Numerical methods for second order singular pertubation problems

This research paper uses two-stage explicit fractional numerical schemes to solve fractional-order initial value problems of ODEs. The proposed methods exhibit structural symmetry in their formulation, contributing to enhanced numerical stability and balanced error behavior across computational steps. The schemes utilize constant and variable step sizes, allowing them to adapt efficiently to solve the considered fractional-order initial value problems. These schemes employ variable step-size control based on error estimation, aiming to minimize computational costs while maintaining good accuracy and stability. We discuss the linear stability of the proposed numerical schemes and observe that a higher-stability region is obtained when the fractional parameter value equals one. We also discuss consistency and convergence analysis of the proposed methods and observe that as the fractional parameter values rise from 0 to 1, the scheme’s convergence rate improves and achieves its maximum at 1. Several numerical test problems are used to demonstrate the efficiency of the proposed methods in solving fractional-order initial value problems with either constant or variable step sizes. The proposed numerical schemes’ results demonstrate better accuracy and convergence behavior than the existing methods used for comparison.

A random walk is a path that is comprised of random steps taken in random directions. Random walks present themselves in a variety of environments, and for this reason they have been a subject of relevance spanning disciplines ranging from economics to biology. A cycle in a random walk consists of the choice of a random direction, followed by a step forward, occurring for some duration. We investigated isotropic and non-isotropic random walks involving the following four cycle variations: 1. constant time, constant step size 2. constant time, varied step size 3. varied time, constant step size 4. varied time, varied step size. In biological terms, these cases of a random walks are of interest in modeling dispersal of organisms. In this study we have attempted to establish a mechanism to better study cases of non-normal diffusion in a population using a simplistic model.

To address the challenge of local optimization and step size factor parameter setting in the full attraction model of the firefly algorithm (FA), this paper introduces level-based attraction and variable step size to FA. In the level-based attraction model, fireflies are firstly grouped in different levels according to the brightness, and each firefly randomly selects two fireflies from a higher level to learn from. By using the variable step size strategy, the searching step size decreases with the number of iterations. Herein, the level-based attracting model can increase the diversity of individual learning and improve the ability to jump out of local optimization. Meanwhile, dynamic adjustment of step size can balance the detection and development ability of the algorithm and improve the optimization accuracy of the algorithm. To evaluate the effectiveness of the proposed algorithm, comparisons are drawn against optimized FA, particle swarm optimization algorithm, artificial bee colony algorithm and differential evolution algorithm on a variety of test function sets.

A variable step size total least squares affine-projection-like algorithm: Formula derivation and performance analysis

Summary form only given. This paper presents a Variable Step Size (VSS) technique that is integrated with a Stop-And-Go (SAG) approach for blind adaptive equalization of Quadrature Amplitude Modulation (QAM) signals. The proposed technique is applied to the Constant-Modulus Algorithm (CMA) and the Multi-Modulus Algorithm (MMA). The SAG approach allows the VSS technique to either vary the step size or to hold the previous one depending on a specific error criterion. This leads to the reduction of the amount of computations involved Simulation results show that algorithms with the VSS technique result in less steady-state Mean-Squared Errors (MSEs) than those obtained for algorithms without the VSS technique for the same speed of convergence.

Aiming at the low computational efficiency of the fixed-step solution of the standard parabolic equation (SPE) in the study of complex environmental radio wave propagation, a variable step size technology for the standard parabolic equation is proposed. Based on the relationship between the factors such as step size and frequency with the calculation error in the SSFT solution, the basic criteria for the SPE variable step size solution are proposed; secondly, the complex environment of SPE application is graded, and the corresponding step size is selected for different complex environment levels under the condition of satisfying the error requirement. Finally, the method is used to simulate the radio wave characteristics of typical complex environment. The calculation results verify the reliability and efficiency of the proposed method. It shows that the proposed method can greatly improve the computational efficiency of parabolic equations under the condition of ensuring the accuracy of calculation.

In wireless sensor networks, clustering is a very crucial problem. Basically clustering means grouping some specific objects based on their behavior and functionality. Clustering can be formulated for different optimization problems, such as nonsmooth, nonconvex problems. This paper is based on the review of the optimization algorithm that was proposed in the paper A Convergent Incremental Gradient Method With Constant Step Size by Blatt et al called Incremental Aggregate Gradient method. A novel algorithm called Incremental Clustered Aggregate Gradient Method was proposed in this paper to counter the shortcomings of the previous one. It has many similarities with the earlier method but it is more efficient for wireless sensor networks. The main aim of Incremental Gradient Method was to minimize the sum of continuously differentiable functions and also it required a single gradient evaluation per iteration and used a constant step size. For quadratic functions, a global linear rate of convergence was proved. It was claimed that it is more suitable for sensor networks. Although the experiments performed in this work confirm the convergence properties of it, it was found that it is not suitable for sensor networks. The proposed method addresses the flaws of the previous method as regards to sensor networks. When both algorithms operate with their respective optimal step sizes, they require approximately the same number of gradient evaluations for convergence.

An adaptive system is a well-known method for removing noise from noise-corrupted speech. The problem arises in many situations such as airplane cockpits and automobiles. The variable step size normalized least mean squares (VSS-NLMS) algorithm incorporating a variable step size parameter whose values are based on the ratio of signal-to-noise power has very good convergence speed and low steady-state misadjustment. This paper investigates the rate of convergence by the wavelet based LMS (WLMS) algorithm to the noise speech problem. Simulation results are presented to compare the performance of the variable step size NLMS algorithm, the wavelet based LMS Algorithm, and other commonly used fixed step size LMS algorithms.

This paper presents a novel approach to Lyapunov stability theory-based adaptive filter (LAF) design. The proposed design is based on the minimization of the Euclidean norm of the difference weight vector under negative definiteness constraint defined over a novel linear Lyapunov function. The proposed fixed step size LAF (FSS-LAF) algorithm is first obtained by using the method of Lagrangian multipliers. The FSS-LAF satisfying asymptotic stability in the sense of Lyapunov provides a significant performance gain in the presence of a measurement noise. The stability of the FSS-LAF algorithm is also statistically analyzed in this study. Moreover, gradient variable step size (VSS) algorithms are adapted to the FSS-LAF algorithm to further enhance the performance for the first time in this paper. These VSS algorithms are Benveniste (BVSS), Mathews and Farhang–Ang (FVSS) algorithms. Simulation results on system identification problems show that the bounds of step size for the FSS-LAF algorithm are verified, and especially, the BVSS-LAF and FVSS-LAF algorithms provide a better trade-off between steady-state mean square deviation error and convergence rate than other proposed algorithms.

Delta Sigma Analog to Digital Converters (ADCs) are superior to their counterparts in achieving high resolution for low to medium bandwidth applications. This paper discusses an improved digital calibration technique for enhancing Signal to Noise ratio (SNR) of Delta Sigma ADC which is directly related to Effective Number of Bits (ENOB). The proposed method uses the Adaptive Line Enhancer (ALE) for improving the SNR of the modulator. Adaptive Line Enhancer filters out the estimate of signal of interest from noisy input signal by updating filter coefficients using Least Mean Square (LMS) Algorithm. Variable step size (VSS) LMS is used for updating filter coefficients in Adaptive Line Enhancer instead of Fixed step size (FSS) LMS to reduce the required number of iterations. The simulation results for second order single loop Delta Sigma modulator show approximately 20dB improvement in SNR after calibration and substitution of VSS LMS with FSS LMS has reduced approximately 4000 iterations.

The step size of least mean square (LMS) algorithm is significant for its performance. To be specific, small step size can get small excess mean square error but results in slow convergence. However, large step size may cause instability. Many variable step size least mean square (VSSLMS) algorithms have been developed to enhance the control performance. In this paper, a new VSSLMS was proposed based on Kwong’s algorithm to evaluate the robustness. The approximate analysis of dynamic and steady-state performance of this developed VSSLMS algorithm was given. An active vibration control system of piezoelectric cantilever beam was established to verify the performance of the VSSLMS algorithms. By comparing with the current VSSLMS algorithms, the proposed method has better performance in active vibration control applications.

Myosin VI dimer walks toward the minus end of the actinfilamentwith a large and variable step size of 25–36 nm. Two competingmodels have been put forward to explain this large step size. TheSpudich model assumes that the myosin VI dimer associates at a distaltail near the cargo-binding domain, which makes two full-length singleα-helix (SAH) domains serve as long legs. In contrast, the Houdusse–Sweeneymodel assumes that the association occurs in the middle (between residues913 and 940) of the SAH domain and that the three-helix bundles unfoldto ensure the large step size. Their consistency with the observationof stepping motion with a large and variable step size has not beenexamined in detail. To compare the two proposed models of myosin VI,we computationally characterized the free energy landscape experiencedby the leading head during the stepping movement along the actin filamentusing the elastic network model of two heads and an implicit modelof the SAH domains. Our results showed that the Spudich model is moreconsistent with the 25–36 nm step size than the Houdusse–Sweeneymodel. The unfolding of the three-helix bundles gives rise to thefree energy bias toward a shorter distance between two heads. Besides,the stiffness of the SAH domain is a key factor for giving strongenergetic bias toward the longer distance of stepping. Free energyanalysis of the stepping motion complements the visual inspectionof static structures and enables a deeper understanding of underlyingmechanisms of molecular motors.

The effects of two different techniques for implementing variable mesh sizes in multistep formulas are investigated. It is proved that one is stable for a greater variation in step sizes than the other with Adams formulas, but that both are stable when the rate of step changing is small with any multistep formula. The practical implications of these results are discussed.

This paper proposes a new application of fuzzy logic to variable step size least mean square (VS-LMS) adaptive beamforming algorithm in CDMA systems. The proposed algorithm adjusts the step size of the least mean square (LMS) by using the application of fuzzy logic in which the increase or decrease of step size depends on the fuzzy inference results of the mean square error (MSE). Computer simulation results show that the proposed algorithm has a better capacity of tracking compared with the conventional LMS algorithms and other variable step size LMS algorithms.

This paper incorporates a correntropy-induced metric (CIM)-based sparsity constraint into variable step size normalized least mean square (VSSNLMS) algorithm for identification of sparse system corrupted by additive noise. The proposed CIM-VSSNLMS algorithm makes a good trade-off between convergence characteristics, filter stability, and steady state error. The proposed (CIM-VSSNLMS) algorithm incorporates a variable step size that accelerates the convergence characteristics and lowers the normalized misalignment (NMSA) error with respect to CIM-NLMS with constant value of step size. An expression for variant step size is derived under the stability condition of proposed algorithm. Finally, the implementation of the proposed algorithm is carried out in MATLAB software to manifest the improved estimated behavior in the identification of sparse system.