Abstract
In the present paper for a stable solution of severely ill-posed problems with perturbed input data, the standard Tikhonov method is applied, and the regularization parameter is chosen according to balancing principle. We establish that the approach provides the order of accuracy on the class of problems under consideration.
Highlights
In the present paper we consider the issue of approximate solving severely ill-posed problems represented by operator equation of the first kind
Where A : X → Y is a linear compact injective operator acting between Hilbert spaces X and Y
Later in the work [6] suggested a general class of regularization methods for solving both linear and non-linear severely ill-posed problems (1.1) with perturbed input data; for choosing regularization parameter, a modification from [14] was employed
Summary
Where A : X → Y is a linear compact injective operator acting between Hilbert spaces X and Y. Later in the work [6] suggested a general class of regularization methods (according to Bakushinski; see, e.g., [1]) for solving both linear and non-linear severely ill-posed problems (1.1) with perturbed input data; for choosing regularization parameter, a modification from [14] was employed. Among the works devoted to the research of approximate methods of solving severely ill-posed problems we should mention [2, 7, 15, 17]. In [15] an approach for solving severely ill-posed problems (1.1) with solutions from Mp1,ρ(A) and exact given operators was proposed. It suggests a combination of the standard Tikhonov regularization with the Morozov discrepancy principle. As opposite to the works mentioned above, our method does not require any additional information about smoothness of the desired solution
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