A class of regularization schemes for linear ill-posed problems in Banach spaces under low order source conditions

Abstract

In this work we consider, in a Banach space framework, the regularization of linear ill-posed problems. Our focus is on the recovery of solutions that have a logarithmic source representation. Such cases typically occur in exponentially ill-posed problems like the backwards heat equation. The mathematical framework on logarithms of operators is provided, and convergence rates for a class of parametric regularization schemes are deduced. This class includes the iterated version of Lavrentiev's method and the method of the abstract Cauchy problem. The presentation includes both a priori and a posteriori parameter choice strategies. Finally, we present an example for a logarithmic source representable function with respect to the integration operator.