Applications of Mathematics, Vol. 63, No. 6, pp. 687-712, 2018


Application of Calderón's inverse problem in civil engineering

Jan Havelka, Jan Sýkora

Received November 20, 2017.   Published online December 3, 2018.

Abstract:  In specific fields of research such as preservation of historical buildings, medical imaging, geophysics and others, it is of particular interest to perform only a non-intrusive boundary measurements. The idea is to obtain comprehensive information about the material properties inside the considered domain while keeping the test sample intact. This paper is focused on such problems, i.e. synthesizing a physical model of interest with a boundary inverse value technique. The forward model is represented here by time dependent heat equation with transport parameters that are subsequently identified using a modified Calderón problem which is numerically solved by a regularized Gauss-Newton method. The proposed model setup is computationally verified for various domains, loading conditions and material distributions.
Keywords:  Calderón problem; finite element method; diffusion equation; boundary inverse value method; Neumann-to-Dirichlet map
Classification MSC:  65M32, 35K05


References:
[1] A. Allers, F. Santosa: Stability and resolution analysis of a linearized problem in electrical impedance tomography. Inverse Probl. 7 (1991), 515-533. DOI 10.1088/0266-5611/7/4/003 | MR 1122034 | Zbl 0736.35141
[2] V. F. Bakirov, R. A. Kline, W. P. Winfree: Discrete variable thermal tomography. AIP Conf. Proc. 700 (2004), 469-476. DOI 10.1063/1.1711659
[3] V. F. Bakirov, R. A. Kline, W. P. Winfree: Multiparameter thermal tomography. AIP Conf. Proc. 700 (2004), 461-468. DOI 10.1063/1.1711658
[4] K.-J. Bathe: Finite Element Procedures. Prentice Hall, Upper Saddle River (2006).
[5] C. A. Berenstein, E. Casadio Tarabusi: Inversion formulas for the $k$-dimensional Radon transform in real hyperbolic spaces. Duke Math. J. 62 (1991), 613-631. DOI 10.1215/S0012-7094-91-06227-7 | MR 1104811 | Zbl 0742.44002
[6] R. S. Blue: Real-time three-dimensional electrical impedance tomography. Ph.D. Dissertation, R.P.I, Troy (1997).
[7] R. S. Blue, D. Isaacson, J. C. Newell: Real-time three-dimensional electrical impedance imaging. Physiological Measurement 21 (2000), 15-26. DOI 10.1088/0967-3334/21/1/303
[8] A. Borsic, W. R. B. Lionheart, C. N. McLeod: Generation of anisotropic-smoothness regularization filters for EIT. IEEE Transactions on Medical Imaging 21 (2002), 579-587. DOI 10.1109/tmi.2002.800611
[9] R. M. Brown, G. A. Uhlmann: Uniqueness in the inverse conductivity problem for nonsmooth conductivities in two dimensions. Commun. Partial Differ. Equations 22 (1997), 1009-1027. DOI 10.1080/03605309708821292 | MR 1452176 | Zbl 0884.35167
[10] A. P. Calderón: On an inverse boundary value problem. Comput. Appl. Math. 25 (2006), 133-138. DOI 10.1590/S0101-82052006000200002 | MR 2321646 | Zbl 1182.35230
[11] S. Campana, S. Piro: Seeing the Unseen. Geophysics and Landscape Archaeology. CRC Press, London (2008). DOI 10.1201/9780203889558
[12] M. Cheney, D. Isaacson, J. C. Newell, S. Simske, J. Goble: NOSER: An algorithm for solving the inverse conductivity problem. Int. J. Imaging Systems and Technology 2 (1990), 66-75. DOI 10.1002/ima.1850020203
[13] K.-S. Cheng, D. Isaacson, J. C. Newell, D. G. Gisser: Electrode models for electric current computed tomography. IEEE Transactions on Biomedical Engineering 36 (1989), 918-924. DOI 10.1109/10.35300
[14] T. Dai, A. Adler: Electrical Impedance Tomography reconstruction using $l_1$ norms for data and image terms. Conf. Proc. IEEE Eng. Med. Biol. Soc. 2008 (2008), 2721-2724. DOI 10.1109/IEMBS.2008.4649764
[15] C. W. Groetsch: Inverse Problems in the Mathematical Sciences. Vieweg Mathematics for Scientists and Engineers, Vieweg, Braunschweig (1993). DOI 10.1007/978-3-322-99202-4 | MR 1247696 | Zbl 0779.45001
[16] S. J. Hamilton, M. Lassas, S. Siltanen: A direct reconstruction method for anisotropic electrical impedance tomography. Inverse Probl. 30 (2014), Article ID 075007, 33 pages. DOI 10.1088/0266-5611/30/7/075007 | MR 3233020 | Zbl 1298.65175
[17] D. S. Holder: Electrical Impedance Tomography: Methods, History and Applications. Series in Medical Physics and Biomedical Engineering, Taylor & Francis, Portland (2004).
[18] C.-H. Huang, S.-C. Chin: A two-dimensional inverse problem in imaging the thermal conductivity of a non-homogeneous medium. Int. J. Heat Mass Transfer 43 (2000), 4061-4071. DOI 10.1016/S0017-9310(00)00044-2 | Zbl 0973.80005
[19] M. R. Jones, A. Tezuka, Y. Yamada: Thermal tomographic detection of inhomogeneities. J. Heat Transfer 117 (1995), 969-975. DOI 10.1115/1.2836318
[20] A. Kirsch: An Introduction to the Mathematical Theory of Inverse Problems. Applied Mathematical Sciences 120, Springer, New York (2011). DOI 10.1007/978-1-4419-8474-6 | MR 3025302 | Zbl 1213.35004
[21] K. Knudsen, M. Lassas, J. L. Mueller, S. Siltanen: Regularized D-bar method for the inverse conductivity problem. Inverse Probl. Imaging 3 (2009), 599-624. DOI 10.3934/ipi.2009.3.599 | MR 2557921 | Zbl 1184.35314
[22] V. Kolehmainen, J. P. Kaipio, H. R. B. Orlande: Reconstruction of thermal conductivity and heat capacity using a tomographic approach. Int. J. Heat Mass Transfer 51 (2008), 1866-1876. DOI 10.1016/j.ijheatmasstransfer.2007.06.043 | Zbl 1140.80396
[23] A. Kučerová, J. Sýkora, B. Rosić, H. G. Matthies: Acceleration of uncertainty updating in the description of transport processes in heterogeneous materials. J. Comput. Appl. Math. 236 (2012), 4862-4872. DOI 10.1016/j.cam.2012.02.003 | MR 2946415 | Zbl 06078396
[24] O. A. Ladyženskaja, V. A. Solonnikov, N. N. Ural'ceva: Linear and Quasi-Linear Equations of Parabolic Type. Translations of Mathematical Monographs 23, American Mathematical Society, Providence (1968). DOI 10.1090/mmono/023 | MR 0241822 | Zbl 0174.15403
[25] C. Lanczos: Linear Differential Operators. Classics in Applied Mathematics 18, Society for Industrial and Applied Mathematics, Philadelphia (1996). DOI 10.1137/1.9781611971187 | MR 1393942 | Zbl 0865.34001
[26] R. E. Langer: An inverse problem in differential equations. Bull. Am. Math. Soc. 39 (1933), 814-820. DOI 10.1090/S0002-9904-1933-05752-X | MR 1562734 | Zbl 0008.04603
[27] Y. Mamatjan, A. Borsic, D. Gürsoy, A. Adler: Experimental/clinical evaluation of EIT image reconstruction with $l_1$ data and image norms. J. Phys., Conf. Ser. 434 (2013), 1-4. DOI 10.1088/1742-6596/434/1/012078
[28] J. L. Mueller, D. Isaacson, J. C. Newell: Reconstruction of conductivity changes due to ventilation and perfusion from EIT data collected on a rectangular electrode array. Physiological Measurement 22 (2001), 97-106. DOI 10.1088/0967-3334/22/1/313
[29] J. L. Mueller, S. Siltanen: Direct reconstructions of conductivities from boundary measurements. SIAM J. Sci. Comput. 24 (2003), 1232-1266. DOI 10.1137/S1064827501394568 | MR 1976215 | Zbl 1031.78008
[30] A. I. Nachman: Global uniqueness for a two-dimensional inverse boundary value problem. Ann. Math. (2) 143 (1996), 71-96. DOI 10.2307/2118653 | MR 1370758 | Zbl 0857.35135
[31] H. Niu, P. Guo, L. Ji, Q. Zhao, T. Jiang: Improving image quality of diffuse optical tomography with a projection-error-based adaptive regularization method. Optics Express 16 (2008), 12423-12434. DOI 10.1364/OE.16.012423
[32] K. Rektorys: Variational Methods in Mathematics, Science and Engineering. D. Reidel Publishing Company, Dordrecht (1980). MR 0596582 | Zbl 0481.49002
[33] F. Santosa, M. Vogelius: A backprojection algorithm for electrical impedance imaging. SIAM J. Appl. Math. 50 (1990), 216-243. DOI 10.1137/0150014 | MR 1036240 | Zbl 0691.65087
[34] S. Siltanen, J. Mueller, D. Isaacson: An implementation of the reconstruction algorithm of A. Nachman for the 2D inverse conductivity problem. Inverse Probl. 16 (2000), 681-699; erratum ibid. 17 (2001), 1561-1563. DOI 10.1088/0266-5611/16/3/310 | MR 1862207 | Zbl 0962.35193
[35] E. Somersalo, M. Cheney, D. Isaacson: Existence and uniqueness for electrode models for electric current computed tomography. SIAM J. Appl. Math. 52 (1992), 1023-1040. DOI 10.1137/0152060 | MR 1174044 | Zbl 0759.35055
[36] E. Somersalo, M. Cheney, D. Isaacson, E. Isaacson: Layer stripping: a direct numerical method for impedance imaging. Inverse Probl. 7 (1991), 899-926. DOI 10.1088/0266-5611/7/6/011 | MR 1140322 | Zbl 0753.35122
[37] J. Sýkora: Modeling of degradation processes in historical mortars. Adv. Eng. Softw. 70 (2014), 203-212. DOI 10.1016/j.advengsoft.2014.01.004
[38] J. Sýkora, T. Krejčí, J. Kruis, M. Šejnoha: Computational homogenization of non-stationary transport processes in masonry structures. J. Comput. Appl. Math. 236 (2012), 4745-4755. DOI 10.1016/j.cam.2012.02.031 | Zbl 1259.80007
[39] J. Sylvester, G. Uhlmann: A global uniqueness theorem for an inverse boundary value problem. Ann. Math. (2) 125 (1987), 153-169. DOI 10.2307/1971291 | MR 0873380 | Zbl 0625.35078
[40] J. Syren: Theoretical and numerical analysis of the Dirichlet-to-Neumann map in EIT. Master Thesis, University of Helsinki (2016).
[41] J. M. Toivanen, T. Tarvainen, J. M. J. Huttunen, T. Savolainen, H. R. B. Orlande, J. P. Kaipio, V. Kolehmainen: 3D thermal tomography with experimental measurement data. Int. J. Heat Mass Transfer 78 (2014), 1126-1134. DOI 10.1016/j.ijheatmasstransfer.2014.07.080
[42] M. Vauhkonen: Electrical impedance tomography and prior information. Ph.D. Dissertation, Kuopio University, Joensuu (2007).
[43] M. Vauhkonen, W. R. B. Lionheart, L. M. Heikkinen, P. J. Vauhkonen, J. P. Kaipio: A MATLAB package for the EIDORS project to reconstruct two-dimensional EIT images. Physiological Measurement 22 (2001), 107-111. DOI 10.1088/0967-3334/22/1/314

Affiliations:   Jan Havelka, Jan Sýkora, Czech Technical University in Prague, Faculty of Civil Engineering, Deparment of Mechanics, Thákurova 7, 166 29 Praha 6, Czech Republic, e-mail: jan.havelka.1@fsv.cvut.cz, jan.sykora.1@fsv.cvut.cz


 
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