Initial Step Size Research Articles

Thresholds for inattentive listeners obtained with adaptive forced-choice procedures

Forced-choice adaptive procedures enjoy widespread use for measurement of detection and discrimination thresholds. Using computer simulations and behavioral data, this paper examines the effect of step size (2 and 4 dB), decision rule (targets =70.7%, 75%, and 79.4% correct) and initial starting level on thresholds obtained under conditions of inattention and complete attention. Simulations of two-alternative forced-choice (AFC) and three-AFC procedures demonstrate that the best general strategy for measuring thresholds under conditions of inattention is to employ a low percentage (e.g., 70.7%) as a target on the psychometric function combined with a large step size. When a track for an inattentive listener is begun far above threshold, we found that using a large step size for the first two turnarounds (reversals in stimulus level direction) and then switching to a step size that is 1/2 the initial step size was less effective than using the large step size for the entire threshold estimation run. Behavioral data obtained in a tonal detection-in-noise task with a 2-AFC procedure support the validity of the simulations.

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.

Optimal variable step-size LMS model and algorithm with independence assumption

To solve the contradiction between convergence rate and steady-state error in least mean square (LMS) algorithm, basing on independence assumption, this paper proposes and proves the optimal step-size theorem from the view of minimizing mean squared error (MSE). The theorem reveals the one-to-one mapping between the optimal step-size and MSE. Following the theorem, optimal variable step-size LMS (OVS-LMS) model, describing the theoretical bound of the convergence rate of LMS algorithm, is constructed. Then we discuss the selection of initial optimal step-size and updating of optimal step-size at the time of unknown system changing. At last an optimal step-size LMS algorithm is proposed and tested in various environments. Simulation results show the proposed algorithm is very close to the theoretical bound.

Adaptive step-size constant modulus algorithm for DS-CDMA receivers in nonstationary environments

A new adaptive step-size constant modulus algorithm for direct-sequence code division multiple access receivers is presented for application in nonstationary channel conditions. The algorithm is derived upon the basis of adapting the step-size to minimise the constant modulus criterion and its convergence is verified analytically. Simulations show the superior performance of the proposed method over similar adaptive receivers and its robustness to different settings of the initial step-size.

Implementation of DIMSIMs for stiff differential systems

Some issues related to the implementation of diagonally implicit multistage integration methods for stiff differential systems are discussed. They include reliable estimation of the local discretization error, construction of continuous interpolants, solution of nonlinear systems of equations by simplified Newton iterations, choice of initial stepsize and order, and step and order changing strategy. Numerical results are presented which indicate that an experimental Matlab code based on type 2 methods of order one, two and three outperforms ode15s code from Matlab ODE suite on problems whose Jacobian has eigenvalues which are close to the imaginary axis.

Performance of two methods for solving separable Hamiltonian systems

We make qualitative comparisons of fixed step symplectic and variable step nonsymplectic integrations of the separable Hénon–Heiles Hamiltonian system. Emphasis is given to interesting numerical phenomena. Particularly, we observe the relationship of the error in the computed Hamiltonian to the presence and absence of chaos, when computing with a symplectic (fixed step) method, qualitative phenomena in the Hamiltonian error for a variable step method, and the sensitivity of the chaotic behavior and of the computation of features in Poincaré sections to very small changes in initial conditions, step sizes and error tolerances.

Factors influencing equal-loudness-level contours

Data on equal-loudness-level contours taken from the literature show considerable differences between the results of different laboratories, even when using the recommended alternative forced-choice (AFC) method. Some of these differences can be attributed to free parameters in the procedures that the experimentator could change. This work investigates the influence of the experimental setup on the resulting curves obtained with an adaptive two-alternative forced-choice procedure for the 30- and 50-phon curves in the range from 200 Hz to 1 kHz. For the adaptive procedure, the amount of change in the resulting equal-loudness-level contours is measured for different free parameters (e.g., initial step size, starting level). It is shown that for a simple adaptive procedure the influence of the starting level is smaller than the range effect when using the method of constant stimuli. The effect of the starting level is reduced when using large initial step size in the adaptive procedure. However, the effect can still amount to 6 dB. This effect diminishes when using an interleaved procedure. The maximum measurement error for the different procedures is estimated. In addition, it is shown that interindividual variance is reduced when the individual threshold is taken into account.

Simplex optimization of protein crystallization conditions

Simplex algorithms have been used to optimize for size, number and morphology of lysozyme and apoferritin crystals. This approach requires fewer experiments than the single-factor-at-a-time method or factorial designs and will be useful in conserving materials on the International Space Station. The simplex method has the possible advantage that it conserves on materials by reducing the number of experiments required to optimize a crystallization system. The process is iterative and exploratory and should allow optimum microgravity conditions to be determined which might very well be different from the optimum conditions on Earth. Because the simplex method uses simple mathematical operations to calculate the next set of crystallization conditions it will be easier for crystal growers to implement than factorial designs. Factorial experiments are based on varying all factors simultaneously at a limited number of factor levels. This results in a model that is used to determine the influence of each factor and their interactions. Factorial design experiments are especially useful at the beginning of an experimental study and as a screening tool to investigate a large number of factors. The simplex method is an optimization method which is model-independent and requires no fitting of models to data. Also, when applied to protein crystal growth the simplex method does not rely on an absolute quality score. Instead, with each iteration a comparison is made to the last experiment and the results are assigned as being “better or worse”. In this study, commercially obtained apoferritin was purified from 65% monomeric apoferritin to 92% monomeric apoferritin by size exclusion chromatography. Simplex optimization found the best apoferritin crystals were obtained at 15 mg/ml apoferritin, 2.0% CdSO 4, 25°C using the hanging drop vapor diffusion method of crystallization and at 24 mg/ml apoferritin, 1.5% CdSO 4, 25°C using the containerless crystallization method. For lysozyme, the simplex method found the best crystals at 19 mg/ml lysozyme, 7.0% (w/v) NaCl, pH 4.0, 25°C using the hanging drop vapor diffusion method of crystallization. For both proteins, the optimum conditions were found with less than ten experiments using very little protein. Finally, we report that the factors to be considered in the successful application of this method to crystallization are the number of variables to be studied, the initial conditions, step size and analysis of crystal quality.

Measures for Symmetric Rank-One Updates

Measures of deviation of a symmetric positive definite matrix from the identity are derived. They give rise to symmetric rank-one, SR1, type updates. The measures are motivated by considering the volume of the symmetric difference of the two ellipsoids, which arise from the current and updated quadratic models in quasi-Newton methods. The measure defined by the problem—maximize the determinant subject to a bound of 1 on the largest eigenvalue—yields the SR1 update. The measure σ(A) = λ1(A)/det(A)1/n yields the optimally conditioned, sized, symmetric rank-one updates. The volume considerations also suggest a ‘correction’ for the initial stepsize for these sized updates. It is then shown that the σ-optimal updates, as well as the Oren-Luenberger self-scaling updates, are all optimal updates for the κ measure, the l2 condition number. Moreover, all four sized updates result in the same largest (and smallest) ‘scaled’ eigenvalue and corresponding eigenvector. In fact, the inversesized BFGS is the mean of the σ-optimal updates, while the inverse of the sized DFP is the mean of the inverses of the σ-optimal updates. The difference between these four updates is determined by the middle n − 2 scaled eigenvalues. The κ measure also provides a natural Broyden class replacement for the SR1 when it is not positive definite.

A fast method for optimal reactive power flow solution

A fast successive linear programming (SLP) method for minimizing transmission losses and improving the voltage profile is proposed. The method uses the same compactly stored, factorized constant matrices in all the LP steps, both for power flow solution and for constructing the LP model. The inherent oscillatory convergence of SLP methods is overcome by proper selection of initial step sizes and their gradual reduction. Detailed studies on three systems, including a 109-bus system, reveal the fast and reliable convergence property of the method.

VODE: A Variable-Coefficient ODE Solver

VODE is a new initial value ODE solver for stiff and nonstiff systems. It uses variable-coefficient Adams-Moulton and Backward Differentiation Formula (BDF) methods in Nordsieck form, as taken from the older solvers EPISODE and EPISODEB, treating the Jacobian as full or banded. Unlike the older codes, VODE has a highly flexible user interface that is nearly identical to that of the ODEPACK solver LSODE.In the process, several algorithmic improvements have been made in VODE, aside from the new user interface. First, a change in stepsize and/or order that is decided upon at the end of one successful step is not implemented until the start of the next step, so that interpolations performed between steps use the more correct data. Second, a new algorithm for setting the initial stepsize has been included, which iterates briefly to estimate the required second derivative vector. Efficiency is often greatly enhanced by an added algorithm for saving and reusing the Jacobian matrix J, as it occurs in the Newton m...

Stimulus step size and heterogeneous stimulus conditions in adaptive psychophysics

This paper reports on simulations of repeated estimates of stimulus thresholds using a two-down, one-up adaptive procedure and forced-choice procedures using 2, 3, and 4 response alternatives. In the first section of the paper, the selection of an initial and final step size for the stimulus is the main focus. For a 2-alternative forced-choice psychometric function, a step size of roughly one-quarter the useful range (60%–90%) provides a reasonable initial step size. A final step size of roughly half that value is recommended. The increase in the variability of threshold estimates is, however, not very great if values near those recommended are selected instead. The change in the variability of the threshold estimates with the change in the number of alternatives used in the forced-choice procedure can be predicted by considering the sweat factor associated with the psychometric function. In the second part of the paper, the focus is on how the variability in threshold estimates increases if the stimulus set is heterogeneous; that is, the stimuli produce different psychometric functions. Stimuli that differ only in their means increase the variability of threshold estimates more than do stimuli that differ only in the slopes of their psychometric functions. Stimuli that contain both forms of heterogeneity, however, increase the variability in threshold estimates by only about 50%.

Comparison of alternating variable search and simplex methods of optimisation for inductively coupled plasma optical emission and atomic fluorescence spectrometry

The performance of several cyclic alternating variable search (AVS) optimisation methods are compared with two simplex methods with respect to the number of changes of variable required to search a model two-factor response space. The roles of the initial step size and of the variable step size are discussed, and the information produced concerning the shape of the factor space is evaluated. An AVS method which starts with a fixed step size and then changes to a variable step size on second and subsequent cycles is compared with a variable step size simplex for the optimisation of an inductively coupled plasma optical emission spectrometer and of the atomiser, source inductively coupled plasmas in atomic fluorescence spectrometry (the ASIA system). The order in which the variables are taken in the AVS method does not affect the value of the optimum eventually found. Both methods perform satisfactorily for the optical emission work, although the AVS method provides information about the shape of the factor space which is easier to interpret than in the simplex method. However, the simplex method was not always able to satisfy the conditions for termination in the case of the atomic fluorescence studies and was much slower to implement than the AVS method as the latter used direct visual feedback from the output of the lock-in amplifier as a measure of the figure of merit (total fluorescence signal).

Starting BDF and Adams codes at optimal order

We present an algorithm based on Runge—Kutta formulas with interpolation to start BDF and Adams multistep codes at optimal order and step size. It first finds automatically and reliably an ‘on scale’ initial step size. Besides being convenient for the user, this results in a more robust and efficient start, especially for stiff problems, because a starting step is taken with an efficient RK formula of moderate order before switching to the multistep formula. Starting the multistep formula at optimal order and step size makes the integration more efficient and reduces difficulties due to mesh distortion. We present the results of a variety of numerical experiments which demonstrate the efficacy of our technique.

Automatic selection of the initial step size for an ODE solver

The choice of initial step size is critical for the reliable numerical solution of the initial value problem for a system of ordinary differential equations. Automatic selection of this step size may lead to a more robust and efficient integration than its provision by a user, and is always more convenient. It is especially important for the reliability of an ODE solver used as a module in a larger software package. Previous approaches to making the selection are combined with some new ideas to produce an effective scheme for the automatic choice of the initial step size. Numerical results illustrate the roles played by the individual phases of the algorithm and show that the whole algorithm is both robust and efficient.

Optimizing the design of an activated sludge/secondary clarifier system via univariate search technique

This article presents a simple and effective univariate search technique which is suitable for optimizing the design of an activated sludge/secondary clarifier system on microcomputers. The usefulness of the proposed optimal design procedure was tested on an IBM personal computer (IBM-PC). The results of this study indicated that the least-cost design of a typical activated sludge/secondary clarifier system can be obtained within 4 min on an IBM-PC. Besides, the proposed procedure was also found capable of obtaining the optimum solution over a wide range of initial step size of search chosen.

Starting step size for an ODE solver

One of the more critical issues in solving ordinary differential equations by a step-by-step process occurs in the starting phase. Somehow the procedure must be supplied with an initial step size that is on scale for the problem at hand. It must be small enough to yield a reliable solution by the process, but not so small as to significantly affect the efficiency of solution. In this paper, we discuss an algorithm for obtaining a good starting step size and present a subroutine which can be readily used in most ODE solvers.

NUMERICAL SOLUTION TECHNIQUE FOR THE TRANSIENT EQUATION OF TRANSFER

Abstract A finite-difference method for the solution of the transient equation of transfer in a plane parallel medium is developed. In this method the adding/doubling method is used by solving a form of the equation of transfer at discrete values of the transient variable. A number of initializations for the adding/doubling method are available. The accuracy of each is dependent on the size of the initial layer used. Once an initialization is chosen, reflection and transmission matrices are built, and both upward and downward radiance are calculated at interior points for all discrete values of the transient variable. The advantages of the method are that (1) calculations of all the transient intensity distributions may be found throughout the layer for specified layer thicknesses, (2) solutions may be obtained for a variety of boundary conditions, and (3) exact solutions are approached as the initial layer size and step size approach zero. Numerous distributions are plotted and compared for two different...

Limiting precision in differential equation solvers. II. Sources of trouble and starting a code

The reasons a class of codes for solving ordinary differential equations might want to use an extremely small step size are investigated. For this class the likelihood of precision difficulties is evaluated and remedies examined. The investigation suggests a way of selecting automatically an initial step size which should be reliably on scale.

On convergence rates of subgradient optimization methods

Rates of convergence of subgradient optimization are studied. If the step size is chosen to be a geometric progression with ratioρ the convergence, if it occurs, is geometric with rateρ. For convergence to occur, it is necessary that the initial step size be large enough, and that the ratioρ be greater than a sustainable ratez(μ), which depends upon a condition numberμ, defined for both differentiable and nondifferentiable functions. The sustainable ratez(μ) is closely related to the rate of convergence of the steepest ascent method for differentiable functions: in fact it is identical if the function is not too well conditioned.