Geodesic
Solution of the Fredholm equation of the first kind by mesh method with Tikhonov regularization
Matematičeskoe modelirovanie, Tome 30 (2018) no. 8, pp. 67-88

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We consider linear ill-posed problem for the Fredholm equation of the first kind. For its regularization, the stabilizer of A.N. Tikhonov is implied. To solve the problem, we use the mesh method in which we replace integral operators by the simplest quadratures and differential ones by the simplest finite differences. We investigate experimentally the influence of the regularization parameter and mesh thickening on the algorithm accuracy. The best performance is provided by the zeroth order regularizer. We explain the reason of this result. We imply the proposed algorithm for an applied problem of recognition of two closely situated stars if the telescope instrument function is known. Also, we show that the stars are clearly distinguished if the distance between them is $\sim$ 0.2 of the instrumental function width and brightness differs by 1–2 stellar magnitude.
Keywords: ill-posed problems, Tikhonov regularization, mesh method. 
@article{MM_2018_30_8_a4,
     author = {A. A. Belov and N. N. Kalitkin},
     title = {Solution of the {Fredholm} equation of the first kind by mesh method with {Tikhonov} regularization},
     journal = {Matemati\v{c}eskoe modelirovanie},
     pages = {67--88},
     publisher = {mathdoc},
     volume = {30},
     number = {8},
     year = {2018},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MM_2018_30_8_a4/}
}
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A. A. Belov; N. N. Kalitkin. Solution of the Fredholm equation of the first kind by mesh method with Tikhonov regularization. Matematičeskoe modelirovanie, Tome 30 (2018) no. 8, pp. 67-88. http://geodesic.mathdoc.fr/item/MM_2018_30_8_a4/