35090 · doi:10.1007/s11425-008-0034-y
[22] Osher S, Sethian J A. Fronts propagating with curvature-dependent speed: algorithms based on Hamilton-Jacobi formulations. J Comput Phys, 79: 12–49 (1988) · Zbl 0659.65132 · doi:10.1016/0021-9991(88)90002-2
[23] Osher S, Fedkiw R. Level set methods and dynamic implicit surfaces. In: Applied Mathematical Sciences, Vol 153. New York: Springer, 2003 · Zbl 1026.76001
[24] Santosa F. A level set approach for inverse problems involving obstacles, ESAIM. Control Optim Calculus Variations, 1: 17–33 (1996) · Zbl 0870.49016 · doi:10.1051/cocv:1996101
[25] Dorn O, Lesselier D. Lever set method for inverse scattering. Inverse Problems, 22: 67–131 (2006) · Zbl 1191.35272 · doi:10.1088/0266-5611/22/4/R01
[26] Chan T F, Tai X C. Level set and total variation regularization for elliptic inverse problems with discontinuous coefficients. J Comput Phys, 193: 40–66 (2003) · Zbl 1036.65086 · doi:10.1016/j.jcp.2003.08.003
[27] Chan T F, Tai X C. Identification of discontinuous coefficients in elliptic problems using total variation regularisation. SIAM J Sci Comput, 25: 881–904 (2003) · Zbl 1046.65090 · doi:10.1137/S1064827599326020
[28] Mumford D, Shah J. Optimal approximation by piecewise smooth functions and associated variational problems. Comm Pure Appl Math, 42: 577–685 (1989) · Zbl 0691.49036 · doi:10.1002/cpa.3160420503
[29] Sokolowski J, Zolésio J P. Introduction to shape optimization: shape sensitivity analysis. In: Springer Series in Computational Mathematics, Vol 16. Berlin: Springer, 1992
[30] Bukhgeim A L, Cheng J, Yamamoto M. Stability for an inverse boundary problem of determining a part of a boundary. Inverse Problems, 15(4): 1021–1032 (1999) · Zbl 0934.35202 · doi:10.1088/0266-5611/15/4/312
[31] Bukhgeim A L, Cheng J, Yamamoto M. Conditional stability in an inverse problem of determining a nonsmooth boundary. J Math Anal Appl, 242(1): 57–74 (2000) · Zbl 0951.35132 · doi:10.1006/jmaa.1999.6654
[32] Cheng J, Hon Y C, Yamamoto M. Conditional stability estimation for an inverse boundary problem with non-smooth boundary in \(\mathbb{R}\)3. Trans Amer Math Soc, 353(10): 4123–4138 (2001) · Zbl 0970.35163 · doi:10.1090/S0002-9947-01-02758-1
[33] Chen Z M, Zou J. An augmented Lagrangian method for identifying discontinuous parameters in elliptic systems. SIAM J Control Optim, 37(3): 892–910 (1999) · Zbl 0940.65117 · doi:10.1137/S0363012997318602
[34] Ito K, Kunisch K. The augmented Lagrangian method for parameter estimation in elliptic systems. SIAM J Control Optim, 28(1): 113–136 (1990) · Zbl 0709.93021 · doi:10.1137/0328006
[35] Cheng J, Yamamoto M. One new strategy for a priori choice of regularizing parameters in Tikhonov’s regularization. Inverse Problems, 16(4): L31–L38 (2000) · Zbl 0957.65052 · doi:10.1088/0266-5611/16/4/101
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.