A scalable sparse minimum-variance open-loop wave-front reconstructor for extreme adaptive optics (ExAO) systems is presented. The reconstructor is based on Ellerbroek’s sparse approximation of the wave-front inverse covariance matrix [J. Opt. Soc. Am. A 19, 1803 (2002)]. The baseline of the numerical approach is an iterative conjugate gradient (CG) algorithm for reconstructing a spatially sampled wave front at grid points on a computational domain of size equal to the telescope’s primary mirror’s diameter that uses a multigrid (MG) accelerator to speed up convergence efficiently and enhance its robustness. The combined MGCG scheme is order and requires only two CG iterations to converge to the asymptotic average Strehl ratio (SR) and root-mean-square reconstruction error. The SR and reconstruction squared error are within standard deviation with figures obtained from a previously proposed MGCG fast-Fourier-transform based minimum-variance reconstructor that incorporates the exact wave-front inverse covariance matrix on a computational domain of size equal to . A cost comparison between the present sparse MGCG algorithm and a Cholesky factorization based algorithm that uses a reordering scheme to preserve sparsity indicates that the latter method is still competitive for real-time ExAO wave-front reconstruction for systems with up to degrees of freedom because the update rate of the Cholesky factor is typically several orders of magnitude lower than the temporal sampling rate.
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