Oberseminar Wissenschaftliches Rechnen (Rommel Real)

We consider the Hanke-Raus heuristic parameter choice rule to the Landweber iteration and
Tikhonov regularization for solving ill-posed problems. This rule does not require the noise level,
which may be inaccessible and prone to improper estimation in many instances. A famous veto
states that convergence in the worst-case scenario cannot be expected in general.
However, by imposing certain conditions on the noisy data, we can establish convergence. Convergence
under this rule also extends with a penalty functional penalty, such as L1 and TV-penalty
terms, which are used to recover solutions with special features, such as sparsity and piecewise
constancy. We discuss some recent progress in this heuristic rule, which also involves Newton-type
methods.