Zeeman-Doppler imaging: old problems and new methods

Zeeman-Doppler Imaging (ZDI) is a powerful inversion method to reconstruct stellar magnetic surface fields. The reconstruction process is usually solved by translating the inverse problem into a regularized least-square or optimization problem. In this contribution we will emphasize that ZDI is an inherent non-linear problem and the corresponding regularized optimization is, like many non-linear problems, potentially prone to local minima. We show how this problem will be exacerbated by using an inadequate forward model. To facilitate a more consistent full radiative transfer driven approach to ZDI we describe a two-stage strategy that consist of a principal component analysis (PCA) based line profile reconstruction and a fast approximate polarized radiative transfer method to synthesize local Stokes profiles. Moreover, we introduce a novel statistical inversion method based on artificial neural networks (ANN) which provide a fast calculation of a first guess model and allows to incorporate better physical constraints into the inversion process.