The instability of some gradient methods for ill-posed problems (Q751191)

Several authors have studied convergence properties of gradient methods like conjugate gradients and steepest descent for operator equations with nonclosed range in a Hilbert space. The authors show that even though these methods converge in the case of exact data the instability makes it impossible in some sense to base a-priori parameter choice regularization methods upon them. However, it is still possible to use some of these methods (e.g. the conjugate gradients) as regularization methods if the parameter n (here representing a stopping rule) is chosen a-posteriori as a function of the perturbed data and the upper bound on the error in the data.