Geodesic
On explicit expression of the solution to the regularizing by Tikhonov optimization problem in~terms of the regularization parameter in the finite-dimensional case
Izvestiya Instituta Matematiki i Informatiki Udmurtskogo Gosudarstvennogo Universiteta, Tome 60 (2022), pp. 90-110.

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

It is well known that using the Tikhonov regularization method for solving operator equations of the first kind one has to minimize a regularized residual functional. The minimizer is determined from so called Euler equation which in finite-dimensional case and at its discretization is written as a one-parametric (depending on the regularization parameter) system of linear algebraic equations of special form. Here, there exist various ways of choosing the regularization parameter. In particular, in the frame of principle of generalized residual, it is necessary to solve the corresponding equation of generalized residual with respect to the regularization parameter. And it implies (when solving this equation numerically), in turn, multifold solving a one-parametric system of linear algebraic equations for arbitrary value of the parameter. In this paper we obtain an explicit simple and effective formula of solution to a one-parametric system for an arbitrary value of the parameter. We give an example of computations by above-mentioned formula and also an example of numerical solution of the Fredholm integral equation of the first kind under usage of this formula which substantiates its effectiveness.
Keywords: Tikhonov regularization method, generalized residual method, one-parametric system of linear algebraic equations, decomposition method. 
@article{IIMI_2022_60_a5,
     author = {A. V. Chernov},
     title = {On explicit expression of the solution to the regularizing by {Tikhonov} optimization problem in~terms of the regularization parameter in the finite-dimensional case},
     journal = {Izvestiya Instituta Matematiki i Informatiki Udmurtskogo Gosudarstvennogo Universiteta},
     pages = {90--110},
     publisher = {mathdoc},
     volume = {60},
     year = {2022},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/IIMI_2022_60_a5/}
}
TY  - JOUR
AU  - A. V. Chernov
TI  - On explicit expression of the solution to the regularizing by Tikhonov optimization problem in~terms of the regularization parameter in the finite-dimensional case
JO  - Izvestiya Instituta Matematiki i Informatiki Udmurtskogo Gosudarstvennogo Universiteta
PY  - 2022
SP  - 90
EP  - 110
VL  - 60
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/IIMI_2022_60_a5/
LA  - ru
ID  - IIMI_2022_60_a5
ER  - 
%0 Journal Article
%A A. V. Chernov
%T On explicit expression of the solution to the regularizing by Tikhonov optimization problem in~terms of the regularization parameter in the finite-dimensional case
%J Izvestiya Instituta Matematiki i Informatiki Udmurtskogo Gosudarstvennogo Universiteta
%D 2022
%P 90-110
%V 60
%I mathdoc
%U http://geodesic.mathdoc.fr/item/IIMI_2022_60_a5/
%G ru
%F IIMI_2022_60_a5
A. V. Chernov. On explicit expression of the solution to the regularizing by Tikhonov optimization problem in~terms of the regularization parameter in the finite-dimensional case. Izvestiya Instituta Matematiki i Informatiki Udmurtskogo Gosudarstvennogo Universiteta, Tome 60 (2022), pp. 90-110. http://geodesic.mathdoc.fr/item/IIMI_2022_60_a5/

[1] Levin E., Meltzer A. Y., “Estimation of the regularization parameter in linear discrete ill-posed problems using the Picard parameter”, SIAM Journal on Scientific Computing, 39:6 (2017), A2741–A2762 | DOI | MR | Zbl

[2] Bj{ö}rck Å., Numerical methods in matrix computations, Springer, Cham, 2015 | DOI | MR

[3] Reichel L., Yu X., “Matrix decompositions for Tikhonov regularization”, Electronic Transactions on Numerical Analysis, 43 (2015), 223–243 https://www.emis.de///journals/ETNA/vol.43.2014-2015/pp223-243.dir/pp223-243.pdf | Zbl

[4] Dykes L., Noschese S., Reichel L., “Rescaling the GSVD with application to ill-posed problems”, Numerical Algorithms, 68:3 (2015), 531–545 | DOI | MR | Zbl

[5] Onunwor E., Reichel L., “On the computation of a truncated SVD of a large linear discrete ill-posed problem”, Numerical Algorithms, 75:2 (2017), 359–380 | DOI | MR | Zbl

[6] Zare H., Hajarian M., “Determination of regularization parameter via solving a multi-objective optimization problem”, Applied Numerical Mathematics, 156 (2020), 542–554 | DOI | MR

[7] Hochstenbach M. E., Reichel L., Rodriguez G., “Regularization parameter determination for discrete ill-posed problems”, Journal of Computational and Applied Mathematics, 273 (2015), 132–149 | DOI | MR | Zbl

[8] Park Y., Reichel L., Rodriguez G., Yu X., “Parameter determination for Tikhonov regularization problems in general form”, Journal of Computational and Applied Mathematics, 343 (2018), 12–25 | DOI | MR | Zbl

[9] Bauer F., Lukas M. A., “Comparingparameter choice methods for regularization of ill-posed problems”, Mathematics and Computers in Simulation, 81:9 (2011), 1795–1841 | DOI | MR | Zbl

[10] Fenu C., Reichel L., Rodriguez G., “GCV for Tikhonov regularization via global Golub–Kahan decomposition”, Numerical Linear Algebra with Applications, 23:3 (2016), 467–484 | DOI | MR | Zbl

[11] Fenu C., Reichel L., Rodriguez G., Sadok H., “GCV for Tikhonov regularization by partial SVD”, BIT Numerical Mathematics, 57:4 (2017), 1019–1039 | DOI | MR | Zbl

[12] Reichel L., Rodriguez G., “Old and new parameter choice rules for discrete ill-posed problems”, Numerical Algorithms, 63:1 (2013), 65–87 | DOI | MR | Zbl

[13] Dykes L., Reichel L., “Simplified GSVD computations for the solution of linear discrete ill-posed problems”, Journal of Computational and Applied Mathematics, 255 (2014), 15–27 | DOI | MR | Zbl

[14] Glasko V. B., Inverse problems of mathematical physics, American Institute of Physics, New York, 1988 | MR | MR | Zbl | Zbl

[15] Voevodin V. V., Kuznetsov Yu. A., Matrices and computations, Nauka, M., 1984 | MR | Zbl

[16] Tikhonov A. N., Goncharsky A. V., Stepanov V. V., Yagola A. G., Numerical methods for the solution of ill-posed problems, Kluwer Academic Publishers, Dordrecht, 1995 | MR | Zbl | Zbl

[17] Petrov Yu. P., Sizikov V. S., Well-posed, ill-posed, and intermediate problems with applications, VSP, Utrecht, 2005 | MR | Zbl

[18] Sumin M. I., Incorrect problems and methods of solving them, Nizhny Novgorod State University, Nizhny Novgorod, 2009

[19] Verzhbitskii V. M., Elements of numerical methods, Vysshaya Shkola, M., 2002

[20] Godunov S. K., Antonov A. G., Kirilyuk O. P., Kostin V. I., Guaranteed accuracy in numerical linear algebra, Kluwer Academic Publishers, Dordrecht, 1993 | MR | MR | Zbl

[21] Surnin Yu. V., “Decomposition and regularization of the solution of ill-conditioned inverse problems in processing of measurement information. Part 1. A theoretical evaluation of the method”, Measurement Techniques, 61:3 (2018), 223–231 | DOI

[22] Zhdanov A. I., “Implicit iterative schemes based on singular decomposition and regularizing algorithms”, Vestnik Samarskogo Gosudarstvennogo Tekhnicheskogo Universiteta. Seriya Fiziko-Matematicheskie Nauki, 22:3 (2018), 549–556 | DOI | Zbl

[23] Buccini A., Pasha M., Reichel L., “Generalized singular value decomposition with iterated Tikhonov regularization”, Journal of Computational and Applied Mathematics, 373 (2020), 112276 | DOI | MR

[24] Bianchi D., Donatelli M., “On generalized iterated Tikhonov regularization with operator-dependent seminorms”, Electronic Transactions on Numerical Analysis, 47 (2017), 73–99 | DOI | MR | Zbl

[25] Vatul'yan A. O., Yavruyan O. V., Methodical instructions for practical tasks on the special course “Inverse problems of mechanics” for students of mechanical-mathematical faculty, Rostov State University, Rostov-on-Don, 2005

[26] Vasil'eva A. B., Tikhonov N. A., Integral equations, Fizmatlit, M., 2002 | Zbl

[27] Chernov A. V., Linear algebra and functional analysis: theory foundations and examples of solving problems, Nizhny Novgorod State Technical University, Nizhny Novgorod, 2010