Self-regularization of projection methods with a posteriori discretization level choice for severely ill-posed problems
- Bruckner, Gottfried
- Pereverzev, Sergei V.
2010 Mathematics Subject Classification
- 65R30 65J20
- severely ill-posed problems, regularization by discretization, projection methods, integral equation of the first kind
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 σ𝑘(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 σ𝑘(A) tend to zero exponentially. The proposed strategy does not require a priori information regarding the solution smoothness and the exact rate of σ𝑘(A).
- Inverse Problems 19, (2003), pp. 147-156