Generalized Convergence Rates Results for Linear Inverse Problems in Hilbert Spaces

Abstract

In recent years, a series of convergence rates conditions for regularization methods has been developed. Mainly, the motivations for developing novel conditions came from the desire to carry over convergence rates results from the Hilbert space setting to generalized Tikhonov regularization in Banach spaces. For instance, variational source conditions have been developed, and they were expected to be equivalent to standard source conditions for linear inverse problems in a Hilbert space setting (see Schuster et al. [13]). We show that this expectation does not hold. However, in the standard Hilbert space setting these novel conditions are optimal, which we prove by using some deep results from Neubauer [11], and generalize existing convergence rates results. The key tool in our analysis is a homogeneous source condition, which we put into relation to the other existing source conditions from the literature. As a positive by-product, convergence rates results can be proven without spectral theory, which is the standard technique for proving convergence rates for linear inverse problems in Hilbert spaces (see Groetsch [7]).