Algorithm for Searching Inhomogeneities in Inverse Nonlinear Diffraction Problems
https://doi.org/10.26907/2541-7746.2024.3.395-406
Abstract
This study aims to solve the inverse problem for determining the heterogeneity of an object. The scattered field was measured outside its boundaries at a set of observation points. Both the radiation source and observation points were assumed to be located outside the object. The scattered field was modeled by solving the direct problem. The inverse problem was solved using a two-step method. Nonlinearities of various types were considered. When introducing the computational grid, the generalized grid method was applied. A numerical method for solving the problem was proposed and implemented. The numerical results obtained illustrate how the problem is solved for specified experimental data.
Keywords
two-step method,
integral equation,
nonlinear diffraction problem,
collocation method,
numerical method
Supporting agencies: This study was supported by the Ministry of Science and Higher Education of the Russian Federation as part of state assignment no. 124020200015-7
About the Authors
A. O. Lapich
Penza State University
Russian Federation
Russian Federation
Penza, 440026
M. Y. Medvedik
Penza State University
Russian Federation
Russian Federation
Penza, 440026
References
1. Korjenevsky A.V. Electrical impedance tomography system suitable for manufacturing in non-standard conditions. Zh. Radioelektron., 2021, no. 9. https://doi.org/10.30898/1684-1719.2021.9.5. (In Russian)
2. Zarafshani A., Bach T., Chatwin C.R., Tang S., Xiang L., Zheng B. Conditioning electrical impedance mammography system. Measurement, 2018, vol. 116, pp. 38–48. https://doi.org/10.1016/j.measurement.2017.10.052.
3. Vladimirov V.S., Zharinov V.V. Uravneniya matematicheskoi fiziki [Equations of Mathematical Physics]. Moscow, Fizmatlit, 2004. 400 p. (In Russian)
4. Sommerfeld A. Die Greensche Funktion der Schwingungsgleichung. Jahresber. Dtsch. Math.-Ver., 1912, Bd. 21, S. 309–353. (In German)
5. Smirnov Yu.G., Tsupak A.A. Matematicheskaya teoriya difraktsii akusticheskikh i elektromagnitnykh voln na sisteme ekranov i neodnorodnykh tel [Mathematical Theory of Diffraction of Acoustic and Electromagnetic Waves by Screens and Inhomogeneous Solids]. Moscow, Ru-Science, 2018. 223 p. (In Russian)
6. Lapich A.O., Medvedik M.Y. Solution of a scalar two-dimensional nonlinear diffraction problem for objects of arbitrary shape. Uchenye Zapiski Kazanskogo Universiteta. Seriya Fiziko-Matematicheskie Nauki, 2023, vol. 165, no. 2, pp. 167–177. https://doi.org/10.26907/2541-7746.2023.2.167-177. (In Russian)
7. Lapich A.O., Medvedik M.Y. Method for restoring the parameters of body inhomogeneities from the results of electromagnetic field measurements. Modeli, Sist., Seti Ekon., Tekh., Prir. O-vo., 2023, no. 4, pp. 142–153. https://doi.org/10.21685/2227-8486-2023-4-9. (In Russian)
8. Lapich A.O., Medvedik M.Y. An iterative scheme for solving a Lippmann–Schwinger nonlinear integral equation by the Galerkin method. Izv. Vyssh. Uchebn. Zaved. Povolzh. Reg. Fiz.-Mat. Nauki, 2023, no. 3, pp.66–73. https://doi.org/10.21685/2072-3040-2023-3-5. (In Russian)
9. Medvedik M.Yu., Smirnov Yu.G., Tsupak A.A. The two-step method for determining a piecewise-continuous refractive index of a 2D scatterer by near field measurements. Inverse Probl. Sci. Eng., 2020, vol. 28, no. 3, pp. 427–447. https://doi.org/10.1080/17415977.2019.1597872.
10. Medvedik M.Yu., Smirnov Yu.G., Tsupak A.A. Non-iterative two-step method for solving scalar inverse 3D diffraction problem. Inverse Probl. Sci. Eng., 2020, vol. 28, no. 10, pp. 1474–1492. https://doi.org/10.1080/17415977.2020.1727466.
11. Medvedik M.Yu. A subhierarchic method for solving the Lippmann–Schwinger integral equation on bodies of complex shapes. J. Commun. Technol. Electron., 2012, vol. 57, no. 2, pp. 158–163. https://doi.org/10.1134/S1064226912010123.
12. Medvedik M.Y. Solution of integral equations by means of subhierarchic method for generalized computational grids. Math. Models Comput. Simul., 2015, vol. 7, no. 6, pp. 570–580. https://doi.org/10.1134/S207004821506006X.
Review
For citations:
Lapich A.O., Medvedik M.Y. Algorithm for Searching Inhomogeneities in Inverse Nonlinear Diffraction Problems. Uchenye Zapiski Kazanskogo Universiteta. Seriya Fiziko-Matematicheskie Nauki. 2024;166(3):395-406. (In Russ.) https://doi.org/10.26907/2541-7746.2024.3.395-406
Views:
236
Email this article 