Step Size Changes Research Articles

Preservation of the growth rates of delay differential equations by Euler schemes with non-uniform step sizes

In this paper, we consider explicit Euler methods which recover the rate of growth to infinity of a highly nonlinear autonomous delay differential equation. The success of the methods rely on the step size changing in response to the state, as it has been shown that Euler methods with constant step size will systematically underestimate the growth rate. It is also shown that the computed solution converges to the true solution on any compact time interval, when a parameter which controls the step size is sent to zero. A second method, which applies to a related ordinary differential equation, and which requires less computational effort, is also presented.

Adaptive Blind Equalization with Variable Step-Size Modified by Attenuation Function

Step-size is one of the important parameters which influence the performance of blind equalization, a variable step-size method control by attenuation function according to instantaneous gradient and iterative times is proposed in this paper. Initially, set larger step-size to obtain faster convergence, with iterative times increasing, step-size would decrease control by attenuation function for instantaneous gradient reducing. Step-size changes according to this method consistent with the demand variable step-size adaptive algorithm that can obtain combination of convergence rate and convergence precision to improve the performance of blind equalization. Simulations show that this method performs better than conventional blind equalization with constant step-size.

A Code Based on the Two-Stage Runge-Kutta Gauss Formula for Second-Order Initial Value Problems

A code based on the two-stage Gauss formula (order four) for second-order initial value problems of a special type is developed. This code can be used to obtain a low- to medium-precision integration for a wide range of problems in the class of oscillatory type, Hamiltonian problems, and time-dependent partial differential equations discretized in space by finite differences or finite elements. The iteration process used in solving for the stage values of the Gauss formula, the selection of the initial step size, and the choice of an appropriate local error estimator for determining the step size change according to a particular tolerance specified by the user are studied. Moreover, a global error estimate and a dense output at equidistant points in the integration interval are supplied with the code. Numerical experiments and some comparisons with certain standard codes on relevant test problems are also given.

Nanoscale effects in carbon structures fabricated using focused ion beam-chemical vapor deposition

Carbon structures were fabricated on silicon by focused ion beam-chemical vapor deposition using phenanthrene gas. The current of the 30 keV gallium beam was set at 10.7 pA. The relationship between the step size of the raster scan and the nanoscale structures obtained was investigated. Changes in step size from 4 to 85 nm on scan areas of 300, 500, and 1000 nm 2 caused variations in the depletion and replenishment of adsorbed gas molecules, which in turn caused several nanoscale effects such as delocalization, round edge, slant sidewall, locally variation in deposition rate, discrete structure, and shadow effect in carbon structures. Changes in the deposition rates and surface morphologies of the carbon deposits were observed and discussed.

Search Control Algorithm Based on Random Step Size Hill-Climbing Method for Adaptive PMD Compensation

Adaptive polarization mode dispersion (PMD) compensation is required for the speed-up and advancement of the present optical communications. The combination of a tunable PMD compensator and its adaptive control method achieves adaptive PMD compensation. In this paper, we report an effective search control algorithm for the feedback control of the PMD compensator. The algorithm is based on the hill-climbing method. However, the step size changes randomly to prevent the convergence from being trapped at a local maximum or a flat, unlike the conventional hill-climbing method. The randomness depends on the Gaussian probability density functions. We conducted transmission simulations at 160 Gb/s and the results show that the proposed method provides more optimal compensator control than the conventional hill-climbing method.

Establishing a Mechanistic Basis for the Large Kinetic Steps of the NS3 Helicase

The NS3 helicase from hepatitis C virus is a prototypical DEx(H/D) RNA helicase. NS3 has been shown to unwind RNA in a discontinuous manner, pausing after long apparent steps of unwinding. We systematically examined the effects of duplex stability and ionic conditions on the periodicity of the NS3 unwinding cycle. The kinetic step size for NS3 unwinding was examined on diverse substrate sequences. The kinetic step size (16 bp/step) was found to be independent of RNA duplex stability and composition, but it exhibited strong dependence on monovalent salt concentration, decreasing to approximately 11 bp/step at low [NaCl]. We addressed this behavior by analyzing the oligomeric state of NS3 at various salt concentrations. Whereas only NS3 oligomers are capable of processive unwinding, we found that monomeric NS3 is an active helicase that unwinds with low processivity. We demonstrate that low salt conditions enhance unwinding by monomeric NS3, which is likely to account for the reduction in apparent step size under low salt conditions. Based on results reported here, as well as available structural and single molecule data, we present an unwinding mechanism that addresses the apparent periodicity of NS3 unwinding, the magnitude of the step size, and that integrates the various stepwise motions observed for NS3. We propose that the large kinetic step size of NS3 unwinding reflects a delayed, periodic release of the separated RNA product strand from a secondary binding site that is located in the NTPase domain (Domain II) of NS3. These findings suggest that the mechanism of product release represents an important and unexplored feature of helicase mechanism.

Some implementation aspects of the general linear methods withinherent Runge-Kutta stability

In this paper we try to put dierent practical aspects of the general linear methods discussed in the papers (1,6,7) under one algorithm to show more details of its implementation. With a proposed initial step size strategy this algorithm shows a better performance in some problems. To illustrate the eciency of the method we consider some standard test problems and report more useful details of step size and order changes, and number of rejected and accepted steps along with relative global errors.

“Rescale and modify” implementation of IRKS methods

In this paper, we implement the “rescale and modify” approach in variable step size mode with both fixed and variable orders for stiff and nonstiff ODEs. A comparison of this approach and the “rescale” approach from the point of stability behavior and also step size and order selection provides very interesting results. To illustrate the efficiency of the method we have considered some standard test problems and report very useful tables and figures for step size and order changes, number of rejected or accepted steps, and also global error. As an optimal implementation, the numerical experiments suggest the application of this approach with both fixed and variable orders for nonstiff, and in variable order for stiff ODEs.

The effect of external forces on discrete motion within holographic optical tweezers

Holographic optical tweezers is a widely used technique to manipulate the individual positions of optically trapped micron-sized particles in a sample. The trap positions are changed by updating the holographic image displayed on a spatial light modulator. The updating process takes a finite time, resulting in a temporary decrease of the intensity, and thus the stiffness, of the optical trap. We have investigated this change in trap stiffness during the updating process by studying the motion of an optically trapped particle in a fluid flow. We found a highly nonlinear behavior of the change in trap stiffness vs. changes in step size. For step sizes up to approximately 300 nm the trap stiffness is decreasing. Above 300 nm the change in trap stiffness remains constant for all step sizes up to one particle radius. This information is crucial for optical force measurements using holographic optical tweezers.

USING EULER ANGLES TO EXPRESS MECHANICAL MODEL OF WALKING ROBOT SYSTEM

Walking robot has complicated structure and strong ability to adapt ground condition, also it is difficult to control. To realize dynamic walking of the humanoid robot, robot dynamic models have to be established, the control algorithm for gait and the stability postures should be designed. The modeling of robot plays an important role in the robot research. The method of using Euler equation to establish the mechanical model of the walking system is discussed, some simplified conditions are proposed, and the rotation and linear motion of components and the kinematic model of the joint places are expressed with Euler equation. Based on these, the confined condition and kinematic equation of the simplified walking robot are investigated while steady walking. For further study kinematics of steady walking robots and even step-size change robots it offers another theory tool.

Linearly-implicit two-step methods and their implementation in Nordsieck form

We consider the Nordsieck representation of well-known linearly-implicit two-step methods in order to facilitate stepsize changes. These methods are treated as implicit schemes with one Newton step with a special predictor. A new predictor with better accuracy is discussed. Methods with 4–7 stages are tested and compared with RODAS. The numerical results show the potential of the Nordsieck implementation of the methods.

Numerical solution of neutral functional differential equations by Adams methods in divided difference form

A variable step and variable order algorithm for the numerical solution of neutral functional differential equations is described. This general purpose algorithm is based on the variable step formulation of the Adams methods represented in divided difference form. The Adams–Bashforth and Adams–Moulton methods are implemented in predictor–corrector mode. The detection of derivative discontinuities relies on the estimates of the local discretization errors. The restarting of the integration at each discontinuity point relies on the step size and order changing strategy based on the estimates of the local discretization errors. This algorithm reduces the overhead cost by implementing the Adams methods in divided difference form. This algorithm increases the reliability and efficiency by obtaining asymptotically correct estimates of the local discretization errors for the lower adjacent orders without any extra function evaluation by using local extrapolation. The results from three test examples are presented.

A Noise Reduction Method Based on Linear Prediction with Variable Step-Size

A noise reduction technique that uses the linear prediction to remove noise components in speech signals has been proposed previously. The noise reduction works well for additive white noise signals, because the coefficients of the linear predictor converge such that the prediction error becomes white. In this method, the linear predictor is updated by a gradient-based algorithm with a fixed step-size. However, the optimal value of the step-size changes with the values of the prediction coefficients. In this paper, we propose a noise reduction system using the linear predictor with a variable step-size. The optimal value of the step-size depends also on the variance of the white noise, however the variance is unknown. We therefore introduce a speech/non-speech detector, and estimate the variance in non-speech segments where the observed signal includes only noise components. The simulation results show that the noise reduction capability of the proposed system is better than that of the conventional one with a fixed step-size.

Flexible Numerical Integration for Efficient Representation of Switching in Real Time Electromagnetic Transients Simulation

In real time digital simulation of electromagnetic transients for equipment testing, solution output is provided at equidistantly spaced time points in order to synchronize the data exchange with external equipment. Switching events, which can occur in between these time points of solution output, are pinpointed by backward interpolation. To recuperate the time interval lost through interpolation, the simulation must be resynchronized with the real time clock. For this purpose, the use of a flexible time step size is desirable. However, in EM transients simulators based on nodal analysis techniques, time step size changes have so far involved changes of the nodal admittance matrix and, thus, imposed a large computational burden. This is avoided through the method flexible integration for readjustment in simulation of transients (FIRST), presented in this paper. Through this method, the characteristics of numerical integration are modulated such that time step size changes are achieved in a computationally very efficient manner and without modifying the nodal admittance matrix.

Adaptive stepsize based on control theory for stochastic differential equations

The numerical solution of stochastic differential equations (SDEs) has been focussed recently on the development of numerical methods with good stability and order properties. These numerical implementations have been made with fixed stepsize, but there are many situations when a fixed stepsize is not appropriate. In the numerical solution of ordinary differential equations, much work has been carried out on developing robust implementation techniques using variable stepsize. It has been necessary, in the deterministic case, to consider the “best” choice for an initial stepsize, as well as developing effective strategies for stepsize control—the same, of course, must be carried out in the stochastic case. In this paper, proportional integral (PI) control is applied to a variable stepsize implementation of an embedded pair of stochastic Runge–Kutta methods used to obtain numerical solutions of nonstiff SDEs. For stiff SDEs, the embedded pair of the balanced Milstein and balanced implicit method is implemented in variable stepsize mode using a predictive controller for the stepsize change. The extension of these stepsize controllers from a digital filter theory point of view via PI with derivative (PID) control will also be implemented. The implementations show the improvement in efficiency that can be attained when using these control theory approaches compared with the regular stepsize change strategy.

A new approach to error estimation for general linear methods

We present a simple, accurate and reliable approach to the estimation of the local discretization error for general linear methods for ordinary differential equations. In this approach the input vector for the next step from x n to x n+1 =x n +h is rescaled and modified accordingly to compensate for the change of stepsize from ¯h to h=r¯h.

Derivation and implementation of Two-Step Runge-Kutta pairs

Explicit Runge-Kutta pairs are known to provide efficient solutions to initial value differential equations with inexpensive derivative evaluations. Two-step Runge-Kutta methods strive to improve the efficiency by utilizing approximations to the solution and its derivatives from the previous step. This article suggests a strategy for computing embeddedpairs of such two-step methods using a smaller number of function evaluations than that required for traditional Runge-Kutta methods of the same order. This leads to the efficient and reliable estimation of local discretization error and a robust step control strategy. The change of stepsize is achieved by a suitable interpolation of stage values from the previous step and does not require any additional function evaluations. Two examples illustrate the features of these pairs.

Spatio-temporal tuning of motion coherence detection at different luminance levels

We studied effects of dark adaptation on spatial and temporal tuning for motion coherence detection. We compared tuning for step size and delay for moving random pixel arrays (RPAs) at two adaptation levels, one light adapted (50 cd/m 2) and the other relatively dark adapted (0.05 cd/m 2). To study coherence detection rather than contrast detection, RPAs were scaled for equal contrast detection at each luminance level, and a signal-to-noise ratio paradigm was used in which the RPA is always at a fixed, supra-threshold contrast level. The noise consists of a spatio-temporally incoherent RPA added to the moving RPA on a pixel-by-pixel basis. Spatial and temporal limits for coherence detection were measured using a single step pattern lifetime stimulus, in which patterns on alternate frames make a coherent step and are being refreshed. Therefore, the stimulus contains coherent motion at a single combination of step size and delay only. The main effect of dark adaptation is a large shift in step size, slightly less than the adjustment of spatial scale required for maintaining equal contrast sensitivity. A similar change of preferred step size occurs also for scaled stimuli at a light-adapted level, indicating that the spatial effect is not directly linked to dark adaptation, but more generally related to changes in the available low-level spatial information. Dark-adaptation shifts temporal tuning by about a factor of 2. Long delays are more effective at low luminance levels, whereas short delays no longer support motion coherence detection. Luminance-invariant velocity tuning curves, as reported previously [Lankheet, M.J.M., van Doorn, A.J., Bouman, M.A., & van de Grind, W.A. (2000) Motion coherence detection as a function of luminance in human central vision. Vision Research, 40, 3599–3611], result from recruitment of different sets of motion detectors, and an adjustment of their temporal properties.

On implicit Euler for high-order high-index DAEs

The Implicit Euler method is seldom used to solve differential–algebraic equations (DAEs) of differential index r⩾3, since the method in general fails to converge in the first r−2 steps after a change of stepsize. However, if the differential equation is of order d= r−1⩾1, an alternative variable-step version of the Euler method can be shown uniformly convergent. For d= r−1, this variable-step method is equivalent to the Implicit Euler except for the first r−2 steps after a change of stepsize. Generalization to DAEs with differential equations of order d> r−1⩾1, and to variable-order formulas is discussed.

A Note on the Step Size Selection in Adams Multistep Methods

In this paper new algorithms for step size prediction in variable step size Adams methods are proposed. It is shown that, when large step size changes are necessary for an efficient integration, the new algorithms provide a prediction that follows more closely the local error estimation than the standard step size prediction. The new predictors can be considered as a easily computable alternative to the step size predictors given by Wille [9] in terms of differential equations.