The ambitious astrophysical objectives of the Extremely Large Telescopes (ELTs) will be achievable only with innovative Adaptive Optics (AO) systems to correct for the wavefront distortions induced by the turbulence in the atmosphere. One of the key components of an AO system is the wavefront reconstruction, which is a real-time estimate of the wavefront distortions above the telescope aperture from data. This reconstruction can be described by an inverse problem approach (IPA),taking advantage of the modeling of second-order statistics of both turbulence and data noise.
First, the benefits of the IPA to wavefront reconstruction is enhanced for two particularities of the ELTs: very high number of estimated parameters (~104) and elongated spots on the sensor for AO using Laser Guide Stars. Moreover, this IPA can be implemented with a fast algorithm for high number of degrees of freedom, which makes it a candidate for the implementation on a future AO system of the E-ELT. The correction performance in closed-loop AO has also been assessed thanks to end-to-end simulations of single-conjugate AO and Ground-Layer AO with Laser Guide Stars on the E-ELT.