**Efficient Computation of Minimum Variance Wavefront ****Reconstructors Using Sparse Matrix Techniques**

Brent L. Ellerbroek, *Gemini Observatory *

The complexity of computing conventional matrix multiply wavefront reconstructors scales as *O*(*n*^{3}) for most adaptive optical (A*O*) systems, where *n* is the number of deformable mirror (DM) actuators. This is impractical for proposed systems with extremely large *n*. It is known that sparse matrix methods improve this scaling for least squares reconstructors, but sparse techniques are not immediately applicable to the minimum variance reconstructors now favored for multi-conjugate adaptive optics (MCAO) systems with multiple wave front sensors (WFS's) and DM's. Complications arise from the non-sparse statistics of atmospheric turbulence, and the global tip-tilt WFS measurement errors associated with laser guide star (LGS) position uncertainty. In this paper, we describe how sparse matrix methods can still be applied by use of a sparse approximation for turbulence statistics, and by recognizing that the non-sparse matrix terms arising from LGS position uncertainty are low rank adjustments that can be evaluated using the matrix inversion lemma. Sample numerical results for AO and MCAO systems illustrate that: The approximation made to turbulence statistics has negligible effect on estimation accuracy, the time to compute the sparse minimum variance reconstructor for a conventional natural guide star (NGS) AO system scales as *O*(*n*^{3/2}) and is only a few seconds for *n *= 3500_{1}, and sparse techniques reduce the reconstructor computations by a factor of 8 for sample MCAO systems with 2417 DM actuators and 4280 WFS subapertures. Extrapolating to 9700 actuators and 17120 subapertures, we predict a reduction by a factor of about 30 or 40 to 1. (c)2002 *Optical Society of America*.

Download the PDF file.

Virginia M. Smith / web@gemini.edu / April 30, 2002