## 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 $$A\in{\mathcal L}(X,Y)$$ 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 $$\| y- y^\delta\|\leq \delta$$. 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 Numerical solutions to equations with linear operators 65J20 Numerical solutions of ill-posed problems in abstract spaces; regularization 47A50 Equations and inequalities involving linear operators, with vector unknowns
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### References:

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