  We consider parameter identification problems in parametrized partial
differential equations (PDE). This leads to nonlinear ill-posed inverse
problems. One way to solve them are iterative regularization methods, which
typically require numerous amounts of forward solutions during the solution
process. In this article we consider the nonlinear Landweber method and want to
couple it with the reduced basis method as a model order reduction technique in
order to reduce the overall computational time. In particular, we consider PDEs
with a high-dimensional parameter space, which are known to pose difficulties
in the context of reduced basis methods. We present a new method that is able
to handle such high-dimensional parameter spaces by combining the nonlinear
Landweber method with adaptive online reduced basis updates. It is then applied
to the inverse problem of reconstructing the conductivity in the stationary
heat equation.
