Currently there is a new generation of large astronomical telescope under construction, e.g. the European Extremely Large Telescope (E-ELT) of the European Southern Observatory (ESO) with a mirror diameter of 39 meters or the Thirty Meter Telescope (TMT), build by a consortium headed by Caltech. The operation of those huge telescopes require new mathematical methods in particular for the Adaptive Optics systems of the telescopes.
The image quality of ground based astronomical telescopes suffers from turbulences in the atmosphere. Adaptive Optics (AO) systems use wavefront sensor measurements of incoming light from guide stars to determine an optimal shape of deformable mirrors (DM) such that the image of the scientific object is corrected after reflection on the DM. The solution of this task involves several inverse problems: First, the incoming wavefronts have to be reconstructed from wavefront sensor measurements. The next step involves the solution of the Atmospheric Tomography problem, i.e., the reconstruction of the turbulence profile in the atmosphere. Finally, the optimal shape of the mirrors has to be determined. As the atmosphere changes frequently, these computations have to be done in real time. In the talk we introduce mathematical models for the elements of different Adaptive Optics system such as Single Conjugate Adaptive Optics (SCAO) or Multi Conjugate Adaptive Optics (MCAO) and present fast reconstruction algorithms as well as related numerical results for each of the sub-tasks that achieve the accuracy and speed required for the operation of ELTs.