OPTIMAL-ALGORITHMS FOR LINEAR ILL-POSED PROBLEMS YIELD REGULARIZATION METHODS

We consider a linear ill-posed operator equation Ax= y in Hilbert spaces. An algorithm Rε: Y→ X for solving this equation with given inexact right-hand side yε, such that ||y — yε|| ≤ ε, 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 Rε, then it is a regularization method, i.e., we have ||x —Rεy ε||→ 0 for ε→ 0 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. © 1990, Taylor & Francis Group, LLC. All rights reserved.