RECONSTRUCTION OF A FUNCTION FROM PARABOLIC INTEGRALS WITH RANDOM EXPONENTIAL ATTENUATION
- Authors
-
-
Uteuliev Niyetbay Uteuliyevich
Doctor of Physical and Mathematical Sciences, Professor, Nukus State Technical University -
Bektemirov Ilkhom Tleubergenovich
Assistant Lecturer, Nukus State Pedagogical Institute -
Seidullaev Abat Kamalovich
PhD in Physical and Mathematical Sciences, Associate Professor, Karakalpak State University, V. I. Romanovskiy Institute of Mathematics of the Academy of Sciences of the Republic of Uzbekistan
-
- Keywords:
- integral geometry parabolic transform random attenuation resolvent Tikhonov regularization
- Abstract
- This paper considers the reconstruction of a deterministic function in a strip from independent repeated measurements on a family of parabolas. The exponential attenuation coefficient follows a known two-point distribution, and its realizations are unobserved. The averaged problem is reduced to a Volterra equation of the second kind; a representation of the solution in terms of the resolvent is obtained, and its uniqueness is proved. The exact covariance structure of the averaging error and the mean-square convergence of Tikhonov regularization are established. An estimate of the systematic error arising when the random coefficient is replaced by its mathematical expectation is derived. The theoretical relations are verified using a synthetic discrete model.
- References
-
[1] R. Gorenflo and S. Vessella, Abel Integral Equations: Analysis and Applications, vol. 1461. in Lecture Notes in Mathematics, vol. 1461. Berlin, Germany: Springer, 1991. doi: 10.1007/BFb0084665.
[2] R. G. Novikov, ‘An inversion formula for the attenuated X-ray transformation’, Arkiv för Matematik, vol. 40, no. 1, pp. 145–167, 2002, doi: 10.1007/BF02384507.
[3] N. Bissantz, T. Hohage, A. Munk, and F. Ruymgaart, ‘Convergence rates of general regularization methods for statistical inverse problems and applications’, SIAM Journal on Numerical Analysis, vol. 45, no. 6, pp. 2610–2636, 2007, doi: 10.1137/060651884.
[4] P. C. Hansen, Discrete Inverse Problems: Insight and Algorithms, vol. 7. in Fundamentals of Algorithms, vol. 7. Philadelphia, PA, USA: SIAM, 2010. doi: 10.1137/1.9780898718836.
[5] N. Uteuliev, G. Djaykov, and A. Seidullaev, ‘Inversion formula for the problem of integral geometry on families of parabolas’, in AIP Conference Proceedings, 2021, p. 070004. doi: 10.1063/5.0056852.
[6] H. Chihara, ‘Inversion of seismic-type Radon transforms on the plane’, Integral Transforms and Special Functions, vol. 31, no. 12, pp. 998–1009, 2020, doi: 10.1080/10652469.2020.1775202.
[7] S. Moon, ‘Inversion of the seismic parabolic Radon transform and the seismic hyperbolic Radon transform’, Inverse Problems in Science and Engineering, vol. 24, no. 2, pp. 317–327, 2016, doi: 10.1080/17415977.2015.1025071.
[8] R. Kress, Linear Integral Equations, 3rd edn, vol. 82. in Applied Mathematical Sciences, vol. 82. New York, NY, USA: Springer, 2014. doi: 10.1007/978-1-4614-9593-2.
[9] A. Abhishek, ‘Minimax optimal estimator in a stochastic inverse problem for exponential Radon transform’, Sankhya A, vol. 85, pp. 980–998, 2023, doi: 10.1007/s13171-022-00285-4.
[10] L. Cavalier, ‘Nonparametric statistical inverse problems’, Inverse Problems, vol. 24, no. 3, p. 034004, 2008, doi: 10.1088/0266-5611/24/3/034004.
[11] A. Kh. Begmatov, A. K. Seidullaev, and A. O. Pirimbetov, ‘Problems of integral geometry in a strip on families of parabolic curves’, Proceedings of the Russian Higher School Academy of Sciences, no. 2(19), pp. 6–15, 2012.
[12] H. W. Engl, M. Hanke, and A. Neubauer, Regularization of Inverse Problems, vol. 375. in Mathematics and Its Applications, vol. 375. Dordrecht, The Netherlands: Kluwer, 1996.
[13] G. M. Djaykov and A. D. Bekbosinov, ‘Solution of parabolic Radon transform with given weight function’, Ilim hám jámiyet (Science and Society), no. 3, pp. 10–12, 2023.
[14] A. N. Tikhonov and V. Y. Arsenin, Solutions of Ill-Posed Problems. Washington, DC, USA: V. H. Winston & Sons, 1977.
[15] J. Kaipio and E. Somersalo, Statistical and Computational Inverse Problems, vol. 160. in Applied Mathematical Sciences, vol. 160. New York, NY, USA: Springer, 2005. doi: 10.1007/b138659.
[16] I. Kharin, ‘Stochastic aspect of the tomographic reconstruction problems in a transport model’. 2010. doi: 10.48550/arXiv.1003.2591.
[17] F. Natterer, The Mathematics of Computerized Tomography, vol. 32. in Classics in Applied Mathematics, vol. 32. Philadelphia, PA, USA: SIAM, 2001. doi: 10.1137/1.9780898719284.