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Zbl 0907.65049
Tautenhahn, Ulrich
Optimality for ill-posed problems under general source conditions. (English)
[J] Numer. Funct. Anal. Optimization 19, No.3-4, 377-398 (1998). ISSN 0163-0563; ISSN 1532-2467/e

The problem of identifying the unknown $x$ of the ill-posed inverse problem $Ax= y$ is studied, where $A\in{\cal L}(X,Y)$ is a linear bounded operator between infinite-dimensional Hilbert spaces $X$ and $Y$ with non-closed range $R(A)$ of $A$ and, $x\in M_{\varphi, E}= \{x\in X$; $x-\overline x=\varphi(A^*A)^{1/2}v$, $\| v\|= E\}$ ($\overline x$ 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 $$\inf\sup\| Ry^\delta- x\|= E\sqrt{\rho^{-1}(\delta^2/E^2)}$$ holds with $\rho(\lambda)= \lambda\varphi^{-1}(\lambda)$, where inf is taken over all methods $R:Y\to X$ and the sup is taken over all $x\in M_{\varphi, E}$ and $\| y- y^\delta\|\le \delta$. 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.
[Costică Moroşanu (Iaşi)]

MSC 2000:: *65J10 Equations with linear operators (numerical methods)
65J20 Improperly posed problems (numerical methods in abstract spaces)
47A50 Equations and inequalities involving linear operators

Keywords: numerical examples; ill-posed inverse problem; Hilbert spaces; regularization methods; Tikhonov methods; spectral methods

Cited in: Zbl 1247.47047 Zbl 1037.65057

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