Regularization by multilevel discretization for linear and nonlinear ill-posed problems

This talk deals with linear and nonlinear ill-posed operator equations and their stabilization by finite-dimensional approximation. After discussing the regularizing effect of projection onto finite dimensional spaces and adressing the question of a posteriori discretization level choice, we consider multigrid methods for such type of problems. Here, due to the adverse eigensystem structure of the forward operator -- small eigenvalues correspond to high frequency eigenfunctions -- appropriate smoothers have to be found. We give theoretical results on level independent contraction factors and V-cycle convergence, that enable us to use the defined multigrid operators in a nested iteration, approaching in a stable way the solution to the original infinite dimensional operator equation. Finally, as an application, the identification of nonlinear B-H curves in magnetics is shown.