Optimal algorithms for linear ill-posed problems yield regularization methods

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Abstract
We consider a linear ill-posed operator equation Ax = y in Hilbert spaces. An algorithm R ε:Y→X for solving this equation with given inexact right-hand side y ε, such that , is called order optimal if it provides best possible error estimates under the assumption that the minimal norm solution x * of this operator equation fulfils some smoothness condition. It is shown that if such an algorithm is slightly modified to then it is a regularization method, i.e., we have without additional conditions on x *. We apply this result to show that the method of conjugate gradients for solving linear ill-posed equations together with a stopping rule yields a regularization method.

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