Optimal discretization of inverse problems in Hilbert scales. Regularization and self-regularization of projection methods

Abstract

We study the efficiency of the approximate solution of ill-posed problems, based on discretized observations, which we assume to be given afore-hand. We restrict ourselves to problems which can be formulated in Hilbert scales. Within this framework we shall quantify the degree of ill-posedness, provide general conditions on projection schemes to achieve the best possible order of accuracy. We pay particular attention on the problem of self-regularization vs. Tikhonov regularization. Moreover, we study the information complexity. Asymptotically, any method, which achieves the best possible order of accuracy must use at least such amount of noisy observations. We accomplish our study with two specific problems, Abel's integral equation and the recovery of continuous functions from noisy coefficients with respect to a given orthonormal system, both classical ill-posed problems