  Accurate and robust wavefront reconstruction methods for pyramid wavefront
sensors are in high demand as these sensors are planned to be part of many
instruments currently under development for ground based telescopes. The
pyramid sensor relates the incoming wavefront and its measurements in a
non-linear way. Nevertheless, almost all existing reconstruction algorithms are
based on a linearization of the model. The assumption of a linear pyramid
sensor response is justifiable in closed loop AO when the measured phase
information is small but may not be reasonable in reality due to unpreventable
errors depending on the system such as non common path aberrations. In order to
solve the non-linear inverse problem of wavefront reconstruction from pyramid
sensor data we introduce two new methods based on the non-linear Landweber and
Landweber-Kaczmarz iteration. Using these algorithms we experience high-quality
wavefront estimation especially for the non-modulated sensor by still keeping
the numerical effort feasible for large-scale AO systems.
