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 <b>19,</b> 1803 (2002)]. The baseline of the numerical approach is an iterative conjugate gradient (CG) algorithm for reconstructing a spatially sampled wave front at <i>N</i> grid points on a computational domain of size equal to the telescope’s primary mirror’s diameter <i>D</i> that uses a multigrid (MG) accelerator to speed up convergence efficiently and enhance its robustness. The combined MGCG scheme is order <i>N</i> 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 2<i>D</i> . 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 N ∼ 10<sup>4</sup> degrees of freedom because the update rate of the Cholesky factor is typically several orders of magnitude lower than the temporal sampling rate.
© 2003 Optical Society of America
L. Gilles, "Order-N sparse minimum-variance open-loop reconstructor for extreme adaptive optics," Opt. Lett. 28, 1927-1929 (2003)