Regularization of linear approximate schemes by the gradient descent (Q2706411)

An abstract approximate scheme for a linear operator equation \(Ax=b\) in a Banach space is considered. The scheme is not supposed to be stable. The approximate equation \(A_nx_n=b_n\) is solved by the gradient method with a stopping rule depending on the approximation error of the scheme. In the authors' previous paper [Zh. Vychisl. Mat. Mat. Fiz. 39, No. 9, 1453-1463 (1999; Zbl 0977.65047)] they proved the convergence of approximate solutions for the case of a continuous operator \(A\) and under some strong approximation conditions on \(\{A_n\}\). NEWLINENEWLINENEWLINEIn this paper similar results concerning regularization properties of the considered method are obtained under weaker approximation assumptions. Moreover, it is not assumed that \(A\) is continuous nor its domain is all the space. The results are obtained under the additional assumption that the solution is sourcewise representable, i.e. \(x\in\text{ im }A\). The problem of existence of sourcewise representable solution is also discussed.