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Optimality for ill-posed problems under general source conditions. (English) Zbl 0907.65049

The problem of identifying the unknown $x$ of the ill-posed inverse problem $Ax=y$ is studied, where $A\in ℒ\left(X,Y\right)$ is a linear bounded operator between infinite-dimensional Hilbert spaces $X$ and $Y$ with non-closed range $R\left(A\right)$ of $A$ and, $x\in {M}_{\varphi ,E}=\left\{x\in X$; $x-\overline{x}=\varphi {\left({A}^{*}A\right)}^{1/2}v$, $\parallel v\parallel =E\right\}$ ($\overline{x}$ denotes an initial approximation for the problem $Ax=y$) with appropriate functions $\varphi$. As regards accuracy which can be obtained for identifying $x$ from ${y}^{\delta }\in Y$ it is proved that under certain conditions

$infsup\parallel R{y}^{\delta }-x\parallel =E\sqrt{{\rho }^{-1}\left({\delta }^{2}/{E}^{2}\right)}$

holds with $\rho \left(\lambda \right)=\lambda {\varphi }^{-1}\left(\lambda \right)$, where inf is taken over all methods $R:Y\to X$ and the sup is taken over all $x\in {M}_{\varphi ,E}$ and $\parallel y-{y}^{\delta }\parallel \le \delta$. In addition, it is proved the optimality of a general class of regularization methods which guarantee this best possible accuracy. In this general class Tikhonov methods and spectral methods are special cases. Different classes of examples are discussed.

MSC:
 65J10 Equations with linear operators (numerical methods) 65J20 Improperly posed problems; regularization (numerical methods in abstract spaces) 47A50 Equations and inequalities involving linear operators, with vector unknowns