Modern ground-based telescopes rely on Adaptive Optics (AO) systems, which compensate for atmospheric turbulences in order to enhance the image quality. The hardware-based technology AO uses measurements of incoming wavefronts from bright light sources and artificially created laser guide stars to reconstruct the turbulence above the telescope and adjusts quickly moving deformable mirrors accordingly. For the up-coming generation of Extremely Large Telescopes (ELTs) with diameters up to 40 m, the computational effort of this technology is strongly increasing as real-time correction at a frequency of around 500 Hertz has to be performed. This results from the fact that the atmospheric turbulences are constant for approximately only 2 ms. Thus the main challenge is having a fast enough algorithm for deriving the shape of the deformable mirror from the measurements. Even though AO correction is used, the quality of astronomical images still is degraded due to the time delay stemming from the wavefront sensor integration time and adjustment of the deformable mirror(s). This results in a blur which can be mathematically described by a convolution of the original image with the point spread function (PSF). The PSF of an astronomical image varies with the position in the observed field, which is a crucial aspect on ELTs.
In this thesis, we focus on two challenges in AO based observations: First, we present new algorithms for the control of modern AO systems, in particular Single Conjugate Adaptive Optics and Ground Layer Adaptive Optics. Our focus is on matrix-free-approaches, taking into account the specific geometry of the system and known statistical properties. Real-life effects, such as spot elongation for laser guide stars and the statistics of the turbulent atmosphere are included in our model and, where possible, also in our reconstruction algorithms. We present results obtained in the end-to-end simulation environment from the European Southern Observatory, OCTOPUS, and show that our methods obtain comparable quality while reducing the computational time significantly compared to established methods. Second, we turn to post-processing and present approaches for PSF reconstruction from wavefront sensor measurements for Single Conjugate Adaptive Optics and Multi-Conjugate Adaptive Optics combining atmospheric tomography and techniques for PSF reconstruction. Existing, verified techniques are fused together in a novel way to deliver accurate field dependent PSFs in very short time. Even though simulation results in OCTOPUS suggest a good agreement between the true and the reconstructed PSF, the reconstructed PSF will never be completely accurate due to the coarse resolution of the wavefront sensor and especially telescope specific effects, such as non common path aberrations. However, the quality of the reconstructed PSF as well as of the observed image can be further improved by using blind deconvolution methods. We opt for a Lanczos-based blind deconvolution scheme to get a fast deconvolution algorithm based on a sparse system. This method is tested to recover information from blurred star images.