Implicit iteration methods in Hilbert scales under general smoothness conditions
Provides a theoretical framework and practical algorithm for regularization of ill-posed problems, but the contribution is incremental as it extends existing Hilbert scale methods to more general smoothness assumptions.
The paper develops implicit iteration methods for solving linear ill-posed problems in Hilbert scales, achieving order optimal error bounds under general smoothness conditions with both a priori and a posteriori parameter choice rules, including a fast algorithm for the discrepancy principle.
For solving linear ill-posed problems regularization methods are required when the right hand side is with some noise. In the present paper regularized solutions are obtained by implicit iteration methods in Hilbert scales. % By exploiting operator monotonicity of certain functions and interpolation techniques in variable Hilbert scales, we study these methods under general smoothness conditions. Order optimal error bounds are given in case the regularization parameter is chosen either {\it a priori} or {\it a posteriori} by the discrepancy principle. For realizing the discrepancy principle, some fast algorithm is proposed which is based on Newton's method applied to some properly transformed equations.