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On application of generalized discrepancy principle to iterative methods for nonlinear ill-posed problems. (English) Zbl 1068.47081

The equation

$F\left(x\right)=f,\phantom{\rule{2.em}{0ex}}\left(1\right)$

where a nonlinear operator $F$ acts on a pair of Hilbert spaces ($f:{H}_{1}\to {H}_{2}$) is considered in the paper. The element $f$ is approximately known, $\parallel {f}_{\delta }{-f\parallel }_{{H}_{2}}<\delta$. The operator $F$ is Fréchet differentiable and satisfies the conditions $\parallel {F}^{\text{'}}\left(x\right)\parallel \le 1$ and $\parallel {F}^{\text{'}}\left(x\right)-{F}^{\text{'}\text{'}}\left(y\right)\parallel \le \parallel x-y\parallel$ for any $x,y\in {H}_{1}$. The original problem is ill-posed, particularly the solution of (1) with the exact data may be nonunique. The following iteratively regularized scheme is used to minimize a functional ${\Phi }\left(x\right)=\parallel F\left(x\right)-{f}_{\delta }{\parallel }_{{H}_{2}}^{2}:{x}_{n+1}=\xi -\theta \left({{F}^{\text{'}}}^{*}\left({x}_{n}\right){F}^{\text{'}}\left({x}_{n}\right),{\alpha }_{n}\right){{F}^{\text{'}}}^{*}\left({x}_{n}\right)\left\{F\left({x}_{n}\right)-{f}_{\delta }-{F}^{\text{'}}\left({x}_{n}\right)\left({x}_{n}-\xi \right)\right\}$. Here $\xi$ is an element of ${H}_{1}$ and a source type condition is fulfilled; $\theta \left(\lambda ,\alpha \right)$ is a function of a spectral parameter $\lambda$ and $\alpha >0$. A novel generalized discrepancy principle

$\parallel F\left({x}_{N}\right)-{f}_{\delta }{\parallel }^{2}\le \tau \delta \le {\parallel F\left({x}_{n}\right)-{f}_{\delta }\parallel }^{2},\phantom{\rule{2.em}{0ex}}\left(2\right)$

where $0\le n\le N,\tau \ge 1$ is suggested in the paper. It is proved (under a source type condition) that if $N=N\left(\delta \right)$ is chosen by (2), then $\underset{\delta \to 0}{lim}\parallel {x}_{N\left(\delta \right)}-\overline{x}\parallel \to 0$, where $\overline{x}$ is a solution of (1). Convergence rates for various generating functions $\theta =\theta \left(\lambda ,\alpha \right)$ are obtained.

##### MSC:
 47J06 Nonlinear ill-posed problems 65F22 Ill-posedness, regularization (numerical linear algebra)
##### Keywords:
discrepancy principle; ill-posed problem; regularization