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.

Voir la notice de l'article provenant de la source Math-Net.Ru

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/}
}
TY  - JOUR
AU  - A. A. Belov
AU  - N. N. Kalitkin
TI  - Solution of the Fredholm equation of the first kind by mesh method with Tikhonov regularization
JO  - Matematičeskoe modelirovanie
PY  - 2018
SP  - 67
EP  - 88
VL  - 30
IS  - 8
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/MM_2018_30_8_a4/
LA  - ru
ID  - MM_2018_30_8_a4
ER  - 
%0 Journal Article
%A A. A. Belov
%A N. N. Kalitkin
%T Solution of the Fredholm equation of the first kind by mesh method with Tikhonov regularization
%J Matematičeskoe modelirovanie
%D 2018
%P 67-88
%V 30
%N 8
%I mathdoc
%U http://geodesic.mathdoc.fr/item/MM_2018_30_8_a4/
%G ru
%F MM_2018_30_8_a4
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/

[1] Jun-Gang Wang, Yan Li, Yu-Hong Ran, “Convergence of Chebyshev type regularization method under Morozov discrepancy principle”, Appl. Math. Lett., 74 (2017), 174–180 | DOI | MR | Zbl

[2] Belov A. A., Kalitkin N. N., “Processing of Experimental Curves by Applying a Regularized Double Period Method”, Doklady Math., 94:2 (2016), 539–543 | DOI | MR | Zbl

[3] Belov A. A., Kalitkin N. N., “Regularization of the Double Period Method for Experimental Data Processing”, Comp. Math. and Math. Phys., 57:11 (2017), 1741–1750 | DOI | MR | Zbl | Zbl

[4] Bakushinsky A. B., Smirnova A., “Irregular operator equations by iterative methods with undetermined reverse connection”, J. Inv. Ill-Posed Problems, 18 (2010), 147–165 | MR | Zbl

[5] Bakushinsky A. B., Smirnova A., “Discrepancy principle for generalized GN iterations combined with the reverse connection control”, J. Inv. Ill-Posed Problems, 18 (2010), 421–431 | MR | Zbl

[6] Jian-guo Tang, “An implicit method for linear ill-posed problems with perturbed operators”, Math. Meth. in the Appl. Sci., 29 (2006), 1327–1338 | DOI | MR

[7] Leonov A. S., Reshenie nekorrktno postavlennykh obratnykh zadach. Ocherk teorii, prakticheskie algoritmy i deomnstratsii v MATLAB, Librokom, M., 2010

[8] Tikhonov A. N., Goncharskii A. V., Stepanov V. V., Iagola A. G., Chislennye metody resheniia nekorrektnykh zadach, Nauka, M., 1990

[9] Gaponenko Iu. L., “On the degree of decidability and the accuracy of the solution of an ill-posed problem for a fixed level of error”, USSR Comp. Math. and Math. Phys., 24 (1984), 96–101 | DOI | MR | Zbl | Zbl

[10] Gaponenko Yu. L., “The accuracy of the solution of a non-linear ill-posed problem for a finite error level”, USSR Comp. Math. and Math. Phys., 25 (1985), 81–85 | Zbl

[11] Hon Y. C., Wei T., “Numerical computation of an inverse contact problem in elasticity”, J. Inv. Ill-Posed Problems, 14 (2006), 651–664 | DOI | MR | Zbl

[12] Ben Ameur H., Kaltenbacher B., “Regularization of parameter estimation by adaptive discretization using refinement and coarsening indicators”, J. Inv. Ill-Posed Problems, 10 (2002), 561–583 | MR | Zbl

[13] Samarskii A. A., Vabishchevich P. N., “Raznostnye skhemy dlia neustoichevykh zadach”, Mat. Modelirovanie, 2:11 (1990), 89–98

[14] Samarskii A. A., “Regularization of difference schemes”, USSR Comp. Math. and Math. Phys., 7 (1967), 79–120 | DOI | MR

[15] Bakushinskii A. B., Leonov A. S., “Novye aposteriornye otsenki tochnosti dlia priblizhennykh reshenii nereguliarnykh operatornykh uravnenii”, Vych. met. programmirovanie, 15:2 (2014), 359–369

[16] Bakushinsky A. B., Smirnova A., Hui Liu, “A posteriori error analysis for unstable models”, J. Inv. Ill-Posed Problems, 20 (2012), 411–428 | DOI | MR | Zbl

[17] Klibanov M. V., Bakushinsky A. B., Beilina L., “Why a minimizer of the Tikhonov functional is closer to the exact solution than the first guess”, J. Inv. Ill-posed Problems, 19 (2011), 83–105 | DOI | MR | Zbl

[18] Goncharskii A. V., Leonov A. S., Yagola A. G., “A generalized discrepancy principle”, USSR Comp. Math. and Math. Phys., 13 (1973), 25–37 | DOI | MR | Zbl

[19] Richardson L. F., Gaunt J. A., “The deferred approach to the limit”, Phil. Trans. A, 226 (1927), 299–349 | DOI | Zbl

[20] Riaben'kii V. S., Fillipov A. F., Ob ustoichivosti raznostnykh uravnenii, Gosudarstvennoe izd-vo tekhniko-teoretichskoi literatury, 1956

[21] Tikhonov A. N., Arsenin V. Ya., Solutions of ill-posed problems, Halsted, New York, 1977 | MR

[22] Kalitkin N. N., Alshin A. B., Alshina E. A., Rogov B. V., Vychisleniia na kvaziravnomernykh setkah, Fizmatlit, M., 2005

[23] A.A. Samarskii, The theory of difference schemes, Marcel Dekker, Inc., New York–Basel, 2001, 761 pp. | MR | Zbl

[24] Rautian S. G., “Realnye spectralnye pribory”, UFN, 66:3 (1958), 475–517 | DOI