  We investigate the ill-posed inverse problem of recovering unknown spatially
dependent parameters in nonlinear evolution PDEs. We propose a bi-level
Landweber scheme, where the upper-level parameter reconstruction embeds a
lower-level state approximation. This can be seen as combining the classical
reduced setting and the newer all-at-once setting, allowing us to,
respectively, utilize well-posedness of the parameter-to-state map, and to
bypass having to solve nonlinear PDEs exactly. Using this, we derive stopping
rules for lower- and upper-level iterations and convergence of the bi-level
method. We discuss application to parameter identification for the
Landau-Lifshitz-Gilbert equation in magnetic particle imaging.
