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On the adaptive selection of the parameter in regularization of ill-posed problems. (English) Zbl 1103.65058
Consider an ill-posed operator equation $Ax=y$ with a linear operator $A\in ℒ\left(X,Y\right)$ between Banach spaces $X$ and $Y$. Let ${y}_{\delta }$ be an available approximation of $y$, $\parallel {y}_{\delta }{-y\parallel }_{Y}\le \delta$. Regularization methods usually replace the generalized inverse ${A}^{+}$ by a family of continuous linear operators $\left\{{R}_{\alpha }\right\}$, which converges pointwise to ${A}^{+}$ such that $\parallel {A}^{+}y-{R}_{\alpha }y\parallel \le \varphi \left(\alpha \right)$, $\parallel {R}_{\alpha }{\parallel \le \lambda \left(\alpha \right)}^{-1}$ and ${lim}_{\alpha \to 0}\varphi \left(\alpha \right)={lim}_{\alpha \to 0}\lambda \left(\alpha \right)=0$. Denote ${x}_{{\alpha }_{i}}^{\delta }={R}_{{\alpha }_{i}}{y}_{\delta }$, ${{\Delta }}_{N}=\left\{{\alpha }_{i}:0<{\alpha }_{0}<\cdots <{\alpha }_{N}\right\}$, ${M}^{+}\left({{\Delta }}_{N}\right)=\left\{{\alpha }_{i}\in {{\Delta }}_{N}:\parallel {x}_{{\alpha }_{i}}^{\delta }-{x}_{{\alpha }_{j}}^{\delta }\parallel \le 4\delta \lambda {\left({\alpha }_{j}\right)}^{-1},j=0,1,\cdots ,i\right\}$ and ${\alpha }_{+}=max\left\{{\alpha }_{i}:{\alpha }_{i}\in {M}^{+}\left({{\Delta }}_{N}\right)\right\}$. Suppose that $\lambda \left({\alpha }_{i}\right)\le q\lambda \left({\alpha }_{i-1}\right)$ for any ${\alpha }_{i}\in {{\Delta }}_{N}$, $i=1,\cdots ,N$. Then one has $\parallel {A}^{+}y-{x}_{\alpha +}^{\delta }\parallel \le 6q\varphi \left({\left(\varphi \lambda \right)}^{-1}\left(\delta \right)\right)$.

##### MSC:
 65J10 Equations with linear operators (numerical methods) 65J20 Improperly posed problems; regularization (numerical methods in abstract spaces) 47A52 Ill-posed problems, regularization 45E10 Integral equations of the convolution type 65R30 Improperly posed problems (integral equations, numerical methods)