THE INVERSE PROBLEM OF BODY’S HETEROGENEITY RECOVERY FOR EARLY DIAGNOSTICS OF DISEASES
USING MICROWAVE TOMOGRAPHY 

Evstigneev Roman Olegovich, Postgraduate student, Penza State University (40 Krasnaya street, Penza, Russia), Roman_cezar@mail.ru
Medvedik Mikhail Yur'evich, Candidate of physical and mathematical sciences, associate professor, sub-department of mathematics and supercomputer modeling, Penza State University (40 Krasnaya street, Penza, Russia), mmm@pnzgu.ru
Smirnov Yuriy Gennad'evich, Doctor of physical and mathematical sciences, professor, head of sub-department of mathematics and supercomputer modeling, Penza State University (40 Krasnaya street, Penza, Russia), mmm@pnzgu.ru
Tsupak Aleksey Aleksandrovich, Candidate of physical and mathematical sciences, associate professor, sub-department of mathematics and supercomputer modeling, Penza State University (40 Krasnaya street, Penza, Russia), mmm@pnzgu.ru

517.968, 517.983.37

10.21685/2072-3040-2017-4-1

Background. The aim of this work is to theoretical and numerical study the inverse scalar problem of diffraction by a volume obstacle characterized by a piecewise Hoelder-continuous function.
Material and methods. The original boundary value problem is considered in the quasiclassical formulation and then reduced to a system of weakly singular integral equations; the properties of the latter system are studied using the potential theory and Fourier transform.
Results. The inverse problem of diffraction is given the integral formulation; the theorem on uniqueness of a piecewise constant solution to the integral equation of the first type is proved; a new two-step algorythm for numerical solving the inverse problem is proposed and implemented; several numerical tests have been carried out.
Conclusions. The obtained theoretical and numerical results confirm high efficiency of the proposed method, which can be applied for solving problems of nearfield tomography.

inverse diffraction problem, reconstruction of refractive index, integral equatons, uniqueness of solutions, integral equations, collocation method

1. Brown B. M., Marlett M., Reyes J. M. J. Differential Equations. 2016, vol. 260, pp. 525–654.
2. Bakushinsky A. B., Kokurin M. Yu. Iterative Methods for Approximate Solution of Inverse Problems. New York: Springer, 2004.
3. Beilina L., Klibanov M. Approximate Global Convergence and Adaptivity for Coefficient Inverse Problems. New York: Springer, 2012.
4. Kabanikhin S. I., Satybaev A. D., Shishlenin M. A. Direct Methods of Solving Multidimensional Inverse Hyperbolic Problems. Utrecht, VSP, 2004.
5. Romanov V. G. Inverse Problems of Mathematical Physics. Utrecht, The Netherlands: VNU, 1986.
6. Colton D., Kress R. Inverse Acoustic and Eleectromagnetic Scattering Theory. Berlin, Heidelberg: Springer-Verlag, 1992.
7. Samokhin A. B. Integral'nye Uravneniya i Iteratsionnye Metody v Elektromagnitnom Rasseyanii [Integral equations and iterative methods in electromagnetic scattering]. Moscow: Radio i svyaz', 1998.
8. Kobayashi K., Shestopalov Yu., Smirnov Yu. SIAM J. Appl. Math. 2009, vol. 70, no. 3, pp. 969–983.
9. Costabel M., Darrigrand E., Kone E. H. Journal of Computational and Applied Mathematics. 2010, vol. 234, no. 6, pp. 1817–1825.
10. Smirnov Yu. G., Tsupak A. A., Valovik D. V. Applicable Analysis. DOI: 10.1080/00036811.2015.1115839.
11. Tsupak A. A. Izvestiya vysshikh uchebnykh zavedeniy. Povolzhskiy region. Fizikomatematicheskie nauki [University proceedings. Volga region. Physical and mathematical sciences]. 2014, no. 1 (29), pp. 30–38.
12. Medvedik M. Yu., Smirnov Yu. G., Tsupak A. A. Zhurnal vychislitel'noy matematiki i matematicheskoy fiziki [The journal of calculus mathematics and mathematical physics]. 2014, vol. 54, no. 8, pp. 1319–1331.
13. Vladimirov V. S. Uravneniya matematicheskoy fiziki [Equations of mathematical physics]. Moscow: Nauka, 1971. 509 p.
14. Kolton D., Kress R. Metody integral'nykh uravneniy v teorii rasseyaniya [Methods of integral equations in the scattering theory]. Moscow: Mir, 1987. 312 p.
15. Natterer F. Matematicheskie aspekty komp'yuternoy tomografii [Mathematical aspects of computer tomography]. Moscow: Mir, 1990. 288 p.