Step-size Strategy Research Articles

Dual‐Inertial Viscosity‐Based Subgradient Extragradient Methods for Equilibrium Problems Over Fixed Point Sets

ABSTRACTThis paper introduces a new algorithmic framework for solving pseudomonotone equilibrium problems and demicontractive fixed‐point problems. Unlike conventional methods that incorporate a single inertial step, our approach employs dual inertia to accelerate convergence while preserving stability. The proposed method combines the viscosity approximation technique with the extragradient method to guarantee strong convergence. Initially, the extragradient method is applied under a Lipschitz continuity assumption on the equilibrium bifunction. Subsequently, this condition is relaxed by adopting a self‐adaptive step size strategy, allowing variable step sizes to be updated iteratively based on the current iterates information. Notably, the algorithm operates without requiring prior knowledge of the Lipschitz constants or any line search procedures. Strong convergence is established under mild condition, and its applicability to variational inequality problems is demonstrated. Numerical experiments validate the effectiveness of the proposed approach, showcasing its capability to handle large‐scale and complex problems efficiently while outperforming traditional single‐inertia methods.

Adaptive Step Size Linear Multistep Method for Small Signal Stability Analysis in Power Systems with Time Delays

Distributed energy resources (DERs) offer a promising solution for enhancing the resilience and efficiency of modern power systems. However, the inherent time delays in their communication and control systems can degrade their small signal stability, affecting reliable operation. This paper presents an Adaptive Step Size Linear Multistep Method (AS-LMS), for analyzing small signal stability of the system in the presence of time delays. The AS-LMS automatically adjusts the step size based on the computed solution behavior, achieving a balance between accuracy and computational efficiency. The essential idea is to use smaller steps in regions where the dynamic is changing rapidly and larger steps in regions where it is relatively smooth. Two adaptive step size strategies are proposed. The first strategy compares the current step size to the initial step size, ensuring consistency across iterations. The second strategy compares the current step size with the previous step size, allowing for more local adjustments throughout the numerical integration process. We evaluate the performance of the proposed AS-LMS method on two standard testing systems, i.e., the IEEE 9-bus and IEEE 30-bus systems. Both systems are modified to converter-interfaced DER units and different delays are applied to each testing system. The results show that the proposed AS-LMS method improves computational efficiency while maintaining accuracy in stability assessments. By balancing the accuracy and computational efficiency, the AS-LMS method provides a potent tool to analyze the stability of power systems with time delay.

BACKWARD DIFFERENTIATION FORMULA FOR SOLVING SECONND ORDER ORDINARY DIFFERENTIAL EQUATIONS (ODEs)

In this paper the derivation of a new numerical method for the numerical integration of second order ordinary differential equations (ODEs) is presented with variable step size strategy, the order of the method is two, and the stability analysis of the method shown that the method is zero stable and it is Convergent, therefore the method can s

DERIVATION OF A 2-POINT VARIABLE STEP SIZE SUPER CLASS OF BLOCK BACKWARD DIFFERENTIATION FORMULA FOR SOLVING SECONND ORDER ORDINARY DIFFERENTIAL EQUATIONS (ODEs)

In this paper the detail of the derivation of a new numerical method for the nummerical integration of second order ODEs is presented with variable step size strategy, the order of the method is two, and the stability analysis of the method shows that the method is zero stable and it is Convergent, therefore the method can serve as an efficient method for solving second order ODEs.

Performance of the fractional step method with various temporal discretization and adaptive time step size schemes for pulsating ventilation

Performance of the fractional step method with various temporal discretization and adaptive time step size schemes for pulsating ventilation

基于改进RRT算法的葡萄采摘机械臂路径规划研究

The robot's operation in a grape orchard environment is often disrupted by obstacles such as vines and leaves, resulting in low fruit picking efficiency. To achieve stable obstacle avoidance, an improved RRT algorithm based on global adaptive step size and target-biased sampling was developed. First, the kinematic equations of the grape-picking robotic arm were established using the PoE method, and both forward and inverse kinematics calculations were performed to determine the robot's workspace. Then, to address the issues of lack of target orientation and other shortcomings in the traditional RRT algorithm when planning collision-free paths, dynamic updating and global adaptive step size strategies were proposed. Simulation experiments conducted using MATLAB software demonstrated that our improved RRT algorithm, compared to the RRT, RRT_informed, and RRT_star algorithms, offered advantages in terms of lower planning time, fewer sampling points, and shorter path lengths in both 2D and 3D map scenarios. Finally, grape-picking experiments were conducted in both a laboratory setting and a real orchard. The results demonstrated that the average path planning time using the proposed algorithm was shorter compared to baseline algorithms, effectively validating the efficiency and practicality of the algorithm.

An Inertial Subgradient Extragradient Method for Efficiently Solving Fixed-Point and Equilibrium Problems in Infinite Families of Demimetric Mappings

The primary objective of this article is to enhance the convergence rate of the extragradient method through the careful selection of inertial parameters and the design of a self-adaptive stepsize scheme. We propose an improved version of the extragradient method for approximating a common solution to pseudomonotone equilibrium and fixed-point problems that involve an infinite family of demimetric mappings in real Hilbert spaces. We establish that the iterative sequences generated by our proposed algorithms converge strongly under suitable conditions. These results substantiate the effectiveness of our approach in achieving convergence, marking a significant advancement in the extragradient method. Furthermore, we present several numerical tests to illustrate the practical efficiency of the proposed method, comparing these results with those from established methods to demonstrate the improved convergence rates and solution accuracy achieved through our approach.

Simulation of obstacle avoidance planning based on improved RRTstar robotic arm

Abstract An obstacle avoidance pathfinding strategy with an augmented RRT* algorithm was presented for the robotic arm’s operation. A simulation model of the robotic arm was created. The Monte Carlo method was employed to solve its workspace. To mitigate the weaknesses of the RRT* algorithm, such as slow convergence and poor orientation, the algorithm was improved by combining gravitational and repulsive force fields from the artificial potential field method with a dynamic step size strategy. Simulation experiments were carried out using MATLAB. The results showed that the revised algorithm lessened the path search time by 52.1% and the path search length by 9.1% as opposed to the original RRT* algorithm. The experiments were verified in the ROS backdrop. The proposed algorithm was found to be more advantageous for robotic arm obstacle avoidance.

MEEF criterion-based spline adaptive filtering algorithm and its application

This paper presents an innovative the minimum error entropy with fiducial points (MEEF)-based spline adaptive filtering (S-AF) algorithm, called SAF-MEEF algorithm, which outperforms the conventional SAF algorithms that use the mean square error (MSE) criterion in reducing non-Gaussian interference. To overcome the limitation of the fixed step-size, a variable step-size strategy is also developed, resulting in the SAF-VMEEF algorithm, which improves the convergence speed and steady-state error performance. Furthermore, the computational complexity and convergence analysis of the SAF-MEEF are discussed. Nonlinear system identification simulations test the performance of the presented algorithms. Furthermore, this article accomplishes the application of nonlinear active noise control (ANC). Their effectiveness and robustness against non-Gaussian noise are demonstrated in different experimental scenarios, including α-stable noise, real-world functional magnetic resonance imaging (fMRI) noise, and real-life server room (SR) noise.

Neo-Hookean modeling of nonlinear coupled behavior in circular plates supported by micro-pillars

In the contemporary era, the enhancement of wearable capacitive sensors is achieved through the utilization of polymeric micropillars as filler materials between electrode plates. To gain a deeper understanding of the dynamic response of the system, nonlinear coupled governing equations of a circular microplate motion resting on an array of polymeric micropillars have been derived. These equations are used to model the system’s behavior. In addition, the squeezing motion of the micro-pillars is characterized using the incompressible Neo-Hookean model. Both static and dynamic responses, including transient and steady-state solutions, are investigated in detail by discretizing over spatial coordinates using a weak formulation approach. A frequency response analysis is conducted using a continuation-based method. This entails expanding the steady-state solution using a Fourier transform and employing the energy balance principle. The unknown coefficients of the expansion series are calculated using a gradient descent-based learning approach that is physically motivated. Furthermore, a dynamic step size strategy for frequency increments is employed to effectively follow the solution path. This strategy is implemented via the ARC length method. In this study, we examine the impact of varying PDMS (polydimethylsiloxane) hydrogel mechanical and geometrical configurations. It can be reasonably concluded that the mechanical properties of the pillars and the geometrical configuration of the circular plate and micropillars have a significant impact on the maximum tolerable pressure, fast transient response, and frequency response analysis.

PSO-FDM (Particle Swarm Optimization-Finite Difference Method)-Based Simulation Model of Temperature and Velocity of Full-Scale Continuous Annealing Furnace

Improving the accuracy of the temperature field prediction model for continuous annealing line strips and enhancing the model’s adaptability to full-size strips are key technical challenges in continuous annealing lines. This paper developed a continuous annealing temperature prediction model based on a variable step-size strategy for the heating section, even-heat section, slow-cooling section, and fast-cooling section of the continuous annealing line. To improve the prediction accuracy for different strip sizes, the PSO optimization algorithm was employed to determine the optimal heat transfer coefficient for each strip size. Additionally, due to the limited production of certain strip gauges, providing insufficient data for optimization, this study introduces a combined file approach to address gauge vacancies. The experimental results indicate that the optimized model with variable step size can control the absolute prediction error to less than 4 °C, improving prediction accuracy by 61.9% and prediction speed by 26.8% compared to the traditional equal-step prediction model. This study verified that the merger method is effective for addressing side gauge vacancies, while the proposed method is suitable for resolving middle gauge vacancies. The main technical contribution of this study is the establishment of a high-precision prediction model for continuous annealing temperature of variable step length strips, ensuring high temperature control accuracy for full-gauge strips when passing through the continuous annealing production line.

Enhancing Accuracy and Efficiency in Stiff ODE Integration Using Variable Step Diagonal BBDF Approaches

Recent advancements in mathematical modelling have uncovered a growing number of systems exhibiting stiffness, a phenomenon that challenges the effectiveness of traditional numerical methods. Motivated by the need for more robust numerical techniques to address this issue, this paper presents an enhanced version of the Diagonally Block Backward Differentiation Formula (BBDF) that incorporates intermediate points, known as off-step points, to improve the accuracy and efficiency of solutions for stiff ordinary differential equations (ODEs). The new scheme leverages an adaptive step-size strategy to refine accuracy and efficiency between regular and off-grid integration steps. Theoretical analysis confirms that the proposed scheme is an A-stable and convergent method, as it satisfies the fundamental criteria of consistency, zero-stability, and A-stability. Numerical experiments on single and multivariable systems across varying time scales demonstrate significant improvements in solving stiff ODEs compared to existing techniques. Therefore, the new proposed method is an effective solver for stiff ODEs.

A new variational integrator for constrained mechanical system dynamics

A new variational integrator is proposed to solve constrained mechanical systems. The main distinguishing feature of the present integrator comes from the distinct discretization of Lagrangians based on the Hamilton's principle in its most general form. Specifically, Hermite interpolation is used for discrete positions, which provides at least C1 continuity for generalized coordinates. The velocities and momentums are interpolated using quadratic polynomials for the consistency, such that the kinematic relation between velocities and positions can be exactly satisfied. Meanwhile, the Gauss-Legendre quadrature rule is employed to guarantee the accuracy of discrete Lagrange equations. To tackle constrained mechanical systems, a coordinate partition approach is used to eliminate the constraint equations. The local incremental rotation vector is exploited to get rid of rotation singularities in spatial problems. Moreover, an adaptive stepsize strategy is implemented to improve the efficiency. The strengths of the new integrator lie in the accessible large step sizes in the simulation and its global second-order accuracy for positions as well as velocities. Several examples are performed and analyzed to validate its accuracy and capabilities.

Structural-information-awareness-based regularization model for infrared image stripe noise removal.

Infrared images play a crucial role in military reconnaissance, security monitoring, fire detection, and other tasks. However, due to the physical limitations of detectors, an infrared image often suffers from significant stripe noise. The presence of stripe noise significantly degrades image quality and subsequent processing, making the removal of such noise indispensable. In this study, we propose, to our knowledge, a novel low-rank decomposition model to separate the stripe noise components in infrared images. In comparison with existing algorithms for removing infrared stripe noise, our method takes into account the distinctiveness between stripe noise and information components. For the stripe noise component, we describe a column gradient domain low-rank prior and standard deviation weighted group sparsity prior. For the image information component, we employ a structure-aware gradient sparsity prior to suppress stripes while preserving the structural features of images. During the iterative solution process, we utilize both an initial solution based on minimizing column differences and an iteration step-size strategy based on variable acceleration to accelerate convergence. To validate the effectiveness of our proposed method, we conduct experiments to compare it with other destriping algorithms, demonstrating the superiority of our method from the perspectives of both subjective evaluation and objective metrics.

A modified feedforward-feedback time-frequency domain hybrid algorithm for multichannel active road noise control system

Multi-channel Active Road Noise Control (ARNC) systems require numerous adaptive filters to reduce noise at multiple positions inside the vehicle. However, road noise conditions are more complex, and the correlation between the reference signals and the target noise is often poor. When the driving conditions become slightly complex, the correlation between the reference signal and the target noise is often poor. As a result, traditional road noise control algorithms applied to multi-channel ARNC systems often suffer from high computational cost and poor noise reduction performance in non-steady-state and high-speed driving conditions. To address this issue,a reference signal selection method based on the genetic algorithm (GA) is proposed to obtain a reference signals combination with stronger coherence. On this basis, we introduce the time–frequency domain structure into the forward-feedback hybrid algorithm and a multi-channel time–frequency domain hybrid algorithm with a novel variable step size method (VSS-TF-HANC) is proposed in this paper.The feedforward part adopts a time–frequency domain structure and introduces a novel variable step-size strategy to reduce computational complexity and improve noise reduction performance. The feedback part adopts a control system based on the Internal Model Control (IMC) architecture to attenuate the interior noise components that have a strong response but poor correlation with the reference signals. Simulation results demonstrate that the proposed algorithm achieves excellent control results under complex driving conditions, with significantly better noise reduction performance compared to the control algorithms that solely employ the feedforward structure, while also exhibiting low computational cost.

Shuffling-type gradient method with bandwidth-based step sizes for finite-sum optimization

Shuffling-type gradient method is a popular machine learning algorithm that solves finite-sum optimization problems by randomly shuffling samples during iterations. In this paper, we explore the convergence properties of shuffling-type gradient method under mild assumptions. Specifically, we employ the bandwidth-based step size strategy that covers both monotonic and non-monotonic step sizes, thereby providing a unified convergence guarantee in terms of step size. Additionally, we replace the lower bound assumption of the objective function with that of the loss function, thereby eliminating the restrictions on the variance and the second-order moment of stochastic gradient that are difficult to verify in practice. For non-convex objectives, we recover the last iteration convergence of shuffling-type gradient algorithm with a less cumbersome proof. Meanwhile, we also establish the convergence rate for the minimum iteration of gradient norms. Under the Polyak-Łojasiewicz (PL) condition, we prove that the function value of last iteration converges to the lower bound of the objective function. By selecting appropriate boundary functions, we further improve the previous sublinear convergence rate results. Overall, this paper contributes to the understanding of shuffling-type gradient method and its convergence properties, providing insights for optimizing finite-sum problems in machine learning. Finally, numerical experiments demonstrate the efficiency of shuffling-type gradient method with bandwidth-based step size and validate our theoretical results.

Dynamic modeling of cable deployment/retrieval based on ALE-ANCF and adaptive step-size integrator

The towing vehicle must maintain a stable distance from the seafloor to ensure the accuracy and safety of deep-towed seismic systems. However, due to equipment limitations, controlling the towing depth is achievable solely by adjusting the length of the towing cable. In this study, an Absolute Nodal Coordinate Formulation (ANCF) with Arbitrary Lagrange-Euler (ALE)description is employed to model variable-length flexible towing cables. The inertial and elastic force formulas for ALE-ANCF, along with their corresponding Jacobian matrices, are derived in detail, and their coefficient matrices are precomputed to avoid Gaussian integration. The proposed adaptive step-size strategy enhances computational efficiency. Detailed investigation into the impact of cable deployment/retrieval on towing depth is conducted, revealing an almost linear relationship between length change and towing depth. In Addition, a proportional, integral, derivative (PID) controller with compensation is proposed for efficiently tracking the target depth and alleviating the perturbations from the mothership, illustrating the feasibility of PID control. These findings hold practical significance for enhancing the operational safety and efficiency of deep-towed seismic systems.

Hydraulic-mechanical-electrical coupled model framework of variable-speed pumped storage system: Measurement verification and accuracy analysis

With the large-scale operation of wind and solar energy, the demand for energy storage in power grids has increased sharply. As a reliable means of long-term energy storage, the variable-speed pumped-storage power station (VSPSU) is a new development direction for pumped storage that has attracted increasing attention owing to its advantages of more rapid and flexible action. However, accurately predicting the transient process of a VSPSU is a key challenge in its design and operation. The inconsistent time step required for the fine simulation of the hydraulic-machine subsystem (millisecond level) and the electrical subsystem (microsecond level) is the main challenge that hinders the coordination of system simulation efficiency and accuracy and restricts further development of the entire coupling system simulation. Simultaneously, a coupling refinement model verified by multi-condition measurements has not been reported. This paper proposes a VSPSU hydraulic-mechanical-electrical coupling model framework that has been verified experimentally. The innovation of the model framework lies in the realization of the fine solution of the hydraulic and electrical systems through the coupling strategy of the variable simulation step size rather than using model simplification. Based on a real micro-VSPSU system, we conducted experiments under various operating conditions and compared the measurements with the numerical simulation results. The results showed that in the generation mode, the maximum simulation error of the proposed coupling model framework on the hydraulic-mechanical side was 2.29 %, and the maximum simulation error on the electrical side was 9.59 %. Finally, we quantified the contribution of key parameters in the model framework to the dynamic response process and clarified the influence of physical parameters on the transient process in the VSPSU. Compared with conventional models, the advantage of the proposed coupling framework is that it can realize the coupling of a hydraulic-mechanical-electrical system without simplification, which has been verified by measuring multiple working conditions. This framework provides a reliable tool for the parameter design and operation optimization of VSPSU.

Stiffly Stable Diagonally Implicit Block Backward Differentiation Formula with Adaptive Step Size Strategy for Stiff Ordinary Differential Equations

Stiffly Stable Diagonally Implicit Block Backward Differentiation Formula with Adaptive Step Size Strategy for Stiff Ordinary Differential Equations

An accelerated stochastic extragradient-like algorithm with new stepsize rules for stochastic variational inequalities

In this paper, we devise a stochastic extragradient-like algorithm incorporated with inertial terms, which requires a single projection onto our feasible set and employs a stochastic approximation version of an Armijo-type line search scheme along a feasible direction, for solving pseudomonotone stochastic variational inequalities. In the algorithm, two different stepsize strategies are employed to update steplength sequences without using the prior knowledge of the Lipschitz constant of involved operators. In the process of the stochastic approximation, we iteratively reduce the variance of stochastic errors. The almost sure convergence, the complexity analysis, and rates are provided in a dimensional space under reasonable conditions. Finally, some numerical experiments with graphical illustrations are reported to demonstrate the applicability and the efficiency of our algorithm in comparison with some projection type methods in the literature.