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.


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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