We propose a new class of projection methods for the solution of ill-posed problems with inaccurately specified coefficients. For methods from this class, we establish the conditions of convergence to the normal solution of an operator equation of the first kind. We also present additional conditions for these methods guaranteeing the convergence with a given rate to normal solutions from a certain set.
Similar content being viewed by others
References
P. Mathe and S. V. Pereverzev, “Discretization strategy for linear ill-posed problems in variable Hilbert scales,” Inverse Probl., 19, 1263–1277 (2003).
S. V. Pereverzev, “Optimization of projection methods for solving ill-posed problems,” Computing, 55, 113–124 (1995).
S. G. Solodky, “A generalized projection scheme for solving ill-posed problems,” J. Inverse Ill-Posed Probl., 7, 185–200 (1999).
S. G. Solodky, “On a quasioptimal regularized projection method for solving operator equations of the first kind,” Inverse Probl., 21, No. 4, 1473–1485 (2005).
S. G. Solodky and E. V. Lebedeva, “Bounds of information expenses in constructing projection methods for solving ill-posed problems,” Comput. Meth. Appl. Math., 6, No. 1, 87–93 (2006).
R. Plato and G. Vainikko, “On the regularization of projection methods for solving ill-posed problems,” Numer. Math., 57, 63–79 (1990).
R. Plato and G. Vainikko, “On the regularization of the Ritz-Galerkin method for solving ill-posed problems,” Uch. Zap. Tartu. Univ., 863, 3–17 (1989).
A. B. Bakushinskii, “A general approach to the construction of a regularizing algorithm for linear ill-posed equations in Hilbert spaces,” Zh. Vychisl. Mat. Mat. Fiz., 7, No. 3, 672–677 (1967).
G. M. Vainikko and A. Yu. Veretennikov, Iterative Procedures in Ill-Posed Problems [in Russian], Nauka, Moscow (1986).
A. V. Goncharskii, A. S. Leonov, and A. G. Yagola, “Generalized discrepancy principle,” Zh. Vychisl. Mat. Mat. Fiz., 13, 294–302 (1973).
E. V. Lebedeva, “On one rule of the choice of discrete information for the approximate solution of ill-posed problems,” Uch. Zap. Tavrich. Nats. Univ., Ser. Mat. Mekh. Inform. Kibernet., 18, No. 1, 47–54 (2005).
M. L. Krasnov, A. I. Kiselev, and G. V. Makarenko, Integral Equations [in Russian], Nauka, Moscow (1976).
Author information
Authors and Affiliations
Additional information
Translated from Ukrains'kyi Matematychnyi Zhurnal, Vol. 60, No. 6, pp. 843–850, June, 2008.
Rights and permissions
About this article
Cite this article
Lebedeva, E.V. On the conditions of convergence for one class of methods used for the solution of ill-posed problems. Ukr Math J 60, 985–994 (2008). https://doi.org/10.1007/s11253-008-0100-1
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11253-008-0100-1