On a numerical quasiconformal mapping method for the medium parameters identification using applied quasipotential tomography

The problem of parameters identification of the bursts of the medium conductivity coefficient according to the tomography of the applied quasipotentials is considered. The method of image reconstruction is suggested, according to which the problem of analysis is reduced to the application of numerical methods of quasiconformal mappings, and the problem of synthesis is reduced to the solving the problem of parametric identification. The results of numerical experiments are presented and their analysis is carried out.

applied quasipotential tomography
quasiconformal mappings
ідентифікація
nonlinear problems
  1. Bomba A. Ya., Kroka L. L. Tchysloviy metod kvazikonformnogo vidobrazhennya rozv'yazannya zadach identifikatsii koyefitsienta elektrichnoi providnosti za danymy tomografii prykladenikh potentsialiv. Volyns'kiy matematychniy visnyk. Seriya prykladna matematika. 11 (20), 24–33 (2014), (in Ukrainian).
  2. Bomba A. Ya., Kroka L. L. Chislovi metody kompleksnogo analizu pry rozv'yazanni odnogo klasu neliniynikh eliptichnikh zadach za umov identifikatsii parametriv. Matematychne ta komp'yuterne modelyuvannya. Seriya: Fizyko-matematychni nauki: zb. nauk. prats'. 10, 24–33 (2014), (in Ukrainian).
  3. Bomba A. Ya., Boichura M. V. One numerical complex analysis method for parameters identification of piecewise homogeneous conductivity media with using applied quasipotential tomographic data. Matematychne ta komp'yuterne modelyuvannya. Seriya: Tekhnichni nauki: zb. nauk. prats'. 14, 5–17 (2016).
  4. Gorb M. S., Gusieva O. V. Vybir matematychnoi modeli ob'yektu doslidzhennya v elektroimpedansniy tomografi. Visnyk NTUU "KPI". Seriya  Radiotekhnika. Radioaparatobuduvannya. 52, 120–128 (2013),  (in Ukrainian).
  5. Sushko Н. O., Rybin O. Н. Vizualizatsiya rozpodilu poverkhnevykh providnostey metodom zon provіdnosti. Visnyk NTUU "KPI". Seriya Radiotekhnika. Radioaparatobuduvannya. 5, 7–17 (2014), (in Ukrainian).
  6. Gavrilov S. V. Iteratsionnyy metod resheniya trekhmernoy zadachi elektroimpedansnoy tomografii v sluchaye kusochno-postoyannoy provodimosti i neskol'kikh izmereniy na granitse. Vychislitel'nyye metody i programmirovaniye: novyye vychislitel'nyye tekhnologii. 14 (1), 26–30 (2013), (in Russian).
  7. Sherina Ye.S., Starchenko A. V. Chislennoye modelirovaniye zadachi elektroimpedansnoy tomografii i issledovaniye podkhoda na osnove metoda konechnykh obyemov. Byulleten' sibirskoy meditsiny. 4, 156–164 (2014), (in Russian).
  8. Denyer C. W. L. Electronics for Real-Time and Three-Dimensional Electrical Impedance Tomographs: PhD Thesis. Oxford (1996).
  9. Holder D. Electrical Impedance Tomography. Methods, History and Applications. London, Institute of Physics (2005).
  10. Hou T. C., Lynch J. P. Electrical Impedance Tomographic Methods for Sensing Strain Fields and Crack Damage in Cementitious Structures. Journal of Intelligent Material Systems and Structures. 20, 1363–1379 (2009).
  11. Haddar H., Kress R. Conformal Mapping and an Inverse Impedance Boundary Value Problem. Journal of Inverse and Ill-posed Problems. 14 (8), 785–804 (2006).
  12. Liu D. Kolehmainen V., Siltanen S. and others. Estimation of Conductivity Changes in a Region of Interest with Electrical Impedance. Inverse Problems and Imaging. 9 (1), 211–229 (2014).
  13. Sun T. Tsuda S., Zauner K. and others. On-chip electrical impedance tomography for imaging biological cells. Biosensors and Bioelectronics. 25 (5), 1109–1115 (2010).
  14. Bomba A. Ya., Kashtan S. S., Prigornits'kiy D. O., Yaroshchak S. V. Metody kompleksnogo analizu: monografiya. Rivne, NUVGP (2013), (in Ukrainian).
  15. Bomba A. Ya., Bulavats'kiy V. M., Skopets'kiy V. V. Neliniyni matematichni modeli protsesiv geogidrodinamiki. Kyiv, Naukova dumka (2007), (in Ukrainian).
  16. Lavrent'yev M. A., Shabat B. V. Metody teorii funktsii kompleksnogo peremennogo. Moscow, Nauka (1973), (in Russian).
  17. Somersalo E., Cheney M., Isaacson D. Existence and uniqueness for electrode models for electric current computed tomography. SIAM J. Appl. Math. 52 (4), 1023–1040 (1992).
  18. Ortega J. M., Rheinboldt W. C. Iterative Solution of Nonlinear Equations in Several Variables. Classics in Appl. Math. SIAM Publications, Philadelphia, 2nd edition (2000).
  19. e Oliveira H. A. Jr., Ingber L., Petraglia A. and others. Stochastic Global Optimization and Its Applications with Fuzzy Adaptive Simulated Annealing. Heidelberg, Springer-Verlag (2012).
  20. Samarskiy A. A. Teoriya raznostnykh skhem. Moscow, Nauka (1977), (in Russian).
  21. Sherina Ye. S., Starchenko A. V. Raznostnyye skhemy na osnove metoda konechnykh ob'yomov dlya zadachi elektroimpedansnoy tomografii. Vestnik tomskogo gosudarstvennogo universiteta. Matematika i mekhanika. 3 (29), 25–38 (2014), (in Russian).