  This paper provides a new regularization method which is particularly
suitable for linear exponentially ill-posed problems. Under logarithmic source
conditions (which have a natural interpretation in terms of Sobolev spaces in
the aforementioned context), concepts of qualifications as well as order
optimal rates of convergence are presented. Optimality results under general
source conditions expressed in terms of index functions are also studied.
  Finally, numerical experiments on three test problems attest the better
performance of the new method compared to the well known Tikhonov method in
instances of exponentially ill-posed problems.
