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On application of generalized discrepancy principle to iterative methods for nonlinear ill-posed problems. (English) Zbl 1068.47081
The equation $$ F(x)=f,\tag1 $$ 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, $ \Vert f_{\delta}-f \Vert _{H_2} <\delta$. The operator $F$ is Fréchet differentiable and satisfies the conditions $\Vert F'(x) \Vert \leq 1$ and $\Vert F'(x)-F''(y)\Vert \leq\Vert x-y \Vert $ 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 $\Phi(x) = \Vert F(x) - f_{\delta} \Vert ^2_{H_2}: x_{n+1} = \xi -\theta ({F'}^*(x_n)F'(x_n), \alpha_n ){F'}^*(x_n) \{F(x_n)-f_{\delta} - F'(x_n)(x_n-\xi) \}$. Here $\xi$ is an element of $H_1$ and a source type condition is fulfilled; $\theta(\lambda, \alpha)$ is a function of a spectral parameter $\lambda$ and $\alpha > 0$. A novel generalized discrepancy principle $$ \Vert F(x_N)- f_{\delta} \Vert ^2 \leq \tau \delta \leq \Vert F(x_n)- f_{\delta} \Vert ^2, \tag2 $$ where $0 \leq n \leq N, \tau \geq 1$ is suggested in the paper. It is proved (under a source type condition) that if $N=N(\delta)$ is chosen by (2), then $ \lim \limits_{\delta \to 0} \Vert x_{N(\delta)}- \overline{x} \Vert \to 0$, where $\overline{x}$ is a solution of (1). Convergence rates for various generating functions $ \theta = \theta(\lambda,\alpha)$ are obtained.

47J06Nonlinear ill-posed problems
65F22Ill-posedness, regularization (numerical linear algebra)
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