Adaptive Step Size Research Articles

Adaptive Stepsize Algorithms for Langevin Dynamics

Adaptive Stepsize Algorithms for Langevin Dynamics

The Performance of Fixed Step Size and Adaptive Step Size Numerical Methods for Solving Deterministic Cell-Growth Models

Deterministic cell-growth models describe the growth of cell populations using fixed mathematical rules, assuming no randomness in the system. These models are often based on differential equations that account for the rates of cell division, death, and other biological processes. The solution to the system is obtained via numerical methods. Most of the developed approaches are based on fixed step sizes. However, fixed step size implementation failed to offer the optimal solutions when dealing with stiff challenges. Fixed step size methods can be unstable for stiff equations, where some components of the solution change much more rapidly than others. The step size, required to maintain stability can become impractically small. Thus, the adaptive step size method is required. Adaptive step size methods adjust the step size dynamically based on the behavior of the solution, aiming to maintain a desired level of accuracy while optimizing computational efficiency. These methods are particularly useful for solving ordinary differential equations (ODEs) where the solution can vary rapidly in some regions and slowly in others. This study is devoted to comparing the implementation of fixed step size and adaptive step size in solving ordinary differential equations (ODEs). The fixed step size and adaptive step size numerical method are solved in this study via the fourth order Runge-Kutta method (RK4) and Runge-Kutta Fehlberg 45 method (RKF45). The performance of both numerical methods used will be analyzed by comparing the numerical results approximated with the actual data. Subsequently, the absolute error, relative error, and rounding-off error will be calculated to compare both approaches. Based on the more precise findings, this work has shown that adaptive step size is predicted to be the optimal representation for solving ODEs. As a result, this may help mathematicians to choose the most effective numerical approach for solving ODEs.

On the Impossibility of Convergence of Mixed Strategies with Optimal No-Regret Learning

We study the limiting behavior of the mixed strategies that result from optimal no-regret learning in a repeated game setting where the stage game is any 2 × 2 competitive game. We consider optimal no-regret algorithms that are mean-based and monotonic in their argument. We show that for any such algorithm, the limiting mixed strategies of the players cannot converge almost surely to any Nash equilibrium. This negative result is also shown to hold under a broad relaxation of these assumptions, including popular variants of Follow-the-Regularized Leader with optimism or adaptive step sizes. Finally, we provide partial evidence that the monotonicity and mean-based assumptions can be removed or relaxed. Our results identify the inherent stochasticity in players’ realizations as a critical factor underlying this divergence, and demonstrate a crucial difference in outcomes between using the opponent’s mixtures and realizations to make updates. Funding: V. Muthukumar was supported by a Simons-Berkeley Research Fellowship, NSF awards IIS-2212182 and CCF-2239151, and generous gifts from Amazon, Adobe, and Google. S. Phade acknowledges the support of the NSF [Grants CNS-1527846 and CCF-1618145] and the NSF Science & Technology Center [Grant CCF-0939370 (Science of Information)]. A. Sahai acknowledges the support of the ML4Wireless center member companies and the NSF [Grants AST-144078 and ECCS-1343398].

5G/GNSS Integrated Vehicle Localization With Adaptive Step Size Kalman Filter

5G/GNSS Integrated Vehicle Localization With Adaptive Step Size Kalman Filter

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.

A multi-strategy improved sparrow search algorithm for indoor AGV path planning

To address the problems of weak search ability, easily falling into local optimal solutions and poor path quality of sparrow search algorithm in AGV path planning, a multi-strategy improved sparrow search algorithm (MISSA) is proposed in this paper. MISSA improves the global search ability by improving the discoverer position update operator and introducing the sine cosine algorithm; adopts the adaptive number of vigilantes and adaptive adjustment step size to improve the convergence speed; introduces the Levy flight variation strategy to reduce the probability of falling into any local optimal solution; optimizes the boundary handling mechanism to prevent the loss of population diversity at a later stage; finally, uses the large-scale neighborhood search strategy and path smoothing mechanism for path optimization to further improve the path quality. The superiority-seeking ability of MISSA was verified by 12 standard test functions, and then 30 simulation experiments were conducted in grid maps with two specifications of 20×20 and 30×30. The experimental results showed that, by using MISSA, the path length was reduced by 44.1% and 63.1%, the number of turns was reduced by 68.4% and 78.4%, and the risk degree was reduced by 61.3% and 77.2%, which verifies the superiority of MISSA in path planning. Finally, MISSA was ported to the QBot2e mobile robot for physical verification to prove its feasibility in practical applications.

Improved active vibration control of a cantilever beam using MFC actuators with Hammerstein model-based hysteresis modeling and VSS-FxLMS control algorithm

This study aims to enhance active vibration control in a cantilever beam using macro fiber composite (MFC) actuators. To address the challenge of accurately modeling the complex hysteresis behavior of MFC actuators, an advanced hysteresis modeling technique is employed, integrating a non-symmetric Bouc–Wen model with an Auto-Regressive model with eXogenous inputs (ARX). The parameter estimation of this model is optimized using an adaptive mutation factor, and multiple mutation strategy differential evolution (AmFMMDE) algorithm. Additionally, a Variable Step Size Filtered-x Least Mean Square (VSS-FxLMS) algorithm with step size adaptation based on normalized error change is utilized for effective vibration control. This approach shows promising potential for practical engineering applications requiring precise and efficient active vibration control.

Robust variable step size least mean fourth with quotient form-based control scheme for DVR operation

ABSTRACT A robust variable step size least mean fourth with a quotient form control scheme (RVSSLMFQ) is implemented for addressing power quality concerns and delivering clean voltage to sensitive loads through a dynamic voltage restorer (DVR). This method estimates the fundamental components from distorted grid voltages, featuring an automated process with adaptive time-varying step sizes and utilizing a quotient structure to reduce mean squared error (MSE) effectively. The suggested control scheme overcomes the stability and performance issues of the least mean fourth (LMF) algorithm by improving convergence and adapting well to varying step sizes. Furthermore, RVSSLMFQ achieves remarkable results with reduced settling time and fewer peak overshoots compared to the least mean square (LMS) and LMF control algorithms. For tuning the load voltage and DC-link voltage proportional-integral (PI) controllers, the improved grey wolf optimization technique (I-GWO) is applied. The results are evaluated through MATLAB simulations.

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 novel hybrid GMPPT method for PV systems to mitigate power oscillations and partial shading detection

The Maximum Power Point Tracking (MPPT) algorithm plays a crucial role in maximizing the power output of a PV system. While uniform shading conditions (USCs) present challenges related to dynamic changes in irradiance, partial shading conditions (PSCs) introduce the complexity of multiple local maxima and rapidly changing shading patterns. Conventional algorithms often struggle to effectively track the global maximum power point (GMPP) during PSCs leading to power losses and oscillations around the MPP. In contrast, advanced algorithms are more complex and require additional computational time causing the algorithm to unnecessarily scan the entire PV curve during USCs, thereby extending the tracking time. This paper proposes a hybrid approach that combines the Modified Perturb and Observe (MP&O) and Modified Coot Optimization Algorithm (MCOA) methods termed as MP&O-MCOA as a robust real time MPPT algorithm. The MP&O method with trapezoidal rule and adaptive step size is employed under USCs, while the MCOA with only one tuning parameter and fewer random numbers is utilized during PSCs to rapidly identify varying conditions. Experimental validation using a boost converter has demonstrated the effectiveness of the proposed method, achieving the fastest average tracking times of 0.24 s under USCs and 0.73 s under PSCs and highest average tracking accuracy above 99 % for all operating conditions tested. The proposed method has proven to outperform other existing MPPT methods under various weather conditions due to its ability to distinguish between USCs and PSCs at exceptional speed and accuracy.

A Numerical Integrator for Kinetostatic Folding of Protein Molecules Modeled as Robots with Hyper Degrees of Freedom

The kinetostatic compliance method (KCM) models protein molecules as nanomechanisms consisting of numerous rigid peptide plane linkages. These linkages articulate with respect to each other through changes in the molecule dihedral angles, resulting in a kinematic mechanism with hyper degrees of freedom. Within the KCM framework, nonlinear interatomic forces drive protein folding by guiding the molecule’s dihedral angle vector towards its lowest energy state in a kinetostatic manner. This paper proposes a numerical integrator that is well suited to KCM-based protein folding and overcomes the limitations of traditional explicit Euler methods with fixed step size. Our proposed integration scheme is based on pseudo-transient continuation with an adaptive step size updating rule that can efficiently compute protein folding pathways, namely, the transient three-dimensional configurations of protein molecules during folding. Numerical simulations utilizing the KCM approach on protein backbones confirm the effectiveness of the proposed integrator.

Suppression of cross-interference in the absorption spectra of gas mixtures

In the quantitative analysis of mixed gases by tunable diode laser absorption spectroscopy, the overlapping of absorption spectra and mutual interference of multi-component gases can lead to problems of large measurement errors and low analysis accuracy. In this paper, an improved firefly algorithm is proposed and applied to the support vector machine regression model to solve this problem. The specific method includes introducing an adaptive step size to balance the local and global searches and using the gradient descent method to accelerate the parameter optimization process so as to improve the model’s generalization ability and prediction accuracy. The experimental results show that the maximum errors of the improved algorithm in the prediction of CH4 and CO gas concentrations are no more than 0.0443 % and 2 ppm, with coefficients of determination, R2, of 0.9994 and 0.99815. The promising results obtained by the system provide theoretical support for the realization of high-precision detection of multicomponent gases with a single source of light, and also demonstrate the high efficiency and feasibility of the method in practical detection.

Painless Stochastic Conjugate Gradient for Large-Scale Machine Learning.

Conjugate gradient (CG), as an effective technique to speed up gradient descent algorithms, has shown great potential and has widely been used for large-scale machine-learning problems. However, CG and its variants have not been devised for the stochastic setting, which makes them extremely unstable, and even leads to divergence when using noisy gradients. This article develops a novel class of stable stochastic CG (SCG) algorithms with a faster convergence rate via the variance-reduced technique and an adaptive step size rule in the mini-batch setting. Actually, replacing the use of a line search in the CG-type approaches which is time-consuming, or even fails for SCG, this article considers using the random stabilized Barzilai-Borwein (RSBB) method to obtain an online step size. We rigorously analyze the convergence properties of the proposed algorithms and show that the proposed algorithms attain a linear convergence rate for both the strongly convex and nonconvex settings. Also, we show that the total complexity of the proposed algorithms matches that of modern stochastic optimization algorithms under different cases. Scores of numerical experiments on machine-learning problems demonstrate that the proposed algorithms outperform state-of-the-art stochastic optimization algorithms.

MCEND: an open-source program for quantum electron-nuclear dynamics

The software MCEND (Multi-Configuration Electron-Nuclear Dynamics) is a free open-source program package which simulates the quantum dynamics of electron-nuclei simultaneously for diatomic molecules. Its formulation, implementation, and usage are described in detail. MCEND uses a grid-based basis representation for the nuclei, and the electronic basis is derived from standard electronic structure basis sets on the nuclear grid. The wave function is represented as a sum over products of electronic and nuclear wave functions, thus capturing correlation effects between electrons, nuclei, and electrons and nuclei. The LiH molecule was used as an example for simulating the molecular properties such as the dipole moment and absorption spectrum. Program summaryProgram Title: MCEND, v.2.6.0CPC Library link to program files:https://doi.org/10.17632/tkb9dwf85t.1Developer's repository link:https://github.com/MCEND-hub/MCEND (https://github.com/MCEND-hub/MCEND-library and https://github.com/MCEND-hub/MCEND-tools are git submodules of MCEND)Licensing provisions: MITProgramming language: Fortran 90 and Python 3External routines/libraries: FFTW, OpenMP, BLAS, LAPACK, PSI4, Matplotlib, mendeleev, NumPy, Pandas, SciPy, PyTablesNature of problem: MCEND is to simulate the quantum dynamics of electrons and nuclei simultaneously at multiconfiguration levels.Solution method: The presented program package solves the time-dependent Schrödinger equation with the wave function represented as sum over configuration products using an 8th-order adaptive step size Runge-Kutta ordinary differential equation (ODE) solver. The software can be extended by supplementing modules on the existing infrastructure.

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.

A modified inertial shrinking projection algorithm with adaptive step size for solving split generalized equilibrium, monotone inclusion and fixed point problems

A modified inertial shrinking projection algorithm with adaptive step size for solving split generalized equilibrium, monotone inclusion and fixed point problems

On the adaptive solution of phase‐field problems with A‐stable explicit last‐stage diagonally implicit Runge–Kutta (ELDIRK) methods

AbstractRunge–Kutta algorithms with adaptive step size control provide reliable tools for the solution of initial value problems with diagonally implicit Runge‐Kutta (DIRK) methods as the most common approach. In this contribution, the new low‐order explicit last‐stage diagonally implicit Runge–Kutta (ELDIRK) methods are investigated, combining implicit schemes with an additional explicit evaluation as an explicit last stage. This results in Butcher tableaus with two solutions of different convergence orders suitable for embedded methods, where the higher‐order solution is achieved by additional explicit evaluations. Thus, the iterative solution of non‐linear systems is omitted for the additional stage, presenting a major reduction in computational cost for the determination of a local error estimate. The key contribution is the application of the novel Butcher tableaux to phase‐field problems, solved with the finite‐element method, leading to substantial numerical investigations with an efficient approach for DIRK schemes. The most important aspects are the investigations of stability properties which lead to the novel class of A‐stable ELDIRK methods. In accordance, the study of the convergence orders is presented. A local error estimator is presented, such that adaptive step size control for the new low‐order embedded schemes based on an empirical approach for error estimation is achieved. A suitable parallel algorithm is presented with conclusive two‐dimensional phase‐field simulations based on a Kobayashi‐Warren‐Carter model. The higher‐order convergence suggested by the novel schemes is confirmed, and their effective results are demonstrated, resulting in a valuable semi‐explicit addition to the family of Runge–Kutta time integration schemes.

Design and development of different adaptive MPPT controllers for renewable energy systems: a comprehensive analysis

As of now, all over the world is focusing on the Electric Vehicle (EV) technology because its features are low environmental pollution, less maitainence cost required, high robustness, and good dynamic response. Also, the EVs work continuously until the input fuel is provided to the fuel stack. Here, a Proton Exchange Membrane Fuel Cell (PEMFC) is used as an input source to the electric vehicle system because of its merits fast startup, and quick response. However, the PEMFC gives nonlinear voltage versus current characteristics. As a result, the extraction of maximum power from the fuel stack is very difficult. The main aim of this work is to study different Maximum Power Point Tracking Techniques (MPPT) for the DC-DC converter-fed PEMFC system. The studied MPPT controllers are Adjusted Step Value of Perturb & Observe (ASV with P&O), Adaptive Step Size with Incremental Conductance (ASS with IC), Radial Basis Functional Network (RBFN), Incremental Step-Fuzzy Logic Controller (IS with FLC), Continuous Step Variation based Particle Swarm Optimization (CSV with PSO), and Adaptive Step Value-Cuckoo Search Algorithm (ASV with CSA). The selected MPPT controllers’ comprehensive study has been in terms of maximum power extraction, tracking speed of the MPP, settling time of the fuel stack output voltage, oscillations across the MPP, and design complexity. From the comprehensive performance results of the hybrid MPPT controllers, the ASV with CSA technique gives superior performance when equated to the other MPPT controllers.

An adaptive fatigue crack growth model at the welded joints of steel bridge

Under the influence of random loads such as those from vehicles, the welded details in steel bridges tend to gradually develop into macrocracks, typically originating from manufacturing defects or stress concentration points. As the service life of the steel bridge increases, the continuous generation and propagation of fatigue cracks lead to a decline in the bridge's service performance, potentially resulting in catastrophic failures such as bridge collapse. Accurately analyzing the fatigue crack growth patterns at welded joints is a prerequisite for reliably assessing the structural service performance. The analysis of fatigue crack propagation depends on both the crack growth step size and the cyclic stress–strain response at the welded joint under loading. Considering the complexity of the material properties and stress concentration at the welded joint, which leads to a lack of analytical solutions for the cyclic stress–strain characteristics at the crack tip, numerical analysis was employed to obtain the cyclic stress–strain response at the crack tip of the welded joints. Using the size of the cyclic plastic zone at the crack tip as an adaptive crack growth step size, a model for fatigue crack growth was developed based on the plastic strain energy of the cyclic plastic zone. Fatigue test specimens extracted from the welded toe of cruciform welded joints were tested, providing cyclic stress–strain (Ramberg-Osgood material parameters) and fatigue damage characteristics (Manson-Coffin material parameters) of the material at the welded toe. High-cycle fatigue tests using cruciform welded joint specimens determined a good correlation between the crack propagation characteristic dimensions obtained from the adaptive numerical analysis model and the experimental results. The experimental findings confirmed the applicability of the established adaptive numerical analysis model for studying fatigue crack growth in cruciform welded joints. The parameters introduced in this model are physically meaningful, easy to obtain, and suitable for engineering applications.

Adaptive stepsize forward–backward pursuit and acoustic emission-based health state assessment of high-speed train bearings

Compressed sensing (CS) is a promising tool for data compression reconstruction. However, fault diagnosis methods for high-speed train bearings based on CS and acoustic emission (AE) technologies have not been reported yet. Notably, the accuracy and speed of CS two-stage reconstruction methods are affected and restricted by prior initial conditions. Therefore, this article proposes adaptive dynamic thresholds applicable to adaptive stepsize forward–backward pursuit (ASFBP), and bearing health state assessment method. First, the adaptive dynamic thresholds for atom selection and deletion are constructed based on the residual feedback mechanism and the atom quality distribution law, which enables ASFBP to realize high-precision rapid reconstruction of signal without any atom priori initial conditions. Second, the initial dictionary length is improved based on the AE hit characteristics. Furthermore, a damage state comprehensive evaluation index (DSCEI) is established using principal component analysis based on AE time-domain hit parameters and compression-domain energy parameter. Compared with the kurtosis index and permutation entropy index, the DSCEI demonstrates better monotonicity and stability in the quantitative evaluation of high-speed train bearing condition. Finally, the validity and stability of the method are verified by testing under complex test conditions resembling actual high-speed train lines, providing valuable insights for the CS-based data-driven bearing fault diagnosis.