Application of Calderón's inverse problem in civil engineering

Jan Havelka; Jan Sýkora

Applications of Mathematics (2018)

  • Volume: 63, Issue: 6, page 687-712
  • ISSN: 0862-7940

Abstract

top
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.

How to cite

top

Havelka, Jan, and Sýkora, Jan. "Application of Calderón's inverse problem in civil engineering." Applications of Mathematics 63.6 (2018): 687-712. <http://eudml.org/doc/294161>.

@article{Havelka2018,
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.},
author = {Havelka, Jan, Sýkora, Jan},
journal = {Applications of Mathematics},
keywords = {Calderón problem; finite element method; diffusion equation; boundary inverse value method; Neumann-to-Dirichlet map},
language = {eng},
number = {6},
pages = {687-712},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Application of Calderón's inverse problem in civil engineering},
url = {http://eudml.org/doc/294161},
volume = {63},
year = {2018},
}

TY - JOUR
AU - Havelka, Jan
AU - Sýkora, Jan
TI - Application of Calderón's inverse problem in civil engineering
JO - Applications of Mathematics
PY - 2018
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 63
IS - 6
SP - 687
EP - 712
AB - 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.
LA - eng
KW - Calderón problem; finite element method; diffusion equation; boundary inverse value method; Neumann-to-Dirichlet map
UR - http://eudml.org/doc/294161
ER -

References

top
  1. Allers, A., Santosa, F., 10.1088/0266-5611/7/4/003, Inverse Probl. 7 (1991), 515-533. (1991) Zbl0736.35141MR1122034DOI10.1088/0266-5611/7/4/003
  2. Bakirov, V. F., Kline, R. A., Winfree, W. P., 10.1063/1.1711659, AIP Conf. Proc. 700 (2004), 469-476. (2004) DOI10.1063/1.1711659
  3. Bakirov, V. F., Kline, R. A., Winfree, W. P., 10.1063/1.1711658, AIP Conf. Proc. 700 (2004), 461-468. (2004) DOI10.1063/1.1711658
  4. Bathe, K.-J., Finite Element Procedures, Prentice Hall, Upper Saddle River (2006). (2006) 
  5. Berenstein, C. A., Tarabusi, E. Casadio, 10.1215/S0012-7094-91-06227-7, Duke Math. J. 62 (1991), 613-631. (1991) Zbl0742.44002MR1104811DOI10.1215/S0012-7094-91-06227-7
  6. Blue, R. S., Real-time three-dimensional electrical impedance tomography, Ph.D. Dissertation, R.P.I, Troy (1997). (1997) 
  7. Blue, R. S., Isaacson, D., Newell, J. C., 10.1088/0967-3334/21/1/303, Physiological Measurement 21 (2000), 15-26. (2000) DOI10.1088/0967-3334/21/1/303
  8. Borsic, A., Lionheart, W. R. B., McLeod, C. N., 10.1109/tmi.2002.800611, IEEE Transactions on Medical Imaging 21 (2002), 579-587. (2002) DOI10.1109/tmi.2002.800611
  9. Brown, R. M., Uhlmann, G. A., 10.1080/03605309708821292, Commun. Partial Differ. Equations 22 (1997), 1009-1027. (1997) Zbl0884.35167MR1452176DOI10.1080/03605309708821292
  10. Calderón, A. P., 10.1590/S0101-82052006000200002, Comput. Appl. Math. 25 (2006), 133-138. (2006) Zbl1182.35230MR2321646DOI10.1590/S0101-82052006000200002
  11. Campana, S., Piro, S., 10.1201/9780203889558, CRC Press, London (2008). (2008) DOI10.1201/9780203889558
  12. Cheney, M., Isaacson, D., Newell, J. C., Simske, S., Goble, J., 10.1002/ima.1850020203, Int. J. Imaging Systems and Technology 2 (1990), 66-75. (1990) DOI10.1002/ima.1850020203
  13. Cheng, K.-S., Isaacson, D., Newell, J. C., Gisser, D. G., 10.1109/10.35300, IEEE Transactions on Biomedical Engineering 36 (1989), 918-924. (1989) MR1080512DOI10.1109/10.35300
  14. Dai, T., Adler, A., 10.1109/IEMBS.2008.4649764, Conf. Proc. IEEE Eng. Med. Biol. Soc. 2008 (2008), 2721-2724. (2008) DOI10.1109/IEMBS.2008.4649764
  15. Groetsch, C. W., 10.1007/978-3-322-99202-4, Vieweg Mathematics for Scientists and Engineers, Vieweg, Braunschweig (1993). (1993) Zbl0779.45001MR1247696DOI10.1007/978-3-322-99202-4
  16. Hamilton, S. J., Lassas, M., Siltanen, S., 10.1088/0266-5611/30/7/075007, Inverse Probl. 30 (2014), Article ID 075007, 33 pages. (2014) Zbl1298.65175MR3233020DOI10.1088/0266-5611/30/7/075007
  17. Holder, D. S., Electrical Impedance Tomography: Methods, History and Applications, Series in Medical Physics and Biomedical Engineering, Taylor & Francis, Portland (2004). (2004) 
  18. Huang, C.-H., Chin, S.-C., 10.1016/S0017-9310(00)00044-2, Int. J. Heat Mass Transfer 43 (2000), 4061-4071. (2000) Zbl0973.80005DOI10.1016/S0017-9310(00)00044-2
  19. Jones, M. R., Tezuka, A., Yamada, Y., 10.1115/1.2836318, J. Heat Transfer 117 (1995), 969-975. (1995) DOI10.1115/1.2836318
  20. Kirsch, A., 10.1007/978-1-4419-8474-6, Applied Mathematical Sciences 120, Springer, New York (2011). (2011) Zbl1213.35004MR3025302DOI10.1007/978-1-4419-8474-6
  21. Knudsen, K., Lassas, M., Mueller, J. L., Siltanen, S., 10.3934/ipi.2009.3.599, Inverse Probl. Imaging 3 (2009), 599-624. (2009) Zbl1184.35314MR2557921DOI10.3934/ipi.2009.3.599
  22. Kolehmainen, V., Kaipio, J. P., Orlande, H. R. B., 10.1016/j.ijheatmasstransfer.2007.06.043, Int. J. Heat Mass Transfer 51 (2008), 1866-1876. (2008) Zbl1140.80396DOI10.1016/j.ijheatmasstransfer.2007.06.043
  23. Kučerová, A., Sýkora, J., Rosić, B., Matthies, H. G., 10.1016/j.cam.2012.02.003, J. Comput. Appl. Math. 236 (2012), 4862-4872. (2012) Zbl06078396MR2946415DOI10.1016/j.cam.2012.02.003
  24. Ladyženskaja, O. A., Solonnikov, V. A., Ural'ceva, N. N., 10.1090/mmono/023, Translations of Mathematical Monographs 23, American Mathematical Society, Providence (1968). (1968) Zbl0174.15403MR0241822DOI10.1090/mmono/023
  25. Lanczos, C., 10.1137/1.9781611971187, Classics in Applied Mathematics 18, Society for Industrial and Applied Mathematics, Philadelphia (1996). (1996) Zbl0865.34001MR1393942DOI10.1137/1.9781611971187
  26. Langer, R. E., 10.1090/S0002-9904-1933-05752-X, Bull. Am. Math. Soc. 39 (1933), 814-820. (1933) Zbl0008.04603MR1562734DOI10.1090/S0002-9904-1933-05752-X
  27. Mamatjan, Y., Borsic, A., Gürsoy, D., Adler, A., 10.1088/1742-6596/434/1/012078, J. Phys., Conf. Ser. 434 (2013), 1-4. (2013) DOI10.1088/1742-6596/434/1/012078
  28. Mueller, J. L., Isaacson, D., Newell, J. C., 10.1088/0967-3334/22/1/313, Physiological Measurement 22 (2001), 97-106. (2001) DOI10.1088/0967-3334/22/1/313
  29. Mueller, J. L., Siltanen, S., 10.1137/S1064827501394568, SIAM J. Sci. Comput. 24 (2003), 1232-1266. (2003) Zbl1031.78008MR1976215DOI10.1137/S1064827501394568
  30. Nachman, A. I., 10.2307/2118653, Ann. Math. (2) 143 (1996), 71-96. (1996) Zbl0857.35135MR1370758DOI10.2307/2118653
  31. Niu, H., Guo, P., Ji, L., Zhao, Q., Jiang, T., 10.1364/OE.16.012423, Optics Express 16 (2008), 12423-12434. (2008) DOI10.1364/OE.16.012423
  32. Rektorys, K., Variational Methods in Mathematics, Science and Engineering, D. Reidel Publishing Company, Dordrecht (1980). (1980) Zbl0481.49002MR0596582
  33. Santosa, F., Vogelius, M., 10.1137/0150014, SIAM J. Appl. Math. 50 (1990), 216-243. (1990) Zbl0691.65087MR1036240DOI10.1137/0150014
  34. Siltanen, S., Mueller, J., Isaacson, D., 10.1088/0266-5611/16/3/310, Inverse Probl. 16 (2000), 681-699 erratum ibid. 17 2001 1561-1563. (2000) Zbl0962.35193MR1862207DOI10.1088/0266-5611/16/3/310
  35. Somersalo, E., Cheney, M., Isaacson, D., 10.1137/0152060, SIAM J. Appl. Math. 52 (1992), 1023-1040. (1992) Zbl0759.35055MR1174044DOI10.1137/0152060
  36. Somersalo, E., Cheney, M., Isaacson, D., Isaacson, E., 10.1088/0266-5611/7/6/011, Inverse Probl. 7 (1991), 899-926. (1991) Zbl0753.35122MR1140322DOI10.1088/0266-5611/7/6/011
  37. Sýkora, J., 10.1016/j.advengsoft.2014.01.004, Adv. Eng. Softw. 70 (2014), 203-212. (2014) DOI10.1016/j.advengsoft.2014.01.004
  38. Sýkora, J., Krejčí, T., Kruis, J., Šejnoha, M., 10.1016/j.cam.2012.02.031, J. Comput. Appl. Math. 236 (2012), 4745-4755. (2012) Zbl1259.80007DOI10.1016/j.cam.2012.02.031
  39. Sylvester, J., Uhlmann, G., 10.2307/1971291, Ann. Math. (2) 125 (1987), 153-169. (1987) Zbl0625.35078MR0873380DOI10.2307/1971291
  40. Syren, J., Theoretical and numerical analysis of the Dirichlet-to-Neumann map in EIT, Master Thesis, University of Helsinki (2016). (2016) 
  41. Toivanen, J. M., Tarvainen, T., Huttunen, J. M. J., Savolainen, T., Orlande, H. R. B., Kaipio, J. P., Kolehmainen, V., 10.1016/j.ijheatmasstransfer.2014.07.080, Int. J. Heat Mass Transfer 78 (2014), 1126-1134. (2014) DOI10.1016/j.ijheatmasstransfer.2014.07.080
  42. Vauhkonen, M., Electrical impedance tomography and prior information, Ph.D. Dissertation, Kuopio University, Joensuu (2007). (2007) 
  43. Vauhkonen, M., Lionheart, W. R. B., Heikkinen, L. M., Vauhkonen, P. J., Kaipio, J. P., 10.1088/0967-3334/22/1/314, Physiological Measurement 22 (2001), 107-111. (2001) DOI10.1088/0967-3334/22/1/314

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.