Self-regularization of projection methods with a posteriori discretization level choice for severely ill-posed problems

it is well known that projection schemes for certain linear ill-posed problems Ax = y can be regularized by a proper choice of the discretization level only, where no additional regularization is needed. The previous study of this self-regularization phenomenon was restricted to the case of so-called moderately ill-posed problems, i.e., when the singular values sigma(k) (A), k = 1, 2,..., of the operator A tend to zero with polynomial rate. The main accomplishment of the present paper is a new strategy for a discretization level choice that provides optimal order accuracy also for severely ill-posed problems, i.e., when sigma(k) (A) tend to zero exponentially. The proposed strategy does not require a priori information regarding the solution smoothness and the exact rate of sigma(k) (A).