The reconstruction of surface magnetic field distributions from time-resolved spectropolarimetric observations -- known as Zeeman-Doppler Imaging (ZDI) or Magnetic-Doppler Imaging (MDI) -- is an inherently non-linear and often ill-posed inverse problem. These kind of problems are most often addressed by translating them into regularized (least-square) minimization problems (i.e. parameter estimation). However, in contrast to linear inverse problems the regularized objective function is no longer strictly convex in the non-linear case, and gradient based methods are therefore prone to local minima.

We will provide some illustrative examples, based on synthetic as well as on real observed Stokes spectra, to show how the effect of local minima and insufficient data availability may lead to quite different surface distributions of the magnetic field vector (i.e. solutions) for one and the same underlying model. Particular emphasis will be put on the spurious appearance of azimuthal and meridional magnetic fields.

In an effort to provide more realism (e.g. incorporation of full polarized radiative transfer, simultaneous modeling of magnetic field and temperature distributions) we introduce our new ZDI code "iMap" which incorporates besides an "iterative" regularized conjugate gradient method a novel artificial neural network inversion which provides a smooth and continuous approximation of the desired inverse function (i.e. the inverse relation from the observed phase-resolved Stokes spectra to the magnetic field surface distribution).