The technique of adaptive optics (AO) is widely used in earth-based large telescopes for compensation of perturbations of light stemming from astronomical objects due to atmospheric turbulence. It is based on measuring the deformations of optical wave-fronts emitted by natural or artificial guide stars via wave-front sensors and subsequent correction using deformable mirrors. The complex setup of such a system, especially the huge amount of data that has to be processed, as well as the real-time demands on performance lead to highly non-trivial requirements on the applied control algorithms, namely in the field of inverse problems. The contents of this thesis result from investigations engaged by the European Southern Observatory (ESO) in the context of development and construction of the European Extremely Large Telescope (EELT). Parts of this research were organized in the Austrian in-kind project "Mathematical algorithms and software for E-ELT Adaptive Optics", which the author of this thesis was a member of. We will give an overview of the various components of an AO system and their interaction, based on the modeling concept of Fourier optics, and subsequently summarize the approaches for control algorithms already known and new ones that were developed by the Austrian team. In particular the research of the author is described in detail, i.e.:
* Investigation of a new method for reconstruction of wave-fronts from measurements of a Shack- Hartmann sensor, based on the singular value decomposition (SVD) of the corresponding forward operator, which was presented in .
* For handling sensor obstruction, a new preprocessing algorithm called MPIM for extension of sensor measurements from active to obstructed sub-apertures is presented.
* Cooperation in development of the new simulation tool MOST as a memory-saving and fast alternative to OCTOPUS.
* Investigation of Strehl ratio evaluation within the established simulation tool OCTOPUS and acceleration, leading to the new tool MAOAM for very fast off-line simulation of MOAO systems.
* Studies about the dependence of tomography performance from chosen reconstruction layer profiles. Though the tests are restricted to an LTAO setup, the results are also relevant for MCAO and MOAO.
The major part of the author's work lies in the first two items, i.e., investigation of the SVD-based method for wave-front reconstruction. Especially symmetry and sparsity properties are found, thus leading to an implementation of acceptable runtime, which is not directly obvious from the rather theoretical representation of the method in . We will show that truncation of the singular system can be used to significantly accelerate the operation especially in the case of low photon ux from the guide star and accordingly increased sensor noise. This is due to the revealed strong interrelation between the singular system and the Fourier domain of the function space containing the wave-fronts. Finally, the implementation of the method is tested on MOST and OCTOPUS and compared to the performance of competitive approaches.