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Multi-object adaptive optics (MOAO) is a novel adaptive optics (AO) technique for wide-field multi-object spectrographs (MOS). MOAO aims at applying dedicated wavefront corrections to numerous separated tiny patches spread over a large field of view (FOV), limited only by that of the telescope. The control of each deformable mirror (DM) is done individually using a tomographic reconstruction of the phase based on measurements from a number of wavefront sensors (WFS) pointing at natural and artificial guide stars in the field. We have developed a novel hybrid, pseudo-analytical simulation scheme, somewhere in between the end-to- end and purely analytical approaches, that allows us to simulate in detail the tomographic problem as well as noise and aliasing with a high fidelity, and including fitting and bandwidth errors thanks to a Fourier-based code. Our tomographic approach is based on the computation of the minimum mean square error (MMSE) reconstructor, from which we derive numerically the covariance matrix of the tomographic error, including aliasing and propagated noise. We are then able to simulate the point-spread function (PSF) associated to this covariance matrix of the residuals, like in PSF reconstruction algorithms. The advantage of our approach is that we compute the same tomographic reconstructor that would be computed when operating the real instrument, so that our developments open the way for a future on-sky implementation of the tomographic control, plus the joint PSF and performance estimation. The main challenge resides in the computation of the tomographic reconstructor which involves the inversion of a large matrix (typically 40 000 × 40 000 elements). To perform this computation efficiently, we chose an optimized approach based on the use of GPUs as accelerators and using an optimized linear algebra library: MORSE providing a significant speedup against standard CPU oriented libraries such as Intel MKL. Because the covariance matrix is symmetric, several optimization schemes can be envisioned to speedup even further the computation. Optimizing the speed of the reconstructor computation is of major interest not only for the design study of MOAO instruments, but also for future routine operations of the system as the reconstructor has to be updated regularly to cope for atmospheric variability.
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