Geodesic
Implicit method for solving a self-adjoint ill-posed problem with approximately operator and a posteriori choice of the regularization parameter
Trudy Instituta matematiki, Tome 23 (2015) no. 2, pp. 76-81.

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The implicit iteration method for solution of the first-kind operator equations with a non-negative self-adjoint bounded operator in the Hilbert space is proposed. Convergence of a method is proved in case of an a posteriori choice of the regularization parameter in ussual norm of Hilbert space, supposing that not only the right part of the equation but the operator as well have errors. The estimation of an error of method and estimation of a posteriori moment of stop are received. 
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O. V. Matysik. Implicit method for solving a self-adjoint ill-posed problem with approximately operator and a posteriori choice of the regularization parameter. Trudy Instituta matematiki, Tome 23 (2015) no. 2, pp. 76-81. http://geodesic.mathdoc.fr/item/TIMB_2015_23_2_a9/

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[2] Vainikko G.M., Veretennikov A.Yu., Iteratsionnye protsedury v nekorrektnykh zadachakh, M., 1986 | MR

[3] Lyusternik L.A., Sobolev V.I., Elementy funktsionalnogo analiza, M., 1965