DK8: Nonlinear regularization methods for the solution of linear ill-posed problems

In the proposed project we aim at studying solution methods for linear ill-posed problems. We want to investigate non-linear methods since they seem to be better suited for the reconstruction of functions which are spatially inhomogeneous. In particular, we will focus on two-step methods where a data preprocessing step is followed by a reconstruction step. Thus, for the operator equation Kf=g and possibly noisy data g^delta a solution is constructed as

  f_{\alpha,\lambda} := T_{\alpha,\lambda} g

with  T_{\alpha,\lambda} = R_\alpha S_\lambda.

Herein S_\lambda: Y -> Y denotes the data estimation defined on the data side and R_\alpha: Y -> X denotes the reconstruction operator.

We are interested in the question which methods are suitable for both steps, how the combination can be achieved and what special results on parameter choice rules, convergence rates and computational costs can be proved.

Supervisor: Prof. Ronny Ramlau