Advertisement

Applications of Mathematics

, Volume 63, Issue 6, pp 687–712 | Cite as

Application of Calderón’s inverse problem in civil engineering

  • Jan Havelka
  • Jan Sýkora
Article
  • 18 Downloads

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 

MSC 2010

65M32 35K05 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    A. Allers, F. Santosa: Stability and resolution analysis of a linearized problem in electrical impedance tomography. Inverse Probl. 7 (1991), 515–533.MathSciNetCrossRefGoogle Scholar
  2. [2]
    V. F. Bakirov, R. A. Kline, W. P. Winfree: Discrete variable thermal tomography. AIP Conf. Proc. 700 (2004), 469–476.CrossRefGoogle Scholar
  3. [3]
    V. F. Bakirov, R. A. Kline, W. P. Winfree: Multiparameter thermal tomography. AIP Conf. Proc. 700 (2004), 461–468.CrossRefGoogle Scholar
  4. [4]
    K.–J. Bathe: Finite Element Procedures. Prentice Hall, Upper Saddle River, 2006.zbMATHGoogle Scholar
  5. [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.MathSciNetCrossRefGoogle Scholar
  6. [6]
    R. S. Blue: Real–time three–dimensional electrical impedance tomography. Ph. D. Dissertation, R. P. I, Troy, 1997.Google Scholar
  7. [7]
    R. S. Blue, D. Isaacson, J. C. Newell: Real–time three–dimensional electrical impedance imaging. Physiological Measurement 21 (2000), 15–26.CrossRefGoogle Scholar
  8. [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.CrossRefGoogle Scholar
  9. [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.MathSciNetCrossRefGoogle Scholar
  10. [10]
    A. P. Calderón: On an inverse boundary value problem. Comput. Appl. Math. 25 (2006), 133–138.MathSciNetCrossRefGoogle Scholar
  11. [11]
    S. Campana, S. Piro: Seeing the Unseen. Geophysics and Landscape Archaeology. CRC Press, London, 2008.Google Scholar
  12. [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.CrossRefGoogle Scholar
  13. [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.CrossRefGoogle Scholar
  14. [14]
    T. Dai, A. Adler: Electrical Impedance Tomography reconstruction using l1 norms for data and image terms. Conf. Proc. IEEE Eng. Med. Biol. Soc. 2008 (2008), 2721–2724.Google Scholar
  15. [15]
    C. W. Groetsch: Inverse Problems in the Mathematical Sciences. Vieweg Mathematics for Scientists and Engineers, Vieweg, Braunschweig, 1993.CrossRefGoogle Scholar
  16. [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.Google Scholar
  17. [17]
    D. S. Holder: Electrical Impedance Tomography: Methods, History and Applications. Series in Medical Physics and Biomedical Engineering, Taylor & Francis, Portland, 2004.CrossRefGoogle Scholar
  18. [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.CrossRefGoogle Scholar
  19. [19]
    M. R. Jones, A. Tezuka, Y. Yamada: Thermal tomographic detection of inhomogeneities. J. Heat Transfer 117 (1995), 969–975.CrossRefGoogle Scholar
  20. [20]
    A. Kirsch: An Introduction to the Mathematical Theory of Inverse Problems. Applied Mathematical Sciences 120, Springer, New York, 2011.CrossRefGoogle Scholar
  21. [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.MathSciNetCrossRefGoogle Scholar
  22. [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.CrossRefGoogle Scholar
  23. [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.MathSciNetCrossRefGoogle Scholar
  24. [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.CrossRefGoogle Scholar
  25. [25]
    C. Lanczos: Linear Differential Operators. Classics in Applied Mathematics 18, Society for Industrial and Applied Mathematics, Philadelphia, 1996.CrossRefGoogle Scholar
  26. [26]
    R. E. Langer: An inverse problem in differential equations. Bull. Am. Math. Soc. 39 (1933), 814–820.MathSciNetCrossRefGoogle Scholar
  27. [27]
    Y. Mamatjan, A. Borsic, D. Gürsoy, A. Adler: Experimental/clinical evaluation of EIT image reconstruction with l1 data and image norms. J. Phys., Conf. Ser. 434 (2013), 1–4.CrossRefGoogle Scholar
  28. [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.CrossRefGoogle Scholar
  29. [29]
    J. L. Mueller, S. Siltanen: Direct reconstructions of conductivities from boundary measurements. SIAM J. Sci. Comput. 24 (2003), 1232–1266.MathSciNetCrossRefGoogle Scholar
  30. [30]
    A. I. Nachman: Global uniqueness for a two–dimensional inverse boundary value problem. Ann. Math. (2) 143 (1996), 71–96.MathSciNetCrossRefGoogle Scholar
  31. [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.CrossRefGoogle Scholar
  32. [32]
    K. Rektorys: Variational Methods in Mathematics, Science and Engineering. D. Reidel Publishing Company, Dordrecht, 1980.zbMATHGoogle Scholar
  33. [33]
    F. Santosa, M. Vogelius: A backprojection algorithm for electrical impedance imaging. SIAM J. Appl. Math. 50 (1990), 216–243.MathSciNetCrossRefGoogle Scholar
  34. [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.MathSciNetCrossRefGoogle Scholar
  35. [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.MathSciNetCrossRefGoogle Scholar
  36. [36]
    E. Somersalo, M. Cheney, D. Isaacson, E. Isaacson: Layer stripping: a direct numerical method for impedance imaging. Inverse Probl. 7 (1991), 899–926.MathSciNetCrossRefGoogle Scholar
  37. [37]
    J. Sýkora: Modeling of degradation processes in historical mortars. Adv. Eng. Softw. 70 (2014), 203–212.CrossRefGoogle Scholar
  38. [38]
    J. Sýkora, T. Krejčí, J. Kruis, M. Šejnoha: Computational homogenization of nonstationary transport processes in masonry structures. J. Comput. Appl. Math. 236 (2012), 4745–4755.CrossRefGoogle Scholar
  39. [39]
    J. Sylvester, G. Uhlmann: A global uniqueness theorem for an inverse boundary value problem. Ann. Math. (2) 125 (1987), 153–169.MathSciNetCrossRefGoogle Scholar
  40. [40]
    J. Syren: Theoretical and numerical analysis of the Dirichlet–to–Neumann map in EIT. Master Thesis, University of Helsinki, 2016.Google Scholar
  41. [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.CrossRefGoogle Scholar
  42. [42]
    M. Vauhkonen: Electrical impedance tomography and prior information. Ph. D. Dissertation, Kuopio University, Joensuu, 2007.Google Scholar
  43. [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.CrossRefGoogle Scholar

Copyright information

© Institute of Mathematics of the Academy of Sciences of the Czech Republic, Praha, Czech Republic 2018

Authors and Affiliations

  1. 1.Faculty of Civil Engineering, Deparment of MechanicsCzech Technical University in PraguePraha 6Czech Republic

Personalised recommendations