  The Landau-Lifshitz-Gilbert equation yields a mathematical model to describe
the evolution of the magnetization of a magnetic material, particularly in
response to an external applied magnetic field. It allows one to take into
account various physical effects, such as the exchange within the magnetic
material itself. In particular, the Landau-Lifshitz-Gilbert equation encodes
relaxation effects, i.e., it describes the time-delayed alignment of the
magnetization field with an external magnetic field. These relaxation effects
are an important aspect in magnetic particle imaging, particularly in the
calibration process. In this article, we address the data-driven modeling of
the system function in magnetic particle imaging, where the
Landau-Lifshitz-Gilbert equation serves as the basic tool to include relaxation
effects in the model. We formulate the respective parameter identification
problem both in the all-at-once and the reduced setting, present reconstruction
algorithms that yield a regularized solution and discuss numerical experiments.
Apart from that, we propose a practical numerical solver to the nonlinear
Landau-Lifshitz-Gilbert equation, not via the classical finite element method,
but through solving only linear PDEs in an inverse problem framework.
