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On the adaptive selection of the parameter in regularization of ill-posed problems. (English) Zbl 1103.65058
Consider an ill-posed operator equation Ax=y with a linear operator A(X,Y) between Banach spaces X and Y. Let y δ be an available approximation of y, y δ -y Y δ. Regularization methods usually replace the generalized inverse A + by a family of continuous linear operators {R α }, which converges pointwise to A + such that A + y-R α yφ(α), R α λ(α) -1 and lim α0 φ(α)=lim α0 λ(α)=0. Denote x α i δ =R α i y δ , Δ N ={α i :0<α 0 <<α N }, M + (Δ N )={α i Δ N :x α i δ -x α j δ 4δλ(α j ) -1 ,j=0,1,,i} and α + =max{α i :α i M + (Δ N )}. Suppose that λ(α i )qλ(α i-1 ) for any α i Δ N , i=1,,N. Then one has A + y-x α+ δ 6qφ((φλ) -1 (δ)).

MSC:
65J10Equations with linear operators (numerical methods)
65J20Improperly posed problems; regularization (numerical methods in abstract spaces)
47A52Ill-posed problems, regularization
45E10Integral equations of the convolution type
65R30Improperly posed problems (integral equations, numerical methods)