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Hybrid topological derivative-gradient based methods for nondestructive testing. (English) Zbl 1470.78009

Summary: This paper is devoted to the reconstruction of objects buried in a medium and their material properties by hybrid topological derivative-gradient based methods. After illustrating the techniques in time-harmonic acoustic problems with different boundary conditions and in electrical impedance tomography problems with continuous Neumann conditions, we extend the hybrid method for a realistic model in tomography where the boundary conditions are given at a discrete set of electrodes.

MSC:

78M99 Basic methods for problems in optics and electromagnetic theory
74J25 Inverse problems for waves in solid mechanics
65N21 Numerical methods for inverse problems for boundary value problems involving PDEs
74P15 Topological methods for optimization problems in solid mechanics
78A46 Inverse problems (including inverse scattering) in optics and electromagnetic theory
78A55 Technical applications of optics and electromagnetic theory
35R30 Inverse problems for PDEs
35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation
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[1] Liseno, A.; Pierre, R., Imaging of voids by means of a physical-optics-based shape-reconstrution algorithm, Journal of the Optical Society of America A, 21, 6, 968-974, (2004)
[2] Devaney, A. J., Geophysical diffraction tomography, IEEE Transactions on Geoscience and Remote Sensing, 22, 1, 3-13, (1984)
[3] Colton, D.; Giebermann, K.; Monk, P., A regularized sampling method for solving three-dimensional inverse scattering problems, SIAM Journal on Scientific Computing, 21, 6, 2316-2330, (2000) · Zbl 0961.35172
[4] Colton, D.; Kirsch, A., A simple method for solving inverse scattering problems in the resonance region, Inverse Problems, 12, 4, 383-393, (1996) · Zbl 0859.35133
[5] Natterer, F.; Wübbeling, F., A propagation-backpropagation method for ultrasound tomography, Inverse Problems, 11, 6, 1225-1232, (1995) · Zbl 0839.35146
[6] Kleinman, R. E.; van den Berg, P. M., A modified gradient method for two-dimensional problems in tomography, Journal of Computational and Applied Mathematics, 42, 1, 17-35, (1992) · Zbl 0757.65133
[7] Hettlich, F., Fréchet derivatives in inverse obstacle scattering, Inverse Problems, 11, 2, 371-382, (1995) · Zbl 0821.35147
[8] Kirsch, A., The domain derivative and two applications in inverse scattering theory, Inverse Problems, 9, 1, 81-96, (1993) · Zbl 0773.35085
[9] Masmoudi, M., Outils pour la conception optimale des formes [Thése d’Etat en Sciences Mathématiques], (1987), Université de Nice
[10] Potthast, R., Fréchet differentiability of the solution to the acoustic Neumann scattering problem with respect to the domain, Journal of Inverse and Ill-Posed Problems, 4, 1, 67-84, (1996) · Zbl 0858.35139
[11] Litman, A.; Lesselier, D.; Santosa, F., Reconstruction of a two-dimensional binary obstacle by controlled evolution of a level-set, Inverse Problems, 14, 3, 685-706, (1998) · Zbl 0912.35158
[12] Santosa, F., A level-set approach for inverse problems involving obstacles, Optimisation et Calcul des Variations, 1, 17-33, (1996) · Zbl 0870.49016
[13] Feijoo, G. R., A new method in inverse scattering based on the topological derivative, Inverse Problems, 20, 6, 1819-1840, (2004) · Zbl 1077.78010
[14] Bonnet, M.; Guzina, B. B., Sounding of finite solid bodies by way of topological derivative, International Journal for Numerical Methods in Engineering, 61, 13, 2344-2373, (2004) · Zbl 1075.74564
[15] Guzina, B. B.; Bonnet, M., Small-inclusion asymptotic of misfit functionals for inverse problems in acoustics, Inverse Problems, 22, 5, 1761-1785, (2006) · Zbl 1105.76055
[16] Samet, B.; Amstutz, S.; Masmoudi, M., The topological asymptotic for the Helmholtz equation, SIAM Journal on Control and Optimization, 42, 5, 1523-1544, (2003) · Zbl 1051.49029
[17] Bonnet, M.; Constantinescu, A., Inverse problems in elasticity, Inverse Problems, 21, 2, R1-R50, (2005) · Zbl 1070.35118
[18] Dorn, O.; Lesselier, D., Level set methods for inverse scattering, Inverse Problems, 22, 4, R67-R131, (2006) · Zbl 1191.35272
[19] Burger, M.; Hackl, B.; Ring, W., Incorporating topological derivatives into level set methods, Journal of Computational Physics, 194, 1, 344-362, (2004) · Zbl 1044.65053
[20] Garreau, S.; Guillaume, P.; Masmoudi, M., The topological asymptotic for PDE systems: the elasticity case, SIAM Journal on Control and Optimization, 39, 6, 1756-1778, (2001) · Zbl 0990.49028
[21] Carpio, A.; Rapún, M.-L., Solving inhomogeneous inverse problems by topological derivative methods, Inverse Problems, 24, 4, (2008) · Zbl 1153.35401
[22] Carpio, A.; Rapún, M.-L., An iterative method for parameter identification and shape reconstruction, Inverse Problems in Science and Engineering, 18, 1, 35-50, (2010) · Zbl 1182.65166
[23] Carpio, A.; Rapún, M.-L., Hybrid topological derivative and gradient-based methods for electrical impedance tomography, Inverse Problems, 28, 9, (2012) · Zbl 1253.35215
[24] Ammari, H.; Garnier, J.; Jugnon, V.; Kang, H., Direct reconstruction methods in ultrasound imaging of small anomalies, Mathematical Modelling in Biomedical Imaging II. Mathematical Modelling in Biomedical Imaging II, Lecture Notes in Mathematics, 2035, (2012), Berlin, Germany: Springer, Berlin, Germany · Zbl 1345.92085
[25] Ammari, H.; Garnier, J.; Jugnon, V.; Kang, H., Stability and resolution analysis for a topological derivative based imaging functional, SIAM Journal on Control and Optimization, 50, 1, 48-76, (2012) · Zbl 1238.35180
[26] Borcea, L.; Papanicolaou, G.; Tsogka, C.; Berryman, J., Imaging and time reversal in random media, Inverse Problems, 18, 5, 1247-1279, (2002) · Zbl 1047.74032
[27] Carpio, A.; Rapún, M.-L., Domain reconstruction using photothermal techniques, Journal of Computational Physics, 227, 17, 8083-8106, (2008) · Zbl 1147.65072
[28] Borcea, L., Electrical impedance tomography, Inverse Problems, 18, 6, R99-R136, (2002) · Zbl 1031.35147
[29] Cheney, M.; Isaacson, D.; Newell, J. C., Electrical impedance tomography, SIAM Review, 41, 1, 85-101, (1999) · Zbl 0927.35130
[30] Brühl, M.; Hanke, M., Numerical implementation of two noniterative methods for locating inclusions by impedance tomography, Inverse Problems, 16, 4, 1029-1042, (2000) · Zbl 0955.35076
[31] Chung, E. T.; Chan, T. F.; Tai, X.-C., Electrical impedance tomography using level set representation and total variational regularization, Journal of Computational Physics, 205, 1, 357-372, (2005) · Zbl 1072.65143
[32] Hintermüller, M.; Laurain, A., Electrical impedance tomography: from topology to shape, Control and Cybernetics, 37, 4, 913-933, (2008) · Zbl 1194.49062
[33] Hintermüller, M.; Laurain, A.; Novotny, A. A., Second-order topological expansion for electrical impedance tomography, Advances in Computational Mathematics, 36, 2, 235-265 , (2012) · Zbl 1243.49049
[34] Hofmann, B., Approximation of the inverse electrical impedance tomography problem by an inverse transmission problem, Inverse Problems, 14, 5, 1171-1187, (1998) · Zbl 0992.78020
[35] Ito, K.; Kunisch, K.; Li, Z., Level-set function approach to an inverse interface problem, Inverse Problems, 17, 5, 1225-1242, (2001) · Zbl 0986.35130
[36] Hanke, M.; Brühl, M., Recent progress in electrical impedance tomography, Inverse Problems, 19, 6, S65-S90, (2003) · Zbl 1048.92022
[37] Borcea, L., A nonlinear multigrid for imaging electrical conductivity and permittivity at low frequency, Inverse Problems, 17, 2, 329-359, (2001) · Zbl 0981.65126
[38] Dines, K. A.; Lytle, R. J., Analysis of electrical conductivity imaging, Geophysics, 46, 7, 1025-1036, (1981)
[39] Yorkey, T. J.; Webster, J. G.; Tompkins, W. J., Comparing reconstruction algorithms for electrical impedance tomography, IEEE Transactions on Biomedical Engineering, 34, 11, 843-852, (1987)
[40] Guzina, B. B.; Chikichev, I., From imaging to material identification: a generalized concept of topological sensitivity, Journal of the Mechanics and Physics of Solids, 55, 2, 245-279, (2007) · Zbl 1419.74149
[41] Abramovitz, M.; Stegun, I. A., Handbook of Mathematical Functions, (1972), New York, NY, USA: Dover, New York, NY, USA
[42] Sokołowski, J.; Żochowski, A., On the topological derivative in shape optimization, SIAM Journal on Control and Optimization, 37, 4, 1251-1272, (1999) · Zbl 0940.49026
[43] Carpio, A.; Rapún, M. L., Topological derivatives for shape reconstruction, Inverse Problems and Imaging. Inverse Problems and Imaging, Lecture Notes in Mathematics, 1943, 85-133, (2008), Berlin, Germany: Springer, Berlin, Germany · Zbl 1144.65307
[44] Feijóo, G. R.; Oberai, A. A.; Pinsky, P. M., An application of shape optimization in the solution of inverse acoustic scattering problems, Inverse Problems, 20, 1, 199-228, (2004) · Zbl 1055.35138
[45] Rapún, M.-L.; Sayas, F.-J., Indirect methods with Brakhage-Werner potentials for Helmholtz transmission problems, Numerical Mathematics and Advanced Applications: Proceedings of ENUMATH 2005, the 6th European Conference on Numerical Mathematics and Advanced Applications Santiago de Compostela, Spain, July 2005, 1146-1154, (2006), Berlin, Germany: Springer, Berlin, Germany · Zbl 1119.65414
[46] Rapún, M.-L.; Sayas, F.-J., A mixed-FEM and BEM coupling for the approximation of the scattering of thermal waves in locally non-homogeneous media, ESAIM: Mathematical Modelling and Numerical Analysis, 40, 5, 871-896, (2006) · Zbl 1123.65115
[47] Carpio, A.; Johansson, B. T.; Rapún, M.-L., Determining planar multiple sound-soft obstacles from scattered acoustic fields, Journal of Mathematical Imaging and Vision, 36, 2, 185-199, (2010)
[48] Colton, D.; Sleeman, B. D., Uniqueness theorems for the inverse problem of acoustic scattering, IMA Journal of Applied Mathematics, 31, 3, 253-259, (1983) · Zbl 0539.76086
[49] Johansson, T.; Sleeman, B. D., Reconstruction of an acoustically sound-soft obstacle from one incident field and the far-field pattern, IMA Journal of Applied Mathematics, 72, 1, 96-112, (2007) · Zbl 1121.76059
[50] Cheney, M.; Isaacson, D.; Somersalo, E., Existence and uniqueness for electrode models for electric current computed tomography, SIAM Journal on Applied Mathematics, 52, 4, 1023-1040, (1992) · Zbl 0759.35055
[51] Lechleiter, A.; Rieder, A., Newton regularizations for impedance tomography: a numerical study, Inverse Problems, 22, 6, 1967-1987, (2006) · Zbl 1109.65100
[52] Dobson, D. C., Convergence of a reconstruction method for the inverse conductivity problem, SIAM Journal on Applied Mathematics, 52, 2, 442-458, (1992) · Zbl 0747.35051
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