  The Multi-conjugate Adaptive Optics RelaY (MAORY) is one of the key Adaptive
Optics (AO) systems on the European Southern Observatory's Extremely Large
Telescope. MAORY aims to achieve a good wavefront correction over a large field
of view, which involves a tomographic estimation of the 3D atmospheric
wavefront disturbance. Mathematically, the reconstruction of turbulent layers
in the atmosphere is severely ill-posed, hence, limits the achievable
reconstruction accuracy. Moreover, the reconstruction has to be performed in
real-time at a few hundred to one thousand Hertz frame rates. Huge amounts of
data have to be processed and thousands of actuators of the deformable mirrors
have to be controlled by elaborated algorithms. Even with extensive
parallelization and pipelining, direct solvers, such as the Matrix Vector
Multiplication (MVM) method, are extremely demanding. Thus, research in the
last years shifted into the direction of iterative methods. In this paper we
focus on the iterative Finite Element Wavelet Hybrid Algorithm (FEWHA). The key
feature of FEWHA is a matrix-free representation of all operators involved,
which makes the algorithm fast and enables on the fly system updates whenever
parameters at the telescope or in the atmosphere change. We provide a
performance analysis of the method regarding quality and run-time for the MAORY
instrument using the AO software package COMPASS.
