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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 \(A\in{\mathcal L}(X,Y)\) 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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