Optimal regularization for ill-posed problems in metric spaces

Abstract

We present a strategy for choosing the regularization parameter (Lepskij-type balancing principle) for ill-posed problems in metric spaces with deterministic or stochastic noise. Additionally we improve the strategy in comparison to the previously used version for Hilbert spaces in some ways. AMS-Classification: 47A52, 65J22, 49J35, 93E25

Consistency of the Tikhonov’s regularization in an ill-posed problem with random data

“…This yields (14). In general, this rate of convergence cannot be achieved (Bauer and Munk, 2007) but in our case we note that the illposedness does not enter in the problem because we have assumed that the sample is i.i.d from the full sequence, then we estimated in an optimal way, using this sequence, the second member of the Eq. (1) and introduced it into the functional of Tikhonov (6).…”

Comparingparameter choice methods for regularization of ill-posed problems

“…This definition requires fewer evaluation steps. It also gives a convergent method [17,105] and performs well in practice. Our experience is that, provided β is big enough so that 24…”

“…As for the MD rule, one has to compute the function f in equation (17). The MDP rule also uses the function ̺ in the bound (8), but this is already required in the derivation of f .…”

Applying Lepskij-Balancing in Practice

“…In both cases, ̺(•) is a monotonically increasing function. Now we will follow the approach presented in [BM07], which already incorporates the (minor) modifications of the balancing principle to make it fit for practice, in particular, by limiting the number of necessary computations.…”