RAS MathematicsЖурнал вычислительной математики и математической физики Computational Mathematics and Mathematical Physics

  • ISSN (Print) 0044-4669
  • ISSN (Online) 3034-533

A NUMERICAL METHOD FOR SOLVING THE MICROWAVE TOMOGRAPHY PROBLEM OF RESTORING INHOMOGENETTES IN A CYLINDRICAL BODY

PII
S3034533S0044466925100108-1
DOI
10.7868/S303453325100108
Publication type
Article
Status
Published
Authors
Volume/ Edition
Volume 65 / Issue number 10
Pages
1746-1758
Abstract
In this paper, a vector three-dimensional inverse diffraction problem on a cylindrical body is solved based on a two-step method. The diffuser is filled with an inhomogeneous nonmagnetic dielectric material. The initial boundary value problem for the Maxwell system of equations is reduced to a system of integro-differential equations. A numerical method for solving a first-order equation in special classes of functions is described. A distinctive feature of the proposed numerical method is its non-iteration, in addition, a two-step method for solving the inverse problem does not require a good initial approximation. The calculation results are presented. It is shown that the two-step method is an effective approach to solving vector problems of microwave tomography.
Keywords
трехмерная векторная обратная задача дифракции восстановление диэлектрической проницаемости интегродифференциальное уравнение двухшаговый метод численный метод
Date of publication
09.12.2025
Year of publication
2025
Number of purchasers
0
Views
5

References

  1. 1. Романов В.Г. Обратные задачи математической физики. Монография. М.: АН СССР, 1984.
  2. 2. Meaney P.M., Goodwin D., Golnabi A.H., Zhou T., Pallone M., Geimer S.D., Burke G., Paulsen K.D. Clinical microwave tomographic imaging of the calcaneus: a first-in-human case study of two subjects // IEEE Trans Biomed Eng. 2012. V. 59. № 12. P. 3304–13.
  3. 3. Kamal A.M., Sakorikar T., Pal U.M., Pandya H.J. Engineering Approaches for Breast Cancer Diagnosis: A Review // IEEE Rev. Biomed. Eng. 2023. V. 16. P. 687–705.
  4. 4. Cherepenin V., Karpov A., Korjenevsky A., Kornienko V., Mazaletskaya A., Mazourov D., Meister D. A 3D electrical impedance tomography (EIT) system for breast cancer detection // Physiol Meas. 2001. V. 22. № 1. P. 9–18.
  5. 5. Tranchida D., Diaz J., Schön P., Schönherr H., Vancso G.J. Scanning Near-Field EllipsometryMicroscopy: imaging nanomaterials with resolution below the diffraction limit // Nanoscale. 2011. V. 3. P. 233–239.
  6. 6. Fleischer M. Near-field scanning optical microscopy nanoprobes // Nanotechnology Reviews. 2012. V. 1. № 4. P. 313–338.
  7. 7. Лаврентьев М.М., Романов В.Г., Шашлинский С.П. Некорректные задачи математической физики и анализа. М.: Наука, 1980.
  8. 8. Бакушинский А.Б., Гончарский А.В. Итеративные методы решения некорректных задач. М.: Наука, 1989.
  9. 9. Тихонов А.Н., Гончарский А.В., Степанов В.В., Ягола А.Г. Численные методы решения некорректных задач. М.: Наука, 1990.
  10. 10. Кабанихин С.И. Обратные и некорректные задачи. Учебник для студентов высших учебных заведений. Новосибирск: Сибирское научное издательство, 2009.
  11. 11. Medvedik M.Y., Smirnov Y.G., Tsupak A.A. Two-step method for solving inverse problem of diffraction by an inhomogeneous body // Springer Proceedings in Mathematics and Statistics 38th. In «Nonlinear and Inverse Problems in Electromagnetics — PIERS 2017». 2018. P. 83–92.
  12. 12. Evstigneev R.O., Medvedik M.Y. Reconstruction of Inhomogeneity Parameters by Measurements of Near Field Outside the Body // Beilina, L., Smirnov, Y. (eds) Nonlinear and Inverse Problems in Electromagnetics. PIERS PIERS 2017. Springer Proceedings in Mathematics & Statistics. 2018. V. 243. P. 29–38.
  13. 13. Medvedik M.Y., Smirnov Y.G., Tsupak A.A. The two-step method for determining a piecewise-continuous refractive index of a 2D scatterer by near field measurements // Inverse Problems in Science and Engineering. 2020. V. 28. № 3. P. 427–447.
  14. 14. Medvedik M.Y., Smirnov Y.G., Tsupak A.A. Non-iterative two-step method for solving scalar inverse 3D diffraction problem // Inverse Problems in Science and Engineering. 2020. V. 28. № 1. P. 1474–1492.
  15. 15. Smirnov Y.G., Tsupak A.A. Direct and inverse scalar scattering problems for the Helmholtz equation in \( \mathbb{R}^m \) // J. Inverse Ill-Posed Probl 2022. V. 30. № 1. P. 101–116.
  16. 16. Evstigneev R.O., Medvedik M.Y. Reconstruction of Inhomogeneities in a Hemisphere from the Field Measurements // Lobachevskii Journal of Mathematics. 2019. V. 40. № 10. P. 1653–1659.
  17. 17. Lapich A.O., Medvedik M.Y. Microwave Tomography Method for Solving the Inverse Problem on Cylindrical Bodies // Tech. Phys. Lett. 2024. V. 50. P. 360–364.
  18. 18. Lapich A.O., Medvedik M.Y. Method of generalized and combined computational grids for restoration of the parameters of inhomogeneities of a body based on the results of measurements of the electromagnetic field // Mathematical Models and Computer Simulations. 2024. V. 16. № 6. P. 806–813.
  19. 19. Lapich A.O., Medvedik M.Yu. Method for reconstruction the parameters of body inhomogeneities from the results of electromagnetic field measurements // Lobachevskii Journal of Mathematics. 2024. V. 45. № 10. P. 4740–4747.
  20. 20. Medvedik M.Y., Smirnov Y.G., Tsupak A.A. Inverse vector problem of diffraction by inhomogeneous body with a piecewise smooth permittivity // J. Inverse Ill-Posed Probl. 2024. V. 32. № 3. P. 453–465.
  21. 21. Colton D., Kress R. Inverse Acoustic and Electromagnetic Scattering Theory. Berlin, Heidelberg: Springer-Verlag, 2013.
  22. 22. Smolkin E., Smirnov Y., Snegur M. Solution of the Vector Three-Dimensional Inverse Problem on an Inhomogeneous Dielectric Hemisphere Using a Two-Step Method // Computation. 2024. V. 12. № 11. P. 213.
  23. 23. Samokhin A.B., Shestopalov Yu.V. Integral Equations and Iteration Methods in Electromagnetic Scattering. Berlin, Boston: De Gruyter, 2001.
  24. 24. Smirnov Yu.G., Tsupak A.A. Diffraction of Acoustic and Electromagnetic Waves by Screens and Inhomogeneous Solids: Mathematical Theory. RU-SCIENCE, Moscow, 2018.
  25. 25. Smirnov Yu.G., Tsupak A.A., Valovik D.V. On the Volume Singular Integro-differential Equation for the Electromagnetic Diffraction Problem // Applicable Analysis: An International Journal. 2017. V. 96. № 2. P. 173–189.
  26. 26. Smirnov Yu.G., Tsupak A.A. Existence and Uniqueness Theorems in Electromagnetic Diffraction on Systems of Lossless Dielectrics and Perfectly Conducting Screens // Applicable Analysis: An International Journal 2017. V. 96. № 8. P. 1326–1341.
  27. 27. Tихонов А.Н., Арсенин В.Я. Методы решения некорректных задач. 2-е изд. М.: Наука, 1979. 288 с.
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