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

We study the efficiency of the approximate solution of ill-posed problems, based on discretized noisy observations, which we assume to be given before hand. A basic purpose of the paper is the consideration of stochastic noise , but deterministic noise is also briefly discussed. We restrict ourselves to problems which can be formulated in Hilbert scales. Within this framewor k we shall quantify the degree of ill-posedness, provide general conditions on projection schemes to achieve the best possible order of accuracy We pa y particular attention on the problem of self-regularization vs. Tikhonov r egularization. Moreover, we study the information complexity Asymptotically, any method wh ich achieves the best possible order of accuracy must use at least such amo unt of noisy observations.