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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(X,Y) is a linear bounded operator between infinite-dimensional Hilbert spaces X and Y with non-closed range R(A) of A and, xM φ,E ={xX; x-x ¯=φ(A * A) 1/2 v, v=E} (x ¯ denotes an initial approximation for the problem Ax=y) with appropriate functions φ. As regards accuracy which can be obtained for identifying x from y δ Y it is proved that under certain conditions

infsupRy δ -x=Eρ -1 (δ 2 /E 2 )

holds with ρ(λ)=λφ -1 (λ), where inf is taken over all methods R:YX and the sup is taken over all xM φ,E and y-y δ δ. 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:
65J10Equations with linear operators (numerical methods)
65J20Improperly posed problems; regularization (numerical methods in abstract spaces)
47A50Equations and inequalities involving linear operators, with vector unknowns