Nesterov-accelerated adaptive momentum estimation-based wavefront distortion correction algorithm

Hui Zhao; Hui Zhao; Jing An; Jing An; Mengjie Yu; Mengjie Yu; Diankai Lv; Diankai Lv; Kaida Kuang; Kaida Kuang; Tianqi Zhang; Tianqi Zhang

doi:10.1364/AO.428465

Applied Optics
Vol. 60,
Issue 24,
pp. 7177-7185
(2021)
•https://doi.org/10.1364/AO.428465

Nesterov-accelerated adaptive momentum estimation-based wavefront distortion correction algorithm

Hui Zhao, Jing An, Mengjie Yu, Diankai Lv, Kaida Kuang, and Tianqi Zhang

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Hui Zhao, Jing An, Mengjie Yu, Diankai Lv, Kaida Kuang, and Tianqi Zhang, "Nesterov-accelerated adaptive momentum estimation-based wavefront distortion correction algorithm," Appl. Opt. 60, 7177-7185 (2021)

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Table of Contents Category
- Atmospheric and Oceanic Optics
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The topics in this list come from the Optics and Photonics Topics applied to this article.

About this Article
History
- Original Manuscript: April 21, 2021
- Revised Manuscript: July 20, 2021
- Manuscript Accepted: July 20, 2021
- Published: August 13, 2021

Abstract

In order to improve the wavefront distortion correction performance of the classical stochastic parallel gradient descent (SPGD) algorithm, an optimized algorithm based on Nesterov-accelerated adaptive momentum estimation is proposed. It adopts a modified second-order momentum and a linearly varying gain coefficient to improve iterative stability. It integrates the Nesterov momentum term and the modified Adam optimizer to further improve the convergence speed, correct the direction of gradient descent in a timely fashion, and avoid falling into local extremum. Besides, to demonstrate the algorithm’s performance, a wavefront sensorless adaptive optics system model is established using a ${{6}} \times {{6}}$ element deformable mirror as wavefront corrector. Simulation results show that, compared with the SPGD algorithm, the proposed algorithm converges faster, and its Strehl ratio after convergence is nearly 6.25 times that of the SPGD algorithm. Also, the effectiveness and superiority of the proposed algorithm are verified by comparing with two existing optimization algorithms.

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Data Availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

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Parameter Settings: Gain coefficients: $γ_{min} = 0.04$ and $γ_{max} = 0.09$ , the exponential decay rates for the moment estimates: $λ_{1} = 0.9$ and $λ_{2} = 0.999$ , the performance metric function $J = J [u_{1}, u_{2}, \dots, u_{N}]$ , and the maximal number of learning iterations $T = 800$ .
Ensure: The optimal parameters $[u_{1}, u_{2}, \dots, u_{N}]$ .
1.	$m_{0} \leftarrow 0$ ; $n_{0} \leftarrow 0$ ; $ε \leftarrow 10^{- 8}$ ; $[u_{1}^{0}, u_{2}^{0}, \dots, u_{N}^{0}] \leftarrow 0$ (Setting initial
values)
2.	Out loop: for $t = 1, \dots ., T$
3.	Randomly generate the values of $δ u_{1}, δ u_{2}, \dots, δ u_{N}$ based on
$\| δ u \| = 1$
4.	$δ J \leftarrow J (u_{1} + δ u_{1}, \dots, u_{j} + δ u_{j}, \dots, u_{N} + δ u_{N})$
$- J (u_{1} - δ u_{1}, \dots, u_{j} - δ u_{j}, \dots, u_{N} - δ u_{N})$
5.	Inner loop: for $j = 1, 2, \dots, N$
6.	$g_{j}^{t} \leftarrow δ J^{t} δ u_{_{j}}^{t}$
7.	$m_{j}^{t} \leftarrow λ_{1} m_{j}^{t - 1} + (1 - λ_{1}) \cdot g_{j}^{t}$
8.	$n_{j}^{t} \leftarrow λ_{2} n_{j}^{t - 1} + (1 - λ_{2}) \cdot (g_{j}^{t})^{2}$
9.	${\tilde{n}}_{j}^{t} \leftarrow max (λ_{2} n_{j}^{t - 1} + (1 - λ_{2}) \cdot {(g_{j}^{t})}^{2}, n_{j}^{t - 1})$
10.	${\hat{m}}_{j}^{t} \leftarrow m_{j}^{t} / [1 - (λ_{1})^{t}]$
11.	${\hat{n}}_{j}^{t} \leftarrow {\tilde{n}}_{j}^{t} / [1 - (λ_{2})^{t}]$
12.	$γ \leftarrow γ_{max} - (\frac{t - 1}{T - 1}) (γ_{max} - γ_{min})$
13.	$u_{j}^{t} \leftarrow u_{j}^{t - 1} + \frac{γ}{(\sqrt{{\hat{n}}_{j}^{t}} + ε)} (λ_{1} {\hat{m}}_{j}^{t} + \frac{(1 - λ_{1}) g_{j}^{t}}{1 - λ_{1}^{t}})$
14.	end for
15.	end for

	Algorithms
	Gain Coefficients
$D / r_{0}$	SPGD	MSPGD	ASPGD	NASPGD
5	0.9	0.9	0.09	0.04–0.09
8	2	2	0.09	0.04–0.09
10	5	5	0.09	0.04–0.09
12	30	30	0.09	0.04–0.09

Turbulence Intensity	Initial SR	SPGD	MSPGD	ASPGD	NASPGD
5	0.4604	0.8152	0.8149	0.7883	0.8154
8	0.1299	0.5648	0.5649	0.5349	0.5649
10	2.4e-04	0.2777	0.2999	0.2856	0.3060
12	1.2e-05	0.0434	0.0803	0.2604	0.2715

Parameter Settings: Gain coefficients: $γ_{min} = 0.04$ and $γ_{max} = 0.09$ , the exponential decay rates for the moment estimates: $λ_{1} = 0.9$ and $λ_{2} = 0.999$ , the performance metric function $J = J [u_{1}, u_{2}, \dots, u_{N}]$ , and the maximal number of learning iterations $T = 800$ .
Ensure: The optimal parameters $[u_{1}, u_{2}, \dots, u_{N}]$ .
1.	$m_{0} \leftarrow 0$ ; $n_{0} \leftarrow 0$ ; $ε \leftarrow 10^{- 8}$ ; $[u_{1}^{0}, u_{2}^{0}, \dots, u_{N}^{0}] \leftarrow 0$ (Setting initial
values)
2.	Out loop: for $t = 1, \dots ., T$
3.	Randomly generate the values of $δ u_{1}, δ u_{2}, \dots, δ u_{N}$ based on
$\| δ u \| = 1$
4.	$δ J \leftarrow J (u_{1} + δ u_{1}, \dots, u_{j} + δ u_{j}, \dots, u_{N} + δ u_{N})$
$- J (u_{1} - δ u_{1}, \dots, u_{j} - δ u_{j}, \dots, u_{N} - δ u_{N})$
5.	Inner loop: for $j = 1, 2, \dots, N$
6.	$g_{j}^{t} \leftarrow δ J^{t} δ u_{_{j}}^{t}$
7.	$m_{j}^{t} \leftarrow λ_{1} m_{j}^{t - 1} + (1 - λ_{1}) \cdot g_{j}^{t}$
8.	$n_{j}^{t} \leftarrow λ_{2} n_{j}^{t - 1} + (1 - λ_{2}) \cdot (g_{j}^{t})^{2}$
9.	${\tilde{n}}_{j}^{t} \leftarrow max (λ_{2} n_{j}^{t - 1} + (1 - λ_{2}) \cdot {(g_{j}^{t})}^{2}, n_{j}^{t - 1})$
10.	${\hat{m}}_{j}^{t} \leftarrow m_{j}^{t} / [1 - (λ_{1})^{t}]$
11.	${\hat{n}}_{j}^{t} \leftarrow {\tilde{n}}_{j}^{t} / [1 - (λ_{2})^{t}]$
12.	$γ \leftarrow γ_{max} - (\frac{t - 1}{T - 1}) (γ_{max} - γ_{min})$
13.	$u_{j}^{t} \leftarrow u_{j}^{t - 1} + \frac{γ}{(\sqrt{{\hat{n}}_{j}^{t}} + ε)} (λ_{1} {\hat{m}}_{j}^{t} + \frac{(1 - λ_{1}) g_{j}^{t}}{1 - λ_{1}^{t}})$
14.	end for
15.	end for

	Algorithms
	Gain Coefficients
$D / r_{0}$	SPGD	MSPGD	ASPGD	NASPGD
5	0.9	0.9	0.09	0.04–0.09
8	2	2	0.09	0.04–0.09
10	5	5	0.09	0.04–0.09
12	30	30	0.09	0.04–0.09

Abstract

Data Availability

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Tables (3)

Equations (18)

Applied Optics