Solution of the Fredholm Equation of the First Kind by the Mesh Method with the Tikhonov Regularization

Abstract: We consider a linear ill-posed problem for the Fredholm equation of the first kind. For its regularization, Tikhonov’s stabilizer is implemented. To solve the problem, we use the mesh method, in which we replace integral operators by the simplest quadratures; and the differential ones, by the simplest finite differences. We investigate experimentally the influence of the regularization parameter and mesh thickening on the algorithm’s accuracy. The best performance is provided by the zeroth-order regularizer. We explain the reason of this result. We use the proposed algorithm for an applied problem of the recognition of two closely situated stars if the telescope instrument function is known. In addition, we show that the stars are clearly distinguished if the distance between them is ~0.2 of the instrumental function’s width and the values of brightness differ by 1–2 stellar magnitudes. © 2019, Pleiades Publishing, Ltd.

Authors
Belov A.A. 1, 2 , Kalitkin N.N.3
Publisher
Pleiades Publishing
Number of issue
2
Language
English
Pages
287-300
Status
Published
Volume
11
Year
2019
Organizations
  • 1 Department of Physics, Moscow State University, Moscow, 119991, Russian Federation
  • 2 Faculty of Physical, Mathematical and Natural Sciences, Peoples’ Friendship University of Russia (RUDN), Moscow, 117198, Russian Federation
  • 3 Keldysh Institute of Applied Mathematics, Russian Academy of Sciences, Moscow, 125047, Russian Federation
Keywords
ill-posed problems; mesh method; Tikhonov regularization
Date of creation
19.07.2019
Date of change
19.07.2019
Short link
https://repository.rudn.ru/en/records/article/record/38730/
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