Small Step Size Research Articles

Antenna Array Design by Artificial Bee Colony Algorithm With Similarity Induced Search Method

Antenna array has been extensively applied to wireless communication systems. Optimal design of antenna array is very useful to attain desired spatial beam and reduce signal degradation. Potential search directions in artificial bee colony (ABC) algorithm are mixed with both large step size and small step size. This could lead to the malfunction of employed bee stage and onlooker bee stage of ABC algorithm. This paper proposes to improve the effectiveness of ABC by a similarity induced search method. This method adaptively adjusts search step size. Employed bees perform large step size search, while onlooker bees do small step size search. Scout bees could probe potential high-quality solutions in neighborhoods. The proposed algorithm is tested on antenna array design problem. Compared with four other algorithms, the results verify the effectiveness of the proposed algorithm. Furthermore, the scalability of the proposed algorithm is also analyzed in terms of array element and computational time. Scalability analysis also shows that the proposed algorithm is suitable for solving antenna array design problems.

Multi-Step Differential Transform Method for Solving the Influenza Virus Model with Disease Resistance

The spread of the influenza virus with disease resistance has been modeled by the SEIR epidemic model. The model is of the form a system of four nonlinear differential equations where its exact solution is difficult to be determined. In this paper, the model is solved by a multi-step differential transform method (MS-DTM), which is a semi-analytical method. In this case, the domain of computation is divided into a finite number of sub-intervals. At each sub-interval, we apply DTM and continuity condition to ensure the continuity solutions. To see the effectiveness of the MS-DTM, we perform some comparative study between the MS-DTM and the MATLAB ode45 routine. Our MS-DTM solutions have good agreement with those of ode45 routine. The accuracy of MS-DTM can be improved by taking a smaller step size or increase the number of terms in each subinterval.

Adaptation and Learning Over Networks Under Subspace Constraints—Part I: Stability Analysis

This paper considers optimization problems over networks where agents have individual objectives to meet, or individual parameter vectors to estimate, subject to subspace constraints that require the objectives across the network to lie in low-dimensional subspaces. This constrained formulation includes consensus optimization as a special case, and allows for more general task relatedness models such as smoothness. While such formulations can be solved via projected gradient descent, the resulting algorithm is not distributed. Starting from the centralized solution, we propose an iterative and distributed implementation of the projection step, which runs in parallel with the stochastic gradient descent update. We establish in this Part I of the work that, for small step-sizes $\mu$, the proposed distributed adaptive strategy leads to small estimation errors on the order of $\mu$. We examine in the accompanying Part II [2] the steady-state performance. The results will reveal explicitly the influence of the gradient noise, data characteristics, and subspace constraints, on the network performance. The results will also show that in the small step-size regime, the iterates generated by the distributed algorithm achieve the centralized steady-state performance.

Discretized fast-slow systems near transcritical singularities

We study a transcritical singularity in a fast-slow system given by the explicit Euler discretization of the corresponding continuous-time normal form. The analysis uses the blow-up method and direct trajectory-based estimates. We prove that the qualitative behaviour is preserved by a time-discretization with sufficiently small step size. This step size is fully quantified relative to the time scale separation. Our proof also yields the continuous-time results as a special case and provides more detailed calculations in the classical (or scaling) chart.

Theoretical and numerical analysis for Volterra integro-differential equations with Itô integral under polynomially growth conditions

In this paper, we theoretically and numerically deal with nonlinear Volterra integro-differential equations with Itô integral under a one-sided Lipschitz condition and polynomially growth conditions. It is proved that both the exact solutions and vector fields are bounded and satisfy a Hölder condition in the pth moment sense. Analogously, the boundedness and Hölder condition in the pth moment sense are preserved by the semi-implicit Euler method for sufficiently small step-size. Moreover, by the local truncated errors, we prove the strong convergence order 1. Finally, numerical simulations on stochastic control models and stochastic Ginzburg–Landau equation illustrate our results.

Stability equivalence between the stochastic differential delay equations driven by [formula omitted]-Brownian motion and the Euler–Maruyama method

Consider a stochastic differential delay equation driven by G-Brownian motion (G-SDDE) dx(t)=f(x(t),x(t−τ))dt+g(x(t),x(t−τ))dB(t)+h(x(t),x(t−τ))d〈B〉(t). Under the global Lipschitz condition for the G-SDDE, we show that the G-SDDE is exponentially stable in mean square if and only if for sufficiently small step size, the Euler–Maruyama (EM) method is exponentially stable in mean square. Thus, we can carry out careful numerical simulations to investigate the exponential stability of the underlying G-SDDE in practice, in the absence of an appropriate Lyapunov function. A numerical example is provided to illustrate our results.

Solution of Linear and Non-Linear Boundary Value Problems Using Population-Distributed Parallel Differential Evolution

Abstract Cases where the derivative of a boundary value problem does not exist or is constantly changing, traditional derivative can easily get stuck in the local optima or does not factually represent a constantly changing solution. Hence the need for evolutionary algorithms becomes evident. However, evolutionary algorithms are compute-intensive since they scan the entire solution space for an optimal solution. Larger populations and smaller step sizes allow for improved quality solution but results in an increase in the complexity of the optimization process. In this research a population-distributed implementation for differential evolution algorithm is presented for solving systems of 2 nd -order, 2-point boundary value problems (BVPs). In this technique, the system is formulated as an optimization problem by the direct minimization of the overall individual residual error subject to the given constraint boundary conditions and is then solved using differential evolution in the sense that each of the derivatives is replaced by an appropriate difference quotient approximation. Four benchmark BVPs are solved using the proposed parallel framework for differential evolution to observe the speedup in the execution time. Meanwhile, the statistical analysis is provided to discover the effect of parametric changes such as an increase in population individuals and nodes representing features on the quality and behavior of the solutions found by differential evolution. The numerical results demonstrate that the algorithm is quite accurate and efficient for solving 2 nd -order, 2-point BVPs.

SOLUSI HAMPIRAN PERSAMAAN LOGISTIK NON-AUTONOMOUS

The aim of this study is to derive the approximation solution of the non-autonomous logistic equation with a non-constant carrying capacity. The solution is found via predictor-corrector method (Adams-Bashforth-Moulton method, Milne method and Hamming method). The approximation solution that obtained, then, is compared to the exact solution. The results show that, for small step size, the approximation solution approximate the exact solution is in good agreement.

Numerical solution to highly nonlinear neutral-type stochastic differential equation

In the paper, our main aim is to investigate the strong convergence of the implicit numerical approximations for neutral-type stochastic differential equations with super-linearly growing coefficients. After providing mean-square moment boundedness and mean-square exponential stability for the exact solution, we show that a backward Euler–Maruyama approximation converges strongly to the true solution under polynomial growth conditions for sufficiently small step size. Imposing a few additional conditions, we examine the p-th moment uniform boundedness of the exact and approximate solutions by the stopping time technique, and establish the convergence rate of one half, which is the same as the convergence rate of the classical Euler–Maruyama scheme. Finally, several numerical simulations illustrate our main results.

Effect of backward facing step on combustion stability in a constant contact area cylindrical meso­scale combustor

This experiment investigates the effect of backward facing step size variation on combustion stability in the cylindrical meso-scale combustor with the constant contact area. The backward facing step was varied by changing the combustor inlet diameter while the combustor diameter at the combustion zone was kept constant, i.e. has a constant contact area. Butane gas (C 4 H 10 ) was used as fuel with air as an oxidizing agent. The result shows that the backward facing step has an important role in the combustion stabilization mechanism. Stable flame could be stabilized inside the meso-scale combustor with the backward facing step. Without the backward facing step, the flame blows out, then stable at the combustor rim. Recirculation flow occurs in the area behind the backward facing step. The increasing backward facing step size leads to an increase in reactant inlet velocity, recirculation flow size and shear stress in the area near the backward facing step. At large backward facing step size, the high reactant inlet velocity together with the large shear stress quenches the flame while the heat recovered by recirculation flow is less sufficient to stabilize flame so that the flame drifts to the downstream position. Hence bigger backward facing step size causes narrower flame stability limit area. The smaller the backward facing step size the wider the flame stability limit which shifts more toward lower equivalence ratio and high reactant velocity regions. Decreasing the backward facing step size decreases the reactant velocity into the combustion reaction zone as well as decreases the recirculation flow and the shear stress, so that the quenching effect decreases. The smaller recirculation flow has a better function for the flame holder to increase flame stability inside the meso-scale combustor. Therefore, small backward facing step size has a very important role in recovering heat energy stabilizing the flame in the meso-scale combustor

Variable step size P&O MPPT algorithm for optimal power extraction of multi-phase PMSG based wind generation system

To eradicate limitations of conventional fixed step-size perturb and observe (P&O) maximum power point tracking (MPPT) based on the variable speed - wind energy conversion system (VS-WECS), this article proposes a fast and an efficient MPPT modular sectors algorithm. The variable-step perturb and observe (VS-PO) MPPT algorithm is implemented to divide the power-speed (P-ω) curve into modular sectors each with specific step-size. The modular sectors are established by comparing a special synthesized ratio, that associates with the theoretical power, with a specified ratio corresponding the required power accuracy. Hence, the proper step-size is selected according to the operating sector location. For the sectors near the maximum power point (MPP), a small step-size is applied. Otherwise, a large step-size is used. To calculate the theoretical power, a wind speed estimation algorithm is utilized for evaluating the wind speed. The system configuration involves a 1.5 MW five-phase permanent magnet synchronous generator (PMSG), and a back-to-back frequency converter with the dc-link capacitor for the grid integration. The simulation results illustrate the VS-PO superiority over both the conventional P&O (CPO) and modified P&O (MPO) techniques. The VS-PO improves the initial speed tracking and minimizes the steady state oscillations. The performance of the proposed control schemes is validated using MATLAB/SIMULINK environment.

On complexity of a new Mehrotra-type interior point algorithm for P_{*}(kappa ) linear complementarity problems

In this paper, a variant of Mehrotra-type predictor–corrector algorithm is proposed for P_{*}(kappa ) linear complementarity problems. In this algorithm, a safeguard step is used to avoid small step sizes and a new corrector direction is adopted. The algorithm has polynomial iteration complexity and the iteration bound is O ((14kappa +11)sqrt{(1+4 kappa )(1+2kappa )}~nlog frac{(x^{0})^{T}s^{0}}{varepsilon } ). Some numerical results are reported as well.

Faster subgradient methods for functions with Hölderian growth

The purpose of this manuscript is to derive new convergence results for several subgradient methods applied to minimizing nonsmooth convex functions with H\"olderian growth. The growth condition is satisfied in many applications and includes functions with quadratic growth and weakly sharp minima as special cases. To this end there are three main contributions. First, for a constant and sufficiently small stepsize, we show that the subgradient method achieves linear convergence up to a certain region including the optimal set, with error of the order of the stepsize. Second, if appropriate problem parameters are known, we derive a decaying stepsize which obtains a much faster convergence rate than is suggested by the classical $O(1/\sqrt{k})$ result for the subgradient method. Thirdly we develop a novel "descending stairs" stepsize which obtains this faster convergence rate and also obtains linear convergence for the special case of weakly sharp functions. We also develop an adaptive variant of the "descending stairs" stepsize which achieves the same convergence rate without requiring an error bound constant which is difficult to estimate in practice.

KSVD-Based Multiple Description Image Coding

In this paper, we present a new multiple description coding scheme, which is based on a sparse dictionary training method called K singular value decomposition (KSVD). In the proposed scheme, each description encodes one source subset with a small quantization stepsize, and other subsets are predictively coded with a large quantization stepsize. The source processed by the KSVD becomes sparse, which can improve the coding efficiency. The proposed scheme is then applied to lapped transform-based multiple description image coding. Finally, image coding results show that the proposed scheme achieves a better performance than the current state-of-the-art multiple description coding methods.

Almost sure exponential stability of stochastic differential delay equations

This paper mainly studies whether the almost sure exponential stability of stochastic differential delay equations (SDDEs) is shared with that of the stochastic theta method. We show that under the global Lipschitz condition the SDDE is pth moment exponentially stable (for p 2 (0; 1)) if and only if the stochastic theta method of the SDDE is pth moment exponentially stable and pth moment exponential stability of the SDDE or the stochastic theta method implies the almost sure exponential stability of the SDDE or the stochastic theta method, respectively. We then replace the global Lipschitz condition with a finite-time convergence condition and establish the same results. Hence, our new theory enables us to consider the almost sure exponential stability of the SDDEs using the stochastic theta method, instead of the method of Lyapunov functions. That is, we can now perform careful numerical simulations using the stochastic theta method with a sufficiently small step size ?t. If the stochastic theta method is pth moment exponentially stable for a sufficiently small p ? (0,1), we can then deduce that the underlying SDDE is almost sure exponentially stable. Our new theory also enables us to show the pth moment exponential stability of the stochastic theta method to reproduce the almost sure exponential stability of the SDDEs.

Convergence and stability of the one-leg θ method for stochastic differential equations with piecewise continuous arguments

The equivalent relation is established here about the stability of stochastic differential equations with piecewise continuous arguments(SDEPCAs) and that of the one-leg ? method applied to the SDEPCAs. Firstly, the convergence of the one-leg ? method to SDEPCAs under the global Lipschitz condition is proved. Secondly, it is proved that the SDEPCAs are pth(p 2 (0; 1)) moment exponentially stable if and only if the one-leg ? method is pth moment exponentially stable for some sufficiently small step-size. Thirdly, the corollaries that the pth moment exponential stability of the SDEPCAs (the one-leg ? method) implies the almost sure exponential stability of the SDEPCAs (the one-leg ? method) are given. Finally, numerical simulations are provided to illustrate the theoretical results.

Deciphering the deformation mechanism in single point incremental forming: experimental and numerical investigation

In the present work, deformation mechanism in single-point incremental forming (SPIF) of drawing quality steel with a fully ferritic microstructure was studied. The effect of tool diameter and vertical step size on the micromechanisms of plastic deformation in SPIF was investigated by observing changes in microstructure and lattice rotation. It was observed that the fraction of grains with {111} ‖ normal direction (ND), which constitutes the gamma fiber in BCC materials, decreased with decrease in tool diameter and vertical step size. It is known that the state of deformation in SPIF is near to plane strain with the direction of major strain being always perpendicular to tool travel direction and negligible strain parallel to tool movement direction. Microstructural evidence for this observation and also for the presence of through thickness shear (TTS) components at smaller step size and tool diameter was observed. Viscoplastic self-consistent (VPSC) simulations revealed that the activity of $$ \left\{112\right\} $$ slip system decreased in comparison to $$ \left\{110\right\} $$ slip system in the presence of TTS which manifested as the deviation from {111} ‖ ND position at smaller step size and tool diameter.

Advanced multi-sector P&O maximum power point tracking technique for wind energy conversion system

This article suggests a new solution for the limitations of conventional fixed step-size perturb and observe (PO) maximum power point tracking (MPPT) of the wind energy conversion systems (WECSs). The proposed algorithm offers a new fast MPPT technique and it keeps the controller searching in the correct region on the power-speed curve. For this issue, a variable-step perturb and observe (VS-PO) MPPT technique based on multisector operation is proposed. The proposed technique divides the operating region into four sectors by comparing the power-speed curve and a special synthesized curve. Hence, the selection of the appropriate step-size is related to the operating sector. For the two sectors close to the maximum power point (MPP), a small step-size is applied. Otherwise, the controller utilizes a large step-size. The large-scale grid-tied WECS is composed of the back-to-back (BTB) converter. The BTB converter is divided into dual sides; the machine side converter and the grid side converter. The proposed technique has been validated over a large scale of 1.5 MW grid-tied PMSG-WECS using MATLAB/SIMULINK. The results prove the efficient operation of the proposed technique based on the improved efficiency and the fast system response compared to conventional P&O techniques.

New feedforward filtered-x least mean square algorithm with variable step size for active vibration control

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.

Deposition Characteristics of SrZrO3 Thermal Barrier Coating Prepared by Solution Precursor Plasma Spray Process

The crystallization kinetics of the dried SrZrO3 precursor powders were investigated by DSC measurements. The crystallization activation energy of ZrO2 in the SrZrO3 precursor powders is 54.9 kJ/mol. And the Avrami parameter value is 2.16, indicating a two-dimensional, diffusion-controlled nucleation and growth mechanism of ZrO2 nuclei in the SrZrO3 precursor powders. The SrZrO3 thermal barrier coating (TBC) was prepared by solution precursor plasma spray (SPPS) process. The coating deposition characteristics were investigated by designing different spray models (single-scan spray, multiple-scan spray, and raster-scan spray) with optimized spray parameters. The surface and cross-sectional morphology of the SrZrO3 coatings were characterized by scanning electron microscope (SEM). The experimental results showed that the atomization pressure of liquid feedstock has strong influence on the trajectory of the precursor droplets in the plasma jet. The inter-pass boundary (IPB) microstructure in the SrZrO3 coating is mainly attributed to the secondary melting of the powdery deposits formed in the cold region of the plasma jet. The influence of the raster-scan step size of plasma torch on the SrZrO3 coating microstructure depends on the stay time of plasma torch on the substrate; the small raster-scan step size is favorable for the formation of IPB structure in this work.