Top

Mathematical Models and Computer Simulations

Published in:

01-03-2019

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

Authors: A. A. Belov, N. N. Kalitkin

Published in: Mathematical Models and Computer Simulations | Issue 2/2019

Activate our intelligent search to find suitable subject content or patents.

search-config

AI-assisted search

Off

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.

previous article Mathematical Model of Generation and Amplification of Radiation in a Multistage Laser

next article Mathematical Model of a Post-Impact Cavitational Deceleration of a Torus in a Liquid

Dont have a licence yet? Then find out more about our products and how to get one now:

Springer Professional "Wirtschaft+Technik"

Online-Abonnement

Mit Springer Professional "Wirtschaft+Technik" erhalten Sie Zugriff auf:

über 102.000 Bücher
über 537 Zeitschriften

aus folgenden Fachgebieten:

Automobil + Motoren
Bauwesen + Immobilien
Business IT + Informatik
Elektrotechnik + Elektronik
Energie + Nachhaltigkeit
Finance + Banking
Management + Führung
Marketing + Vertrieb
Maschinenbau + Werkstoffe
Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

inform now

Springer Professional "Technik"

Online-Abonnement

Mit Springer Professional "Technik" erhalten Sie Zugriff auf:

über 67.000 Bücher
über 390 Zeitschriften

aus folgenden Fachgebieten:

Automobil + Motoren
Bauwesen + Immobilien
Business IT + Informatik
Elektrotechnik + Elektronik
Energie + Nachhaltigkeit
Maschinenbau + Werkstoffe

Jetzt Wissensvorsprung sichern!

inform now

Springer Professional "Wirtschaft"

Online-Abonnement

Mit Springer Professional "Wirtschaft" erhalten Sie Zugriff auf:

über 67.000 Bücher
über 340 Zeitschriften

aus folgenden Fachgebieten:

Bauwesen + Immobilien
Business IT + Informatik
Finance + Banking
Management + Führung
Marketing + Vertrieb
Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

inform now

Jun-Gang Wang, Yan Li, and Yu-Hong Ran, “Convergence of Chebyshev type regularization method under Morozov discrepancy principle,” Appl. Math. Lett. 74, 174–180 (2017).MathSciNetCrossRefMATH

A. A. Belov and N. N. Kalitkin, “Processing of experimental curves by applying a regularized double period method,” Dokl. Math. 94, 539–543 (2016).MathSciNetCrossRefMATH

A. A. Belov and N. N. Kalitkin, “Regularization of the double period method for experimental data processing,” Comput. Math. Math. Phys. 57, 1741–1750 (2017).MathSciNetCrossRefMATH

A. B. Bakushinsky and A. Smirnova, “Irregular operator equations by iterative methods with undetermined reverse connection,” J. Inv. Ill-Posed Probl. 18, 147–165 (2010).MathSciNetMATH

A. B. Bakushinsky and A. Smirnova, “Discrepancy principle for generalized GN iterations combined with the reverse connection control,” J. Inv. Ill-Posed Probl. 18, 421–431 (2010).MathSciNetMATH

Jian-guo Tang, “An implicit method for linear ill-posed problems with perturbed operators,” Math. Meth. Appl. Sci. 29, 1327–1338 (2006).MathSciNetCrossRefMATH

A. S. Leonov, Solving Ill-Posed Inverse Problems: Essays on the Theory, Practical Algorithms and Demonstrations in MATLAB (Librokom, Moscow, 2010) [in Russian].

A. N. Tikhonov, A. V. Goncharskii, V. V. Stepanov, and A. G. Yagola, Numerical Methods for the Solution of Ill-Posed Problems, Vol. 328 of Math. Appl. (Nauka, Moscow, 1990; Springer, Dordrecht, 1995).

Yu. L. Gaponenko, “On the degree of decidability and the accuracy of the solution of an ill-posed problem for a fixed level of error,” USSR Comput. Math. Math. Phys. 24, 96–101 (1984).CrossRefMATH

10.

Yu. L. Gaponenko, “The accuracy of the solution of a non-linear ill-posed problem for a finite error level,” USSR Comput. Math. Math. Phys. 25, 81–85 (1985).MathSciNetCrossRefMATH

11.

Y. C. Hon and T. Wei, “Numerical computation of an inverse contact problem in elasticity,” J. Inv. Ill-Posed Probl. 14, 651–664 (2006).MathSciNetCrossRefMATH

12.

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

13.

A. A. Samarskii and P. N. Vabishchevich, “Difference schemes for unstable problems,” Mat. Model. 2 (11), 89–98 (1990).MathSciNet

14.

A. A. Samarskii, “Regularization of difference schemes,” USSR Comput. Math. Math. Phys. 7, 79–120 (1967).MathSciNetCrossRef

15.

A. B. Bakushinskii and A. S. Leonov, “New a posteriori accuracy estimates for approximate solutions of irregular operator equations,” Vychisl. Metody Programmir. 15, 359–369 (2014).

16.

A. B. Bakushinsky, A. Smirnova, and Hui Liu, “A posteriori error analysis for unstable models,” J. Inv. Ill-Posed Probl. 20, 411–428 (2012).MathSciNetMATH

17.

M. V. Klibanov, A. B. Bakushinsky, and L. Beilina, “Why a minimizer of the Tikhonov functional is closer to the exact solution than the first guess,” J. Inv. Ill-Posed Probl. 19, 83–105 (2011).MathSciNetMATH

18.

A. V. Goncharskii, A. S. Leonov, and A. G. Yagola, “A generalized discrepancy principle,” USSR Comput. Math. Math. Phys. 13, 25–37 (1973).CrossRef

19.

L. F. Richardson and J. A. Gaunt, “The deferred approach to the limit,” Philos. Trans., A 226, 299–349 (1927).

20.

V. S. Ryaben’kii and A. F. Fillipov, On the Stability of Difference Equations (Gos. Izd. Tekh.-Teor. Liter., Moscow, 1956) [in Russian].

21.

A. N. Tikhonov and V. Ya. Arsenin, Solutions of Ill-Posed Problems (Halsted, New York, 1977; Nauka, Moscow, 1979).

22.

N. N. Kalitkin, A. B. Alshin, E. A. Alshina, and B. V. Rogov, Computation on Quasi-Uniform Meshes (Fizmatlit, Moscow, 2005) [in Russian].

23.

A. A. Samarskii, The Theory of Difference Schemes (Marcel Dekker, New York, Basel, 2001; Nauka, Moscow, 1989), eng. p. 761.

24.

S. G. Rautian, “Real spectral instruments,” Sov. Phys. Usp. 1, 245 (1958).CrossRef

Title: Solution of the Fredholm Equation of the First Kind by the Mesh Method with the Tikhonov Regularization
Authors: A. A. Belov
N. N. Kalitkin
Publication date: 01-03-2019
Publisher: Pleiades Publishing
Published in: Mathematical Models and Computer Simulations / Issue 2/2019
Print ISSN: 2070-0482
Electronic ISSN: 2070-0490
DOI: https://doi.org/10.1134/S2070048219020042

Springer Professional

Abstract

Please log in to get access to your license.

Dont have a licence yet? Then find out more about our products and how to get one now:

Springer Professional "Wirtschaft+Technik"

Springer Professional "Technik"

Springer Professional "Wirtschaft"

Other articles of this Issue 2/2019

On the Distribution of the First Break Down in the Wireless D2D Communication with Cashing

Galactic Structures in a Viscous Gas Flow in a Channel with Solid Walls

Mathematical Model of a Post-Impact Cavitational Deceleration of a Torus in a Liquid

On a Parallel Version of a Second-Order Incomplete Triangular Factorization Method

Simulation of Turbulent Mixing by the CABARET Algorithm for the Case of a Richtmyer–Meshkov Instability

Optimal Die Profile for the Plane Strain Drawing of Sheets

Premium Partner