Abstract

Future extreme adaptive optics (ExAO) systems have been suggested with up to 105 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 128×128 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.

© 2008 Optical Society of America

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2007 (1)

2006 (2)

2005 (3)

2004 (2)

2003 (4)

D. G. MacMartin, “Local, hierarchic, and iterative reconstructors for adaptive optics,” J. Opt. Soc. Am. A 20, 1084-1093 (2003).
[CrossRef]

L. Gilles, “Order-N sparse minimum-variance open-loop reconstructor for extreme adaptive optics,” Opt. Lett. 28, 1927-1929 (2003).
[CrossRef] [PubMed]

B. L. Ellerbroek and C. R. Vogel, “Simulations of closed-loop wavefront reconstruction for multiconjugate adaptive optics on giant telescopes,” Proc. SPIE 5169, 206-217 (2003).
[CrossRef]

D. T. Gavel and D. Wiberg, “Toward Strehl-optimizing adaptive optics controllers,” Proc. SPIE 4839, 890-901 (2003).
[CrossRef]

2002 (3)

2000 (1)

B. L. Ellerbroek and F. J. Rigaut, “Scaling multiconjugate adaptive optics performance estimates to extremely large telescopes,” Proc. SPIE 4007, 1088-1099 (2000).
[CrossRef]

1993 (1)

1983 (1)

Anderson, D. J.

Brase, J. M.

Britton, M.

Britton, M. C.

M. C. Britton, “Arroyo,” Proc. SPIE 5497, 290-300 (2004).
[CrossRef]

M. C. Britton, “Arroyo C++ library: object oriented class libraries for the simulation of electromagnetic wave propagation through turbulence,” http://eraserhead.caltech.edu/arroyo/arroyo.html (April 8, 2008).

Conan, J.-M.

Dekany, R.

Ellerbroek, B.

L. Gilles, B. Ellerbroek, and C. Vogel, “A comparison of multigrid V-cycle versus Fourier domain preconditioning for laser guide star atmospheric tomography,” in Adaptive Optics: Analysis and Methods/Computational Optical Sensing and Imaging/Information Photonics/Signal Recovery and Synthesis Topical Meetings on CD-ROM, OSA Technical Digest (CD) (Optical Society of America, 2007), paper JTuA1.
[PubMed]

Ellerbroek, B. L.

Fusco, T.

Gavel, D. T.

Gilles, L.

L. Gilles, “Closed-loop stability and performance analysis of least-squares and minimum-variance control algorithms for multiconjugate adaptive optics,” Appl. Opt. 44, 993-1002 (2005).
[CrossRef] [PubMed]

L. Gilles, “Order-N sparse minimum-variance open-loop reconstructor for extreme adaptive optics,” Opt. Lett. 28, 1927-1929 (2003).
[CrossRef] [PubMed]

L. Gilles, C. R. Vogel, and B. L. Ellerbroek, “Multigrid preconditioned conjugate-gradient method for large-scale wave-front reconstruction,” J. Opt. Soc. Am. A 19, 1817-1822 (2002).
[CrossRef]

L. Gilles, B. Ellerbroek, and C. Vogel, “A comparison of multigrid V-cycle versus Fourier domain preconditioning for laser guide star atmospheric tomography,” in Adaptive Optics: Analysis and Methods/Computational Optical Sensing and Imaging/Information Photonics/Signal Recovery and Synthesis Topical Meetings on CD-ROM, OSA Technical Digest (CD) (Optical Society of America, 2007), paper JTuA1.
[PubMed]

Hardy, J. W.

J. W. Hardy, Adaptive Optics for Astronomical Telescopes (Oxford U. Press, 1998).

Kulcsár, C.

Le Roux, B.

Macintosh, B. A.

MacMartin, D. G.

Mugnier, L. M.

Oosterlee, C. W.

U. Trottenberg, A. Schüller, and C. W. Oosterlee, Multigrid Methods (Academic, 2000).

Paschall, R. N.

Piatrou, P.

P. Piatrou and M. Roggemann, “Performance analysis of Kalman filter and minimum variance controllers for multiconjugate adaptive optics,” Proc. SPIE 5894, 288-296 (2005).

Poyneer, L. A.

Raynaud, H.-F.

Ren, H.

Rigaut, F. J.

B. L. Ellerbroek and F. J. Rigaut, “Scaling multiconjugate adaptive optics performance estimates to extremely large telescopes,” Proc. SPIE 4007, 1088-1099 (2000).
[CrossRef]

Roggemann, M.

P. Piatrou and M. Roggemann, “Performance analysis of Kalman filter and minimum variance controllers for multiconjugate adaptive optics,” Proc. SPIE 5894, 288-296 (2005).

Schüller, A.

U. Trottenberg, A. Schüller, and C. W. Oosterlee, Multigrid Methods (Academic, 2000).

Tatarskii, V. I.

V. I. Tatarskii, Wave Propagation in a Turbulent Medium (McGraw-Hill, 1961).

Trottenberg, U.

U. Trottenberg, A. Schüller, and C. W. Oosterlee, Multigrid Methods (Academic, 2000).

Véran, J.-P.

Vogel, C.

L. Gilles, B. Ellerbroek, and C. Vogel, “A comparison of multigrid V-cycle versus Fourier domain preconditioning for laser guide star atmospheric tomography,” in Adaptive Optics: Analysis and Methods/Computational Optical Sensing and Imaging/Information Photonics/Signal Recovery and Synthesis Topical Meetings on CD-ROM, OSA Technical Digest (CD) (Optical Society of America, 2007), paper JTuA1.
[PubMed]

Vogel, C. R.

Wallner, E. P.

Wiberg, D.

D. T. Gavel and D. Wiberg, “Toward Strehl-optimizing adaptive optics controllers,” Proc. SPIE 4839, 890-901 (2003).
[CrossRef]

Yang, Q.

Appl. Opt. (5)

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (6)

Opt. Lett. (1)

Proc. SPIE (5)

D. T. Gavel and D. Wiberg, “Toward Strehl-optimizing adaptive optics controllers,” Proc. SPIE 4839, 890-901 (2003).
[CrossRef]

P. Piatrou and M. Roggemann, “Performance analysis of Kalman filter and minimum variance controllers for multiconjugate adaptive optics,” Proc. SPIE 5894, 288-296 (2005).

B. L. Ellerbroek and C. R. Vogel, “Simulations of closed-loop wavefront reconstruction for multiconjugate adaptive optics on giant telescopes,” Proc. SPIE 5169, 206-217 (2003).
[CrossRef]

M. C. Britton, “Arroyo,” Proc. SPIE 5497, 290-300 (2004).
[CrossRef]

B. L. Ellerbroek and F. J. Rigaut, “Scaling multiconjugate adaptive optics performance estimates to extremely large telescopes,” Proc. SPIE 4007, 1088-1099 (2000).
[CrossRef]

Other (5)

L. Gilles, B. Ellerbroek, and C. Vogel, “A comparison of multigrid V-cycle versus Fourier domain preconditioning for laser guide star atmospheric tomography,” in Adaptive Optics: Analysis and Methods/Computational Optical Sensing and Imaging/Information Photonics/Signal Recovery and Synthesis Topical Meetings on CD-ROM, OSA Technical Digest (CD) (Optical Society of America, 2007), paper JTuA1.
[PubMed]

J. W. Hardy, Adaptive Optics for Astronomical Telescopes (Oxford U. Press, 1998).

V. I. Tatarskii, Wave Propagation in a Turbulent Medium (McGraw-Hill, 1961).

U. Trottenberg, A. Schüller, and C. W. Oosterlee, Multigrid Methods (Academic, 2000).

M. C. Britton, “Arroyo C++ library: object oriented class libraries for the simulation of electromagnetic wave propagation through turbulence,” http://eraserhead.caltech.edu/arroyo/arroyo.html (April 8, 2008).

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Figures (7)

Fig. 1
Fig. 1

Sensor ( s i j ) and actuator ( a i j ) arrangement for a Fried geometry. We measure a noisy gradient of the phase at s i j , and the goal is to estimate the phase at a i j .

Fig. 2
Fig. 2

Error measures for least-squares reconstruction using a guide star of magnitude M = 8 and the seeing conditions from Subsection 5A. The exact inverse was computed and averaged over 1000 independent runs. In these conditions, the optimal sensor sampling rate is about 417 Hz , and the minimum expected relative error is just under 10 4 .

Fig. 3
Fig. 3

SNR variation for various guide star magnitudes M as a function of the sensor sampling rate.

Fig. 4
Fig. 4

Bandwidth-error trade-off curve for various star magnitude values ( M ) using least-squares and minimum-variance reconstructions. The SNR at the minimum points are 26.2 for M = 5 , 9.7 for M = 8 , and 4.5 for M = 10 . We averaged 1000 independent runs.

Fig. 5
Fig. 5

Convergence plots comparing simple MG, with conjugate-gradient methods using either a MG-PCG or FD-PCG preconditioner. MG methods use Gauss–Seidel (GS) or Jacobi (J) smoothers. The pair ( ν 1 , ν 2 ) is the number of presmoothing and postsmoothing steps. Most methods converge in a few iterations with comparable computational effort. We averaged 1000 independent runs.

Fig. 6
Fig. 6

Plots comparing converged values of various methods using the open-loop warm-start technique. For every method except MG-J(1,0) and FD-PCG, only one iteration per measurement is required for minimum error. The best method to choose is simply the one that has the smallest iteration cost. We averaged 1000 independent runs.

Fig. 7
Fig. 7

Closed-loop time series in cold start (top), warm start (middle), and warm-started POLC (bottom); 500 time steps were simulated, performing one iteration per time step. We tested 15 different iterative schemes (only 4 are plotted). All were stable and performed equally well, with the exception of MG-J(1,0) and FD-PCG in the cold-start standard case.

Tables (3)

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Table 1 List of Iterative Schemes a

Tables Icon

Table 2 Cost Comparison (Iterating to Convergence) a

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Table 3 Cost Comparison (Fewest Possible Iterations) a

Equations (24)

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y ( s 12 ) = 1 2 [ ϕ ( a 13 ) ϕ ( a 12 ) + ϕ ( a 23 ) ϕ ( a 22 ) ϕ ( a 13 ) + ϕ ( a 12 ) ϕ ( a 23 ) ϕ ( a 22 ) ] + [ v 1 v 2 ] .
y = G ϕ + v ,
y = G x + v .
σ 2 = ( 3 π 2 K g 8 ) 2 n ph + n bg + N D σ e 2 n ph 2 ,
n ph = η μ ph A b , n bg = η μ bg A b ,
σ 2 = ( 3 π 2 K g 8 ) 2 ( η ( μ ph + μ bg ) A b + N D σ e 2 ) ( b η μ ph A ) 2 .
y = G x + v .
minimize y G x ̂ 2 subject to V T x ̂ = 0 .
x ̂ = ( G T G + V V T ) 1 G T y .
( G T G + V V T ) x ̂ = G T y .
C 1 L L T ,
minimize E ( x x ̂ 2 y ) subject to V T x ̂ = 0 ,
x ̂ = ( G T G + σ 2 C 1 + V V T ) 1 G T y .
( G T G + σ 2 L L T + V V T ) x ̂ = G T y .
μ = 0.9405 × 10 10 0.4 M ,
y 1 ( i ) = G x 1 ( i ) + v 1 ( i ) ,
x ̂ 1 ( i ) = A 1 G T y 1 ( i ) .
1. Lag error: 1 M i = 1 M x 1 ( i ) x 2 ( i ) 2 1 M i = 1 M x 2 ( i ) 2 ,
2. Noise error: 1 M i = 1 M x ̂ 1 ( i ) x 1 ( i ) 2 1 M i = 1 M x 2 ( i ) 2 ,
3. Total error: 1 M i = 1 M x ̂ 1 ( i ) x 2 ( i ) 2 1 M i = 1 M x 2 ( i ) 2 .
SNR = ( E G x 2 E v 2 ) 1 2 .
y t = G ( x t u t 1 ) + v t , y t = G ( x t u t 1 ) + v t ,
e ̂ t = K y t , x ̂ t = K ( y t + G u t 1 ) ,
u t + 1 = u t + β e ̂ t , u t + 1 = u t + β ( x ̂ t u t 1 ) .

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