Multigrid preconditioned conjugate-gradient method for large-scale wave-front reconstruction

Gilles, Luc; Vogel, Curtis R.; Ellerbroek, Brent L.

doi:10.1364/josaa.19.001817

Cited by 77 publications

(65 citation statements)

References 3 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Sparse matrix factorization methods [2] have been used with a conjugate-gradient iterative scheme paired with either a multigrid (MG) [3,4] or a Fourier [5] preconditioner. This provides convergence only in a small number of iterations.…”

Section: Introductionmentioning

confidence: 99%

Warm-started wavefront reconstruction for adaptive optics

et al. 2008

View full text Add to dashboard Cite

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 128ϫ 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.

show abstract

Section: Introductionmentioning

confidence: 99%

Warm-started wavefront reconstruction for adaptive optics

et al. 2008

View full text Add to dashboard Cite

show abstract

“…However, the local nature of the simplex B-spline basis polynomials allows the definition of a D-SABRE algorithm that can theoretically obtain a computational complexity of ON 2 ∕G 2 when a total of G parallel processors are used. Additionally, the sparse matrix methods from [8,9] and the multigrid preconditioned conjugate-gradient methods from [10,11] can be used with the SABRE to further increase computational efficiency. Two numerical experiments are conducted using a Fourier optics based simulation of a SH lenslet array.…”

Section: Discussionmentioning

confidence: 99%

“…Fueled by the development of a new generation of extremely large optical telescopes, innovations in FD methods are mainly focused on increasing their computational efficiency [6,7]. Examples of such innovations are a computationally efficient sparse matrix inversion method by Ellerbroek [8] and Vogel [9] and a multigrid preconditioned conjugate-gradient method by Gilles et al [10] and Vogel and Yang [11]. More recently, Rosensteiner presented a cumulative reconstruction method based on line integrals that further reduces computational complexity [12].…”

Section: Introductionmentioning

confidence: 99%

Wavefront reconstruction in adaptive optics systems using nonlinear multivariate splines

Visser

Verhaegen

2012

J. Opt. Soc. Am. A

View full text Add to dashboard Cite

This paper presents a new method for zonal wavefront reconstruction (WFR) with application to adaptive optics systems. This new method, indicated as Spline based ABerration REconstruction (SABRE), uses bivariate simplex B-spline basis functions to reconstruct the wavefront using local wavefront slope measurements. The SABRE enables WFR on nonrectangular and partly obscured sensor grids and is not subject to the waffle mode. The performance of SABRE is compared to that of the finite difference (FD) method in numerical experiments using data from a simulated Shack-Hartmann lenslet array. The results show that SABRE offers superior reconstruction accuracy and noise rejection capabilities compared to the FD method.

show abstract

“…Many faster methods have been proposed and analyzed using computer simulations. Examples include the conjugate-gradient (CG) [1], Fourier-domain (FD) [2], blended FD-CG [3], and sparse methods [4]. Recently, it was shown through simulation that a single-iteration multigrid (SIMG) method is as effective as CG methods for both multiconjugate [5] and single-conjugate adaptive optics (MCAO and SCAO, respectively) [6].…”

mentioning

confidence: 99%

“…It is worth noting that this simple trick destroys the block-toeplitz with toeplitz block structure of G T G. Although this does not affect the SIMG in any way, methods such as FD reconstruction [2] or Fourierbased preconditioning [11] rely on a shift-invariant structure. To adapt to a circular aperture they must either use a heuristic to correct for edge effects [2] or use an enlarged computational domain [1,11]. No special provisions were made to account for the central obscured region; these measurements are simply zeroed.…”

mentioning

confidence: 99%

Experimental validation of single-iteration multigrid wavefront reconstruction at the Palomar Observatory

et al. 2008

View full text Add to dashboard Cite

Single-iteration multigrid (SIMG) wavefront reconstruction schemes were implemented and validated on the adaptive optics system at the Hale 5.1 m telescope at the Palomar Observatory. Results indicate that even the simplest such method produces a performance indistinguishable from that of the standard leastsquares reconstructor for both bright and dim guide stars. SIMG provides a dramatic reduction in computational cost when compared to vector-matrix multiplication and can be implemented in parallel, making it the obvious choice for reconstruction in future large-scale adaptive optics systems. [6]. In this Letter, we detail the experimental validation of this work on a real SCAO system.Two computationally efficient methods that have previously been tested on-sky are the FD method [7] and a sparse method [4]. These methods are O͑n log n͒. The SIMG is O͑n͒ and shows no performance degradation when compared to the leastsquares reconstructor.A good model for the wavefront sensor (WFS) is [6,8] where x is the wavefront phase, y are the measurements, v is white noise, and G is a sparse influence matrix. The least-squares reconstruction matrix is found by taking the pseudoinverse. In practice, we can compute it by evaluating K = ͑G T G + ⑀I͒ −1 G T for a small ⑀. This ensures that unobservable modes such as piston and waffle are zeroed out. The SIMG method [6] uses a single multigrid sweep with an initial guess of 0 to obtain an approximate solution to the equationIf the measurements are taken in open-loop, x is the wavefront phase. In this case, x 0 = 0 is a bad guess, so multiple iterations are required to achieve acceptable convergence [6]. However, when we operate in closedloop, x is the change in wavefront phase between successive time steps, and only one iteration is required. Our tests were performed on the Palomar Adaptive Optics (PALAO) system on the Hale 5.1 m telescope [9]. The PALAO system has a deformable mirror (DM) with 241 active actuators and a ShackHartmann WFS array with 256 subapertures, producing a total of 512 measurements. The DM and WFS are aligned in a Fried geometry.The AO system collects measurements y at up to 2 kHz. Tip and tilt are removed from the wavefront using a fast-steering mirror (FSM) and proportionalintegral (PI) controller. The rest of the wavefront offset x is reconstructed by VMM: x = Ky. This estimate is fed back through a second PI loop to the DM. The closed-loop corrected wavefront is split using a dichroic mirror. The near-infrared portion is sent to the Palomar High Angle Resolution Observer (PHARO) [10]. See Fig. 1.To implement the SIMG on the PALAO system, virtual actuators were added to fill the 17ϫ 17 grid containing the circular actuator arrangement, and the influence matrix G was augmented with zeros appropriately, as in [3]. The new system is y = Ḡ x, where Ḡ is 512ϫ 289 instead of 512ϫ 241. This new system is September 15, 2008 / Vol. 33, No. 18 / OPTICS LETTERS 2047 0146-9592/08/182047-3/$15.00

show abstract

Multigrid preconditioned conjugate-gradient method for large-scale wave-front reconstruction

Cited by 77 publications

References 3 publications

Warm-started wavefront reconstruction for adaptive optics

Warm-started wavefront reconstruction for adaptive optics

Wavefront reconstruction in adaptive optics systems using nonlinear multivariate splines

Experimental validation of single-iteration multigrid wavefront reconstruction at the Palomar Observatory

Contact Info

Product

Resources

About