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We consider generalized inverses and linear ill-posed problems in Banach spaces, and the concept of pseudo-optimal parameter choices and stopping rules for regularization methods is presented. The pseudo-optimality of the discrepancy principle for iterative methods like the Richardson iteration is shown, as well as the pseudo-optimality of diierent… (More)
In this paper resolvent estimates for Abel integral operators are provided. These estimates are applied to deduce regularizing properties of Lavrentiev's m-times iterated method as well as iterative schemes ? with the discrepancy principle as corresponding parameter choice or stopping rule, respectively for solving the corresponding Abel integral equations… (More)
For the numerical solution of the Galerkin equations associated with linear ill-posed problems that are symmetric and positive semideenite, the method of conjugate residuals is considered. An a posteriori stopping rule is introduced, and associated estimates for the approximations are provided which are order-optimal with respect to noise in the right-hand… (More)
In Hilbert spaces we consider linear ill-posed problems with corresponding operators that are symmetric, positive semideenite and compact, and we review some results on the conjugate residual method which by deenition minimizes the residual over Krylov subspaces and which can be realized by a conjugate gradient-type algorithm. In particular , for a… (More)
The repeated trapezoidal method was considered by P. Eggermont for the numerical solution of weakly singular Volterra integral equations of the first kind with exactly given right-hand sides (). In the present paper we consider the regularizing properties of this method for perturbed right-hand sides. Finally, numerical results are presented.
We consider an ill-posed problem T u = f in Hilbert spaces and suppose that the linear bounded operator T is approximately available, with a known estimate for the operator perturbation at the solution. As a numerical scheme the CGNRRmethod is considered, this is, the method of conjugate gradients for solving the associated normal equations. Two a… (More)