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Zbl 1154.65042
Kaltenbacher, B.
A note on logarithmic convergence rates for nonlinear Tikhonov regularization.
(English)
[J] J. Inverse Ill-Posed Probl. 16, No. 1, 79-88 (2008). ISSN 0928-0219; ISSN 1569-3945/e

A nonlinear ill-posed operator equation $F(x)= y$, from the domain $D(F)\subseteq X$ of the Hilbert space $X$ into a Hilbert space $Y$, is considered in case of only noisy data $y^\delta$ are available, with the assumption $\Vert y- y\Vert\le\delta$, $\delta > 0$. The Tikhonov regularization method consists in using the Tikhonov functional $$J_\alpha(x)=\Vert F(x)- y^\delta\Vert^2+ \alpha\Vert x- x_0\Vert^2.$$ The convergence rates for this type of regularization under a mild regularity assumption on the solution, namely source conditions of logarithmic type, are proved. For the choice of the regularization parameter a priori or a posteriori strategies according to the discrepancy can be used. Restrictions on the nonlinearity of the forward operator are made unless the initial error is sufficiently smooth.
[Jiří Vaníček (Praha)]
MSC 2000:
*65J15 Equations with nonlinear operators (numerical methods)
65J20 Improperly posed problems (numerical methods in abstract spaces)
47J06 Nonlinear ill-posed problems

Keywords: convergence; nonlinear operator equations; Hilbert space; ill-posed operator equation; Tikhonov regularization

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