Expanding the applicability of Tikhonov's regularization and iterative approximation for ill-posed problems
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Abstract
Recently, Vasin [J. Inverse Ill-Posed Probl. 21 (2013), 109–123] considered a new iterative method for approximately solving nonlinear ill-posed operator equation in Hilbert spaces. In this paper we introduce a modified form of the method considered by Vasin. This paper weakens the conditions needed in the existing results. We use a center-type Lipschitz condition in our convergence analysis instead of a Lipschitz-type condition used in [J. Inverse Ill-Posed Probl. 21 (2013), 109–123]. This way a tighter convergence analysis is obtained and under less computational cost, since the more precise and easier to compute center-Lipschitz instead of the Lipschitz constant is used in the convergence analysis. Order optimal error bounds are given in case the regularization parameter is chosen a priori and by the adaptive method of Pereverzev and Schock [SIAM J. Numer. Anal. 43 (2005), 2060–2076]. A numerical example of a nonlinear integral equation proves the efficiency of the proposed method.
Funding source: Russia Federation Government
Award Identifier / Grant number: Agreement No. 11. G 34.31.0064
Funding source: Ural Branch of RAS
Award Identifier / Grant number: 12-P-15-2019
Funding source: RFBF
Award Identifier / Grant number: 12-01-00106
© 2014 by De Gruyter
Articles in the same Issue
- Frontmatter
- A modified quasi-boundary value method for an ultraparabolic ill-posed problem
- An inverse problem for the quadratic pencil of non-self-adjoint matrix operators on the half-line
- Numerical inversion of the spherical Radon transform and the cosine transform using the approximate inverse with a special class of locally supported mollifiers
- A spherical x-ray transform and hypercube sections
- Regularization of autoconvolution and other ill-posed quadratic equations by decomposition
- An adaptive algorithm for determination of source terms in a linear parabolic problem
- Expanding the applicability of Tikhonov's regularization and iterative approximation for ill-posed problems
Articles in the same Issue
- Frontmatter
- A modified quasi-boundary value method for an ultraparabolic ill-posed problem
- An inverse problem for the quadratic pencil of non-self-adjoint matrix operators on the half-line
- Numerical inversion of the spherical Radon transform and the cosine transform using the approximate inverse with a special class of locally supported mollifiers
- A spherical x-ray transform and hypercube sections
- Regularization of autoconvolution and other ill-posed quadratic equations by decomposition
- An adaptive algorithm for determination of source terms in a linear parabolic problem
- Expanding the applicability of Tikhonov's regularization and iterative approximation for ill-posed problems
