Warm-started wavefront reconstruction for adaptive optics

Laurent Lessard; Matthew West; Douglas MacMynowski; Sanjay Lall

doi:10.1364/JOSAA.25.001147

Journal of the Optical Society of America A
Vol. 25,
Issue 5,
pp. 1147-1155
(2008)
•https://doi.org/10.1364/JOSAA.25.001147

Warm-started wavefront reconstruction for adaptive optics

Laurent Lessard, Matthew West, Douglas MacMynowski, and Sanjay Lall

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Laurent Lessard, Matthew West, Douglas MacMynowski, and Sanjay Lall, "Warm-started wavefront reconstruction for adaptive optics," J. Opt. Soc. Am. A 25, 1147-1155 (2008)

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Abstract

Future extreme adaptive optics (ExAO) systems have been suggested with up to $10^{5}$ sensors and actuators. We analyze the computational speed of iterative reconstruction algorithms for such large systems. We compare a total of 15 different scalable methods, including multigrid, preconditioned conjugate-gradient, and several new variants of these. Simulations on a $128 \times 128$ square sensor/actuator geometry using Taylor frozen-flow dynamics are carried out using both open-loop and closed-loop measurements, and algorithms are compared on a basis of the mean squared error and floating-point multiplications required. We also investigate the use of warm starting, where the most recent estimate is used to initialize the iterative scheme. In open-loop estimation or pseudo-open-loop control, warm starting provides a significant computational speedup; almost every algorithm tested converges in one iteration. In a standard closed-loop implementation, using a single iteration per time step, most algorithms give the minimum error even in cold start, and every algorithm gives the minimum error if warm started. The best algorithm is therefore the one with the smallest computational cost per iteration, not necessarily the one with the best quasi-static performance.

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Number	Type	Smoother	Cycle	Cost per Iteration
1	MG	GS(1,0)	V	$3.406 \times 10^{5}$
2	MG	GS(1,1)	V	$4.723 \times 10^{5}$
3	MG	GS(2,2)	V	$7.358 \times 10^{5}$
4	MG	J(1,0)	V	$3.406 \times 10^{5}$
5	MG	J(1,1)	V	$4.723 \times 10^{5}$
6	MG	J(2,2)	V	$7.358 \times 10^{5}$
7	MG	GS(1,0)	W	$5.050 \times 10^{5}$
8	MG	GS(1,1)	W	$7.003 \times 10^{5}$
9	MG-PCG	SGS(1,1)	V	$6.377 \times 10^{5}$
10	MG-PCG	SGS(2,2)	V	$9.012 \times 10^{5}$
11	MG-PCG	J(1,1)	V	$6.377 \times 10^{5}$
12	MG-PCG	J(2,2)	V	$9.012 \times 10^{5}$
13	MG-PCG	J(1,1)	W	$8.657 \times 10^{5}$
14	MG-PCG	SGS(1,1)	W	$8.657 \times 10^{5}$
15	FD-PCG	—	—	$12.984 \times 10^{5}$

Iterative Scheme	Iteration Cost (Multiplications)	Iterations to Convergence	Total Cost (Multiplications)
FD-PCG	$1.30 \times 10^{6}$	9	$1.17 \times 10^{7}$
MG, J(1,0)	$3.41 \times 10^{5}$	6	$2.04 \times 10^{6}$
MG-PCG, J(2,2)	$9.01 \times 10^{5}$	2	$1.80 \times 10^{6}$
MG, GS(2,2)	$7.36 \times 10^{5}$	2	$1.47 \times 10^{6}$
MG, GS(1,1)	$4.72 \times 10^{5}$	3	$1.42 \times 10^{6}$
MG, GS(1,0)	$3.41 \times 10^{5}$	4	$1.36 \times 10^{6}$
MG-PCG, J(1,1)	$6.38 \times 10^{5}$	2	$1.27 \times 10^{6}$
MG, GS(1,0), W-cycle	$5.05 \times 10^{5}$	2	$1.01 \times 10^{6}$

Iterative Scheme	Iteration Cost (Multiplications)	Iterations per Time Step	Total Cost (Multiplications)
FD-PCG	$1.30 \times 10^{6}$	2	$2.60 \times 10^{6}$
MG-PCG, J(2,2)	$9.01 \times 10^{5}$	1	$9.01 \times 10^{5}$
MG, GS(2,2)	$7.36 \times 10^{5}$	1	$7.36 \times 10^{5}$
MG, J(1,0)	$3.41 \times 10^{5}$	2	$6.81 \times 10^{5}$
MG-PCG, J(1,1)	$6.38 \times 10^{5}$	1	$6.38 \times 10^{5}$
MG, GS(1,0), W-cycle	$5.05 \times 10^{5}$	1	$5.05 \times 10^{5}$
MG, GS(1,1)	$4.72 \times 10^{5}$	1	$4.72 \times 10^{5}$
MG, GS(1,0)	$3.41 \times 10^{5}$	1	$3.41 \times 10^{5}$

Number	Type	Smoother	Cycle	Cost per Iteration
1	MG	GS(1,0)	V	$3.406 \times 10^{5}$
2	MG	GS(1,1)	V	$4.723 \times 10^{5}$
3	MG	GS(2,2)	V	$7.358 \times 10^{5}$
4	MG	J(1,0)	V	$3.406 \times 10^{5}$
5	MG	J(1,1)	V	$4.723 \times 10^{5}$
6	MG	J(2,2)	V	$7.358 \times 10^{5}$
7	MG	GS(1,0)	W	$5.050 \times 10^{5}$
8	MG	GS(1,1)	W	$7.003 \times 10^{5}$
9	MG-PCG	SGS(1,1)	V	$6.377 \times 10^{5}$
10	MG-PCG	SGS(2,2)	V	$9.012 \times 10^{5}$
11	MG-PCG	J(1,1)	V	$6.377 \times 10^{5}$
12	MG-PCG	J(2,2)	V	$9.012 \times 10^{5}$
13	MG-PCG	J(1,1)	W	$8.657 \times 10^{5}$
14	MG-PCG	SGS(1,1)	W	$8.657 \times 10^{5}$
15	FD-PCG	—	—	$12.984 \times 10^{5}$

Iterative Scheme	Iteration Cost (Multiplications)	Iterations to Convergence	Total Cost (Multiplications)
FD-PCG	$1.30 \times 10^{6}$	9	$1.17 \times 10^{7}$
MG, J(1,0)	$3.41 \times 10^{5}$	6	$2.04 \times 10^{6}$
MG-PCG, J(2,2)	$9.01 \times 10^{5}$	2	$1.80 \times 10^{6}$
MG, GS(2,2)	$7.36 \times 10^{5}$	2	$1.47 \times 10^{6}$
MG, GS(1,1)	$4.72 \times 10^{5}$	3	$1.42 \times 10^{6}$
MG, GS(1,0)	$3.41 \times 10^{5}$	4	$1.36 \times 10^{6}$
MG-PCG, J(1,1)	$6.38 \times 10^{5}$	2	$1.27 \times 10^{6}$
MG, GS(1,0), W-cycle	$5.05 \times 10^{5}$	2	$1.01 \times 10^{6}$

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Equations (24)

Journal of the Optical Society of America A