Convergence rates for inverse Problems in Hilbert spaces: A Comparative Study

Abstract

In this paper, we apply a new kind of smoothness concept, i.e. H\"older stability estimates for the determination of convergence rates of Tikhonov regularization for linear and non-linear inverse problems in Hilbert spaces. For linear inverse problems, we obtain the convergence rates without incorporating the classical concept of spectral theory and for non-linear inverse problems, we obtain the convergence rates without incorporating any additional non-linearity estimate. Further, we employ the smoothness concept of inhomogeneous variational inequalities to deduce the convergence rates for non-linear inverse problems. In addition to Tikhonov regularization, we also consider Lavrentiev's regularization method for non-linear inverse problems and determine its convergence rates by incorporating the H\"older stability estimates as well as inhomogeneous variational inequalities. And finally, we discuss the co-action between the variational inequalities and the H\"older stability estimates.