Step Size Selection Research Articles

Automatic Step Size Selection of the PO MPPT Algorithm to Improve Wind Power Generation

Perturb and Observe (P&O) is a commonly used algorithm for Maximum Power Point Tracking (MPPT) in wind turbines. MPPT plays a critical role in enhancing wind turbine efficiency by dynamically adjusting operating parameters to adapt to fluctuating wind conditions. Although P&O is favored for its simplicity and adaptability, its performance is hindered by step size selection issues, which lead to inefficiency, oscillations, and slow convergence. To overcome these limitations, this research proposes a modified P&O algorithm that automates step size selection based on divided sectors of wind speed and normalized power in region two. Additionally, an integration of the pitch-angle control from region three was employed to maintain the optimal power output under variable wind conditions. The proposed approach reduces tracking time, minimizes perturbation errors, and ensures a stable power output. The proposed modifications enhance the efficiency and reliability of Wind Energy Conversion Systems (WECS) by addressing the shortcomings of the conventional P&O methods.

Stability and robustness of time-discretization schemes for the Allen-Cahn equation via bifurcation and perturbation analysis

The Allen-Cahn equation is a fundamental model for phase transitions, offering critical insights into the dynamics of interface evolution in various physical systems. This paper investigates the stability and robustness of frequently utilized time-discretization numerical schemes for solving the Allen-Cahn equation, with focuses on the Backward Euler, Crank-Nicolson (CN), convex splitting of modified CN, and Diagonally Implicit Runge-Kutta (DIRK) methods. Our stability analysis reveals that the Convex Splitting of the Modified CN scheme exhibits unconditional stability, allowing greater flexibility in time step size selection, while the other schemes are conditionally stable. Additionally, our robustness analysis highlights that the Backward Euler method converges to correct physical solutions regardless of initial conditions. In contrast, all other methods studied in this work show sensitivity to initial conditions and may converge to incorrect physical solutions if the initial conditions are not carefully chosen. This study introduces a comprehensive approach to assessing stability and robustness in numerical methods for solving the Allen-Cahn equation, providing a new perspective for evaluating numerical techniques for general nonlinear differential equations.

Adaptive time step selection for spectral deferred correction

AbstractSpectral Deferred Correction (SDC) is an iterative method for the numerical solution of ordinary differential equations. It works by refining the numerical solution for an initial value problem by approximately solving differential equations for the error, and can be interpreted as a preconditioned fixed-point iteration for solving the fully implicit collocation problem. We adopt techniques from embedded Runge-Kutta Methods (RKM) to SDC in order to provide a mechanism for adaptive time step size selection and thus increase computational efficiency of SDC. We propose two SDC-specific estimates of the local error that are generic and do not rely on problem specific quantities. We demonstrate a gain in efficiency over standard SDC with fixed step size and compare efficiency favorably against state-of-the-art adaptive RKM.

W‐schemes in spectral methods for reaction‐diffusion equations

AbstractWe consider reaction‐diffusion equations, which are parabolic systems of partial differential equations. Assuming periodic boundary conditions in space, a multidimensional Fourier transform can be used. A spectral method yields an initial value problem of a stiff system of ordinary differential equations for time‐dependent Fourier coefficients. We investigate the application of Rosenbrock‐Wanner W‐schemes in the time integration of the ordinary differential equations including step size selection. Results of numerical computations are presented using a Turing system in two space dimensions.

Ablation and molten layer flow simulation for plate model of SiO2f/SiO2 composite material using particle method

In this paper, the moving particle semi-implicit method (MPS) is extended from calculating free mobility to simulating the extremely viscous and temperature-dependent molten layer flow of SiO2f/SiO2 composite material under aerodynamic heating conditions, which includes strong heating and shear of incoming flow. A method for applying heat flux and airflow shear, based on the conceptual particle approach, has been established. Heat transfer, melting, solidification, and evaporation behaviors are considered, with temperature-dependent viscosity variations also accounted for. The ablative regression of the plate model is verified using experimental results of the SiO2f/SiO2 composite material, and results from convergence analysis demonstrate the accuracy of the space step size selection. Surface morphology analysis through three-dimensional computation indicates that the extended particle method also accurately describes the surface morphology of SiO2f/SiO2 composite material under aerodynamic heating conditions. Thus, the extended particle method accurately simulates both the ablation process and the surface morphology of the SiO2f/SiO2 composite material. The influences of acceleration and surface tension are discussed. Ablative recession, when subject to acceleration, is smaller than that observed in its absence. When exposed to surface tension, the liquid layer tends to form a spherical shape, and the particles behave as a cohesive unit, resulting in smaller ablative recession than in the absence of surface tension.

Sensitivity analysis of frequency response functions with imaginary parts decoupling based on multicomplex-step perturbation

Imaginary perturbation is used in the complex step differentiation method to compute first-order derivatives, widely known as an effective approach for sensitivity analysis in structural dynamics. However, coupling of imaginary parts occurs in the damped frequency response functions when employing this method. To mitigate this coupling, a novel approach for sensitivity analysis based on multicomplex-step perturbation is proposed in this paper, for sensitivity analysis of Frequency Response Functions in structural dynamics. The structural parameters are perturbed in multicomplex domain, the dimensions of structural matrices are expanded using the Cauchy Riemann matrix representation, the equation of motion for sensitivity analysis in frequency domain is transformed to matrix operation in field of real numbers, imaginary term will not exist in the equation of motion for sensitivity analysis, the imaginary part of the frequency response function and the imaginary part of the perturbation are decoupled, the structural frequency response functions and the corresponding sensitivities are obtained from the dimension-expanded equation of motion. A truss structure and a solar wing are adopted to verify the accuracy of the proposed method. Results show that the sensitivity of FRFs can be effectively calculated using the proposed method. Compare to the finite difference method, the proposed method is not depended on the step-size selection procedure. The multi-order and mixed-order sensitivity matrices, especially Hessian matrix can also be obtained using the proposed method.

Efficient sensitivity analysis of the thermal profile in powder bed fusion of metals using hypercomplex automatic differentiation finite element method

Rapid cyclic temperature fluctuation occurring in powder bed fusion of metals using a laser beam (PBF-LB/M) influences the formation of flaws in printed parts. Consequently, there is a pressing need to enhance the quality of printed parts by developing innovative methodologies that can predict thermal histories and help uncover the intricate relationships between process parameters and thermal profiles. Sensitivity Analysis (SA) emerges as an essential tool for this, offering the potential for process optimization and enhanced quality control. Nonetheless, conventional SA methodologies often incur in excessive computational costs and potential numerical approximation errors. To address this technical challenge, we present a novel method for SA that integrates the HYPercomplex-based Automatic Differentiation (HYPAD) technique with transient thermal simulations conducted via the finite element method (FEM). Leveraging this methodology, we efficiently and accurately perform SA for PBF-LB/M processes in a post-processing step. Compared to traditional methods like Finite Differences (FD), HYPAD-FEM required 96 % less computational time for obtaining sensitivities for 22 process parameters, under a comparative study conducted within the context of the 2018–02 AM benchmark of the National Institute of Standards and Technology. In summary, HYPAD-FEM offers superior efficiency and accuracy in SA over conventional methods, delivering the best sensitivity of a model without the need for step-size selection and problem or parameter-based implementations.

Molecular interactions of the Omicron, Kappa, and Delta SARS-CoV-2 spike proteins with quantum dots of graphene oxide.

The Omicron, Kappa, and Delta variants are different strains of the SARS-CoV-2 virus. Graphene oxide quantum dots (GOQDs) represent a burgeoning class of oxygen-enriched, zero-dimensional materials characterized by their sub-20-nm dimensions. Exhibiting pronounced quantum confinement and edge effects, GOQDs manifest exceptional physical-chemical attributes. This study delves into the potential of graphene oxide quantum dots, elucidating their inherent properties pertinent to the surface structures of SARS-CoV-2, employing an integrated computational approach for the repositioning of inhibitory agents. Following rigorous adjustment tests, a spectrum of divergent bonding conformations emerged, with particular emphasis placed on identifying the conformation exhibiting optimal adjustment scores and interactions. The investigation employed molecular docking simulations integrating affinity energy evaluations, electrostatic potential clouds, molecular dynamics encompassing average square root calculations, and the computation of Gibbs-free energy. These values quantify the strength of interaction between GOQDs and SARS-CoV-2 spike protein variants. The receptor structures were optimized using the CHARM-GUI server employing force field AMBERFF14SB. The algorithm embedded in CHARMM offers an efficient interpolation scheme and automatic step size selection, enhancing the efficiency of the optimization process. The 3D structures of the ligands are constructed and optimized with density functional theory (DFT) method based on the most stable conformer of each binder. Autodock Vina Software (ADV) was utilized, where essential parameters were specified. Electrostatic potential maps (MEPs) provide a visual depiction of molecules' charge distributions and related properties. After this, molecular dynamics simulations employing the CHARM36 force field in Gromacs 2022.2 were conducted to investigate GOs' interactions with surface macromolecules of SARS-CoV-2 in an explicit aqueous environment. Furthermore, our investigation suggests that lower values indicate stronger binding. Notably, GO-E consistently showed the most negative values across interactions with different variants, suggesting a higher affinity compared to other GOQDs (GO-A to GO-D).

Interior point algorithm for second-order cone optimization based on a new kernel function

Second-order cone optimization (SOCO) problems are crucial in various fields such as engineering, finance, and machine learning due to their ability to model complex convex optimization tasks efficiently. In this article, we propose an interior point algorithm tailored for SOCO, leveraging a novel kernel function to enhance optimization performance. Traditional interior point methods for SOCO often encounter challenges in scalability and efficiency, particularly with large-scale problems. These methods typically rely on barrier functions that may suffer from slow convergence rates, especially when dealing with highly ill-conditioned or degenerate cones. To address these limitations, our approach introduces a new kernel function that exploits the geometric properties of second-order cones, thereby improving convergence behavior and computational efficiency. The key innovation of our algorithm lies in the construction of the kernel function, which incorporates insights from the structure of second-order cones to efficiently capture the curvature information of the optimization problem. By exploiting this curvature information, our algorithm can effectively navigate the optimization landscape, leading to faster convergence and enhanced robustness compared to traditional methods. Furthermore, our algorithm is equipped with advanced techniques for handling various challenges encountered in SOCO, such as dealing with degenerate cones and exploiting problem structure to accelerate convergence. Additionally, we incorporate strategies for warm-starting and adaptive step-size selection to further improve efficiency, particularly in iterative optimization processes. Finally, our proposed interior point algorithm for second-order cone optimization, based on a new kernel function, derives the iteration bounds O (Nlog (N/∊)) for large update methods. Here denotes N the number of second-order cones in the problem formulation and ∊ the desired accuracy. These iteration bounds are currently the best known bounds for such methods.

Effectiveness of Constant Stepsize in Markovian LSA and Statistical Inference

In this paper, we study the effectiveness of using a constant stepsize in statistical inference via linear stochastic approximation (LSA) algorithms with Markovian data. After establishing a Central Limit Theorem (CLT), we outline an inference procedure that uses averaged LSA iterates to construct confidence intervals (CIs). Our procedure leverages the fast mixing property of constant-stepsize LSA for better covariance estimation and employs Richardson-Romberg (RR) extrapolation to reduce the bias induced by constant stepsize and Markovian data. We develop theoretical results for guiding stepsize selection in RR extrapolation, and identify several important settings where the bias provably vanishes even without extrapolation. We conduct extensive numerical experiments and compare against classical inference approaches. Our results show that using a constant stepsize enjoys easy hyperparameter tuning, fast convergence, and consistently better CI coverage, especially when data is limited.

Accuracy improvement of the load forecasting in the district heating system by the informer-based framework with the optimal step size selection

Accurate load forecasting is crucial for effectively regulating regional heat network systems. However, existing forecasting methods often rely on subjective experience to determine the forecasting step, which is limited by the presence of thermal inertia, leading to suboptimal accuracy. To address this limitation, an optimal step size selection method based on the Informer-based framework is proposed to enhance load forecasting accuracy in heat exchange stations. This method leverages the Attention mechanism within the Informer model, enabling the capture of global information in a single step. To verify the effectiveness of the proposed method, real operational data from a typical thermal power plant in North China is utilized to analyze and test the impact of data distribution and prediction step size on the model's prediction capability. The performance is evaluated using Mean Square Error (MSE), Mean Absolute Error (MAE), and Mean Absolute Percentage Error (MAPE). Comparative analysis against Support Vector Regression (SVR) and Long Short-Term Memory (LSTM) models demonstrates that the Informer algorithm with optimal prediction step size achieves the highest prediction accuracy. Notably, the proposed method achieved a minimum reduction of 62.7 %, 46.5 %, and 42.9 % in MSE, MAE, and MAPE, respectively, significantly surpassing the performance of alternative prediction methods.

Step Size Selection for Tap Changers in Converter Transformers for Line Commutated HVDC Systems

This paper deals with the tap changer design for line-commutated HVDC converter transformers. It examines the impacts of tap changer switching to provide criteria for selecting an optimal step size. When switching the tap changer the firing angles in the converter change and with them the reactive power consumption. The dependence of the changes on the step size of the tap-changer is linear, so that a maximum step size of the tap changer can be calculated based on criteria for maximum allowable changes of firing angle and reactive power. For implementation into an existing AC system the analysis of tap changer switching has to consider the interaction between converter reactive power and AC voltage iteratively to derive a stable operating point of the HVDC after switching. Both effects are influenced by the reaction of the AC-voltage.

A variable metric proximal stochastic gradient method: An application to classification problems

Due to the continued success of machine learning and deep learning in particular, supervised classification problems are ubiquitous in numerous scientific fields. Training these models typically involves the minimization of the empirical risk over large data sets along with a possibly non-differentiable regularization. In this paper, we introduce a stochastic gradient method for the considered classification problem. To control the variance of the objective's gradients, we use an automatic sample size selection along with a variable metric to precondition the stochastic gradient directions. Further, we utilize a non-monotone line search to automatize step size selection. Convergence results are provided for both convex and non-convex objective functions. Extensive numerical experiments verify that the suggested approach performs on par with state-of-the-art methods for training both statistical models for binary classification and artificial neural networks for multi-class image classification. The code is publicly available at https://github.com/koblererich/lisavm.

On the construction of a quartically convergent method for high-dimensional Black-Scholes time-dependent PDE

There exist several numerical methods for finding the resolution of high dimensional option pricing Black-Scholes PDE with variable coefficients. Here we use radial basis functions to produce differentiation stencils (abbreviated by RBF-FD) on uniform point sets. In fact, the target is to propose a fully quartically convergent scheme via the RBF-FD methodology along both time and space directions. We too present a stability analysis for the selection of the time step size when the spatial step sizes vary. Furthermore, we draw a conclusion by several challenging computational experiments in high dimensions and reveal the superiority and the robustness of the scheme.

A mini-batch algorithm for large-scale learning problems with adaptive step size

The step size selection in stochastic optimization methods is a crucial aspect that holds significant importance in theoretical analysis and practical applications. We propose two stochastic optimization methods based on the competitive Barzilai-Borwein (BB) step size in both the inner and outer loops of the mini-batch semi-stochastic gradient descent (mS2GD) algorithm. The competitive BB step size is automatically updated using the latest and most accurate secant equation. We introduce two algorithms: mS2GD-CBB, which updates the step size in the outer loop, and mS2GD-RCBB, which updates the step size in the inner loop. We evaluate the performance of the proposed algorithms on the classical optimization problem and compare them against existing methods. Experimental results demonstrate that the methods exhibit favorable convergence properties and can effectively handle the challenges of big data arising from the fields of signal processing, statistics, and machine learning. The methods offer enhanced adaptability, flexibility, and exploration capabilities, making them promising approaches for a wide range of optimization problems.

A novel adaptive approach based on over-distance interval searching and precise step size determining for first-order structural reliability analysis

The first-order second-moment reliability method (FORM) is a prevalent method in the reliability analyzing community owing to its accuracy and efficiency. However, it may encounter unstable solutions such as periodic oscillation and non-convergence when a high-dimensional problem with complex performance functions is involved. In this study, a novel adaptive approach based on over-distance interval searching and precise step size determining is presented to work out a solution for the above issues. First, the relationship between the state of the iterative rotation angle and the ideal step size was noticed. Therefore, a rotation control condition is proposed to dictate the selection of step size for the algorithm. After that, the determination of the step size of the proposed method here can be realized in two stages: In the first stage, an over-distance step size interval searching approach is established to define an initial exploration interval of step size with a length of 1. In the second stage, a determination approach of adaptive precise step size is then proposed to modify the fixed step sizes to make it adaptable to changes, therefore, get the final step sizes for the proposed method. Together, an over-distance adaptive precise searching approach of step size that can generate self-adaptive step sizes both greater and less than 1 is performed by interplaying the above two stages, thus preventing an excessive amount of redundant workload and speeding up the convergence while reducing the risk of oscillations. Besides, an iterative direction modification approach is given to orient the first iteration more accurately. In addition to that, the proposed method is tested on a offshore jacket platform and other seven examples to verify its performance. The results demonstrate that the proposed method provides excellent robustness and efficiency for both low and high-nonlinearity problems, showing significant merits.

A Modified Inexact SARAH Algorithm with Stabilized Barzilai-Borwein Step-Size in Machine learning

The Inexact SARAH (iSARAH) algorithm as a variant of SARAH algorithm, which does not require computation of the exact gradient, can be applied to solving general expectation minimization problems rather than only finite sum problems. The performance of iSARAH algorithm is frequently affected by the step size selection, and how to choose an appropriate step size is still a worthwhile problem for study. In this paper, we propose to use the stabilized Barzilai-Borwein (SBB) method to automatically compute step size for iSARAH algorithm, which leads to a new algorithm called iSARAH-SBB. By introducing this adaptive step size in the design of the new algorithm, iSARAH-SBB can take better advantages of both iSARAH and SBB methods. We analyse the convergence rate and complexity of the modified algorithm under the usual assumptions. Numerical experimental results on standard data sets demonstrate the feasibility and effectiveness of our proposed algorithm.

Computationally efficient direction of arrival estimation using adaptive grid selection

AbstractThe authors propose a computationally efficient approach to estimate the directions of arrival of far‐field sources impinging on a sensor array. The proposed estimator is formed using a sparse reconstruction framework, employing a novel adaptive grid selection technique to reduce the dimensionality of the used dictionary matrix. The method further makes use of a SPICE‐inspired dictionary to adaptively select an appropriate regularisation parameter and to select the active grid set. An adaptive stepsize selection strategy is also introduced to reduce computation complexity further. The proposed method allows for accurate direction‐of‐arrival (DOA) estimates using even a single snapshot, also from weak sources. Numerical simulations indicate that the method offers preferable performance in comparison to existing state‐of‐the‐art DOA estimation algorithms.

An optimal step-size simulation framework for large-scale heat-electric integrated energy system considering fault states

Dynamic simulation plays a crucial role in the development of integrated energy systems, which are typically governed by a set of partial differential equations. The selection of an appropriate simulation step-size is essential for balancing the accuracy and speed of the system simulation. However, determining the optimal simulation step-size is challenging, particularly for large-scale and fault-state systems. To address this issue, this paper proposes a novel approach for determining the optimal step-size in simulations of large-scale systems and fault-state systems. Firstly, an iteration simulation approach and a multi-step-size simulation framework are introduced to accelerate the simulation of large-scale systems without compromising accuracy. Additionally, a fault hazard evaluation model and a fault-state system variable-step-size simulation framework are developed to ensure accurate simulation results for fault-state systems. To validate the proposed methodology, comprehensive analyses are conducted on different scenarios using a 4536-node and a 38-node testbed. The results demonstrate that the proposed large-scale system simulation strategy reduces simulation time by 69.1%, while the fault-state simulation approach improves simulation accuracy by 61.99%.

Linear convergence rate analysis of a class of exact first-order distributed methods for weight-balanced time-varying networks and uncoordinated step sizes

We analyze a class of exact distributed first order methods under a general setting on the underlying network and step-sizes. In more detail, we allow simultaneously for time-varying uncoordinated step sizes and time-varying directed weight-balanced networks, jointly connected over bounded intervals. The analyzed class of methods subsumes several existing algorithms like the unified Extra and unified DIGing (Jakovetić in IEEE Trans Signal Inf Process Netw 5(1):31–46, 2019), or the exact spectral gradient method (Jakovetić et al. in Comput Optim Appl 74:703–728, 2019) that have been analyzed before under more restrictive assumptions. Under the assumed setting, we establish R-linear convergence of the methods and present several implications that our results have on the literature. Most notably, we show that the unification strategy in Jakovetić (2019) and the spectral step-size selection strategy in Jakovetić et al. (2019) exhibit a high degree of robustness to uncoordinated time-varying step sizes and to time-varying networks.