Regularization of exponentially ill-posed problems

Abstract

Linear and nonlinear inverse problems which are exponentially ill-posed arise in heat conduction, satellite gradiometry, potential theory and scattering theory. For these problems logarithmic source conditions have natural interpretations whereas standard Hölder-type source conditions are far too restrictive. This paper provides a systematic study of convergence rates of regularization methods under logarithmic source conditions including the case that the operator is given onlyapproximately. We also extend previous convergence results for the iteratively regularized Gauß-Newton method to operator approximations.