Gradient Norm Minimization of Nesterov Acceleration: o(1/k3)o(1/k^3)

In the history of first-order algorithms, Nesterov's accelerated gradient descent (NAG) is one of the milestones. However, the cause of the acceleration has been a mystery for a long time. It has not been revealed with the existence of gradient correction until the high-resolution differential equation framework proposed in [Shi et al., 2021]. In this paper, we continue to investigate the acceleration phenomenon. First, we provide a significantly simplified proof based on precise observation and a tighter inequality for $L$-smooth functions. Then, a new implicit-velocity high-resolution differential equation framework, as well as the corresponding implicit-velocity version of phase-space representation and Lyapunov function, is proposed to investigate the convergence behavior of the iterative sequence $\{x_k\}_{k=0}^{\infty}$ of NAG. Furthermore, from two kinds of phase-space representations, we find that the role played by gradient correction is equivalent to that by velocity included implicitly in the gradient, where the only difference comes from the iterative sequence $\{y_{k}\}_{k=0}^{\infty}$ replaced by $\{x_k\}_{k=0}^{\infty}$. Finally, for the open question of whether the gradient norm minimization of NAG has a faster rate $o(1/k^3)$, we figure out a positive answer with its proof. Meanwhile, a faster rate of objective value minimization $o(1/k^2)$ is shown for the case $r > 2$.Comment: 16 page

Wind turbine asymmetrical load reduction with pitch sensor fault compensation

Offshore wind turbines suffer from asymmetrical loading (blades, tower, etc), leading to enhanced structural fatigue. As well as asymmetrical loading different faults (pitch system faults etc.) can occur simultaneously, causing degradation of load mitigation performance. Individual pitch control (IPC) can achieve rotor asymmetric loads mitigation, but this is accompanied by an enhancement of pitch movements leading to the increased possibility of pitch system faults, which exerts negative effects on the IPC performance. The combined effects of asymmetrical blade and tower bending together with pitch sensor faults are considered as a “co‐design” problem to minimize performance deterioration and enhance wind turbine sustainability. The essential concept is to attempt to account for all the “fault effects” in the rotor and tower systems, which can weaken the load reduction performance through IPC. Pitch sensor faults are compensated by the proposed fault‐tolerant control (FTC) strategy to attenuate the fault effects acting in the control system. The work thus constitutes a combination of IPC‐based load mitigation and FTC acting at the pitch system level. A linear quadratic regulator (LQR)‐based IPC strategy for simultaneous blade and tower loading mitigation is proposed in which the robust fault estimation is achieved using an unknown input observer (UIO), considering four different pitch sensor faults. The analysis of the combined UIO‐based FTC scheme with the LQR‐based IPC is shown to verify the robustness and effectiveness of these two systems acting together and separately

Actuator fault tolerant offshore wind turbine load mitigation control

Offshore wind turbine (OWT) rotors have large diameters with flexible blade structures which are subject to asymmetrical loads caused by blade flapping and turbulent or unsteady wind flow. Rotor imbalance inevitably leads to enhanced fatigue of blade rotor hub and tower structures. Hence, to enhance the life of the OWT and maintain good power conversion the unbalanced loading requires a reliable mitigation strategy, typically using a combination of Individual Pitch Control (IPC) and Collective Pitch Control (CPC). Increased pitch motion resulting from IPC activity can increase the possibility of pitch actuator faults and the resulting load imbalance results in loss of power and enhanced fatigue. This has accelerated the emergence of new research areas combining IPC with the fault tolerant control (FTC)-based fault compensation, a so-called FTC and IPC “co-design” system. A related research challenge is the clear need to enhance the robustness of the FTC IPC “co-design” to some dynamic uncertainty and unwanted disturbance. In this work a Bayesian optimization-based pitch controller using Proportional–Integral (PI) control is proposed to improve pitch control robustness. This is achieved using a systematic search for optimal controller coefficients by evaluating a Gaussian process model between the designed objective function and the coefficients. The pitch actuator faults are estimated and compensated using a robust unknown input observer (UIO)-based FTC scheme. The robustness and effectiveness of this “co-design” scheme are verified using Monte Carlo simulations applied to the 5MW NREL FAST WT benchmark system. The results show clearly (a) the effectiveness of the load mitigation control for a wide range of wind loading conditions, (b) the effect of actuator faults on the load mitigation performance and (c) the recovery to normal load mitigation, subject to FTC action

Comparison of the short term therapeutic effects by different incision triple surgery in treating angle-closure glaucoma with cataract

AIM:To compare the therapeutic effects of different incision phacoemulsification with foldable intraocular lens implantation combined with trabeculectomy in treating angle-closure glaucoma complicated with cataract. <p>METHODS: The retrospective study analyzed 70 patients(98 eyes)who performed phacoemulsification with foldable intraocular lens implantation combined with trabeculectomy. Single incision surgery(one-site approach)was performed on 34 patients(50 eyes), and double incision surgery(two-site approaches)was done on 36 patients(48 eyes). The differences of intraocular pressure control,filtering bleb, the density and preservation of corneal endothelium cells both pre-surgery and 1 month post-surgery or later complication. in the two groups were compared and analyzed.Follow-up was 12-24 months with a mean of 18.2 months.<p>RESULTS: The average postoperative intraocular pressure in one-site groups was(10.16±4.31)mmHg. The average postoperative intraocular pressure in two-site groups was(11.38±3.55)mmHg. There were no statistically significant differences between the two groups(<i>P</i>>0.05)regarding the postoperative intraocular pressure and the formation of filtering blebs(<i>P</i>>0.05). The density and area of corneal endothelium cells in the two-incision group pre-operation were comparable(<i>P</i>>0.05). However, in 1 month post-operation, the rate of the corneal endothelial cells loss in double incision group was superior compared to the single incision group(<i>P</i><0.01).<p>CONCLUSION: Both single incision and double incision approach phacotrabeculectomy are effective in reducing intraocular pressure and can maintain the function of filtering blebs well. There are no statistically significant differences in the intraocular pressure reduction between the two approaches. The double incision approach is superior in minimizing the corneal endothelial cells loss compared to the single incision approach

Absence of barren plateaus in finite local-depth circuits with long-range entanglement

Ground state preparation is classically intractable for general Hamiltonians. On quantum devices, shallow parameterized circuits can be effectively trained to obtain short-range entangled states under the paradigm of variational quantum eigensolver, while deep circuits are generally untrainable due to the barren plateau phenomenon. In this Letter, we give a general lower bound on the variance of circuit gradients for arbitrary quantum circuits composed of local 2-designs. Based on our unified framework, we prove the absence of barren plateaus in training finite local-depth circuits for the ground states of local Hamiltonians. These circuits are allowed to be deep in the conventional definition of circuit depth so that they can generate long-range entanglement, but their local depths are finite, i.e., there is only a finite number of non-commuting gates acting on individual qubits. This fact suggests that long-range entangled ground states, such as topologically ordered states, are in general possible to be prepared efficiently on quantum devices via variational methods. We validate our analytical results with extensive numerical simulations and demonstrate the effectiveness of variational training using the generalized toric code model.Comment: 28 pages, 7 figure

Study of Antenna Superstrates Using Metamaterials for Directivity Enhancement Based on Fabry-Perot Resonant Cavity

Metamaterial superstrate is a significant method to obtain high directivity of one or a few antennas. In this paper, the characteristics of directivity enhancement using different metamaterial structures as antenna superstrates, such as electromagnetic bandgap (EBG) structures, frequency selective surface (FSS), and left-handed material (LHM), are unifiedly studied by applying the theory of Fabry-Perot (F-P) resonant cavity. Focusing on the analysis of reflection phase and magnitude of superstrates in presently proposed designs, the essential reason for high-directivity antenna with different superstrates can be revealed in terms of the F-P resonant theory. Furthermore, a new design of the optimum reflection coefficient of superstrates for the maximum antenna directivity is proposed and validated. The optimum location of the LHM superstrate which is based on a refractive lens model can be determined by the F-P resonant distance