A Perturbation Analysis Approach to Phantom Estimators for Waiting Times in the G/G/1 Queue

We study gradient estimation for waiting times in the G/G/1 queue. We propose a new estimator based on a synthesis of perturbation analysis and weak differentiation. More specifically, we combine the perturbation propagation rules from perturbation analysis with perturbation generation rules from weak differentiation. This leads to an on-line phantom estimator. Numerical experiments show that this estimator has smaller work normalized variance than IPA.     

A Note on Gradient Estimation for Maintenance Systems

This technical note addresses gradient estimation for the cost performance of a two-component maintenance system with respect to the threshold parameter of an age replacement policy. We derive a new gradient estimator based on the measure-valued differentiation (MVD) approach. The performance of the phantom estimator is compared with that of the known smoothed perturbation analysis (SPA) estimator. We show that the phantom estimator has a lower variance, and requires less computational effort than the SPA estimator.     

Maximum likelihood estimation of a stochastic integrate-and-fire neural encoding model

We examine a cascade encoding model for neural response in which a linear filtering stage is followed by a noisy, leaky, integrate-and-fire spike generation mechanism. This model provides a biophysically more realistic alternative to models based on Poisson (memoryless) spike generation, and can effectively reproduce a variety of spiking behaviors seen in vivo. We describe the maximum likelihood estimator for the model parameters, given only extracellular spike train responses (not intracellular voltage data). Specifically, we prove that the log-likelihood function is concave and thus has an essentially unique global maximum that can be found using gradient ascent techniques. We develop an efficient algorithm for computing the maximum likelihood solution, demonstrate the effectiveness of the resulting estimator with numerical simulations, and discuss a method of testing the model's validity using time-rescaling and density evolution techniques.     

Implementation and solution of the diffusion–reaction equation using high-order methods

The orthogonal collocation, Galerkin, tau and least-squares methods are applied to solve a diffusion–reaction problem. In general, the least-squares method suffers from lower accuracy than the other weighted residual methods. The least-squares method holds the most complex linear algebra theory and is thus associated with the most complex implementation. On the other hand, an advantage of the least-squares method is that it always produces a symmetric and positive definite system matrix which can be solved with an efficient iterative technique such as the conjugate gradient method or its preconditioned version. For the present problem, neither the Galerkin, tau and orthogonal collocation techniques produce symmetric and positive definite system matrices, hence the conjugate gradient method is not applicable for these numerical techniques.     

Uncertainty analysis of structural systems by perturbation techniques

The formulation of an efficient method to evaluate the uncertainty of the structural response by applying perturbation techniques is described. Structural random variables are defined by their mean values, standard deviations and correlations. The uncertainty of structural behaviour is evaluated by the covariance matrix of response according to the developed perturbation methodology. It is also presented the procedure used to implement this method in a structural finite element framework. The implemented computational program allows, in only one structural analysis, to evaluate the mean value and the standard deviation of the structural response, defined in terms of displacements or forces. The proposed method is exact for problems with linear design functions and normal-distributed random variables. Results remain accurate for non-linear design functions if they can be approximated by a linear combination of the basic random variables.     

Smoothed perturbation analysis for a class of discrete-eventsystems

A gradient-estimation procedure for a general class of stochastic discrete-event systems is developed. In contrast to most previous work, the authors focus on performance measures whose realizations are inherently discontinuous (in fact, piecewise constant) functions of the parameter of differentiation. Two broad classes of finite-horizon discontinuous performance measures arising naturally in applications are considered. Because of their discontinuity, these important classes of performance measures are not susceptible to infinitesimal perturbation analysis (IPA). Instead, the authors apply smoothed perturbation analysis, formalizing it and generalizing it in the process. Smoothed perturbation analysis uses conditional expectations to smooth jumps. The resulting gradient estimator involves two factors: the conditional rate at which jumps occur, and the expected effect of a jump. Among the types of performance measures to which the methods can be applied are transient state probabilities, finite-horizon throughputs, distributions on arrival, and expected terminal cost     

Application of Vasileva's singular perturbation method to a problem in ecology

The simultaneous presence of fast and slow processes in ecological models leads to the formulation of model equations in a form amenable for the analysis by singular perturbation method. The method as originally proposed by Vasileva (1963) is applied to a specific species-resource logistic model and an algorithm useful for implementation on a digital computer is given. The solutions are obtained for zeroth, first and second order approximations. The accuracy of the solutions is examined by considering the integral-squared-error between the exact solutions and the perturbed solutions. The variation of this error with respect to the small parameter associated with the perturbation method is studied. An important result due to Tikhnov (1950) in the singular perturbation theory is illustrated. It is seen that Vasileva's singular perturbation method is a powerful analytical tool for investigations of various phenomena in the natural sciences.     

Flow estimation of boundary layers using DNS-based wall shear information

This article investigates the problem of obtaining a state-space model of the disturbance evolution that precedes turbulent flow across aerodynamic surfaces. This problem is challenging since the flow is governed by nonlinear, partial differential-algebraic equations for which there currently exists no efficient controller/estimator synthesis techniques. A sequence of model approximations is employed to yield a linear, low-order state-space model, to which standard tools of control theory can be applied. One of the novelties of this article is the application of an algorithm that converts a system of differential-algebraic equations into one of ordinary differential equations. This enables straightforward satisfaction of boundary conditions whilst dispensing with the need for parallel flow approximations and velocity–vorticity transformations. The efficacy of the model is demonstrated by the synthesis of a Kalman filter that clearly reconstructs the characteristic features of the flow, using only wall velocity gradient information obtained from a high-fidelity nonlinear simulation.     

Hopfield Neural Networks for Parametric Identification of Dynamical Systems

In this work, a novel method, based upon Hopfield neural networks, is proposed for parameter estimation, in the context of system identification. The equation of the neural estimator stems from the applicability of Hopfield networks to optimization problems, but the weights and the biases of the resulting network are time-varying, since the target function also varies with time. Hence the stability of the method cannot be taken for granted. In order to compare the novel technique and the classical gradient method, simulations have been carried out for a linearly parameterized system, and results show that the Hopfield network is more efficient than the gradient estimator, obtaining lower error and less oscillations. Thus the neural method is validated as an on-line estimator of the time-varying parameters appearing in the model of a nonlinear physical system.     

An efficient implementation of LU decomposition in C

An implementation of the well-known LU decomposition method for solving systems of linear equations is presented. Operation counts typically stated for this method are derived from a theoretical analysis of the algorithm and ignore the practical aspects of the implementation. The overhead associated with assessing elements of the two-dimensional coefficient array are explored herein, and seen to be substantial. This overhead is incorporated into the analysis of LU decomposition and new operation counts are developed. By exploiting the addressing capabilities of C the cost of array access is significantly reduced, resulting in an efficient implementation. The techniques employed here can be applied to a wide variety of C programs which utilize multi-dimensional arrays.     

Optimizing discrete event dynamic systems via the gradient surface method

In this paper we propose a gradient surface method (GSM) for the optimization of discrete event dynamic systems. GSM combines the advantages of response surface methodology (RSM) and efficient derivative estimation techniques like perturbation analysis (PA) or likelihood ratio method (LR). In GSM, the gradient estimation is obtained by PA (or LR), and the performance gradient surface is obtained from observations at various points in a fashion similar to the RSM. Zero points of the successive approximating gradient surface are then taken as the estimates of the optimal solution. GSM is characterized by several attractive features: it is a single-run method and more efficient than RSM; it uses at each iteration step the information from all data points rather than just the local gradient; it tries to capture the global features of the gradient surface and thereby quickly arrives at the vicinity of the optimal solution. A number of examples are exhibited to illustrate this method.This work was supported by the Office of Naval Research Grants Nos. N00014-90-K-1093 and N00014-89-J-1023, by National Science Foundation Grant No. ECS-85-15449 and by Army Grant No. DAAL-03-86-K-0171.     

An efficient iterated method for mathematical biology model

The purpose of this study is to introduce an efficient iterated homotopy perturbation transform method (IHPTM) for solving a mathematical model of HIV infection of CD4+ T cells. The equations are Laplace transformed, and the nonlinear terms are represented by He’s polynomials. The solutions are obtained in the form of rapidly convergent series with elegantly computable terms. This approach, in contrast to classical perturbation techniques, is valid even for systems without any small/large parameters and therefore can be applied more widely than traditional perturbation techniques, especially when there do not exist any small/large quantities. A good agreement of the novel method solution with the existing solutions is presented graphically and in tabulated forms to study the efficiency and accuracy of IHPTM. This study demonstrates the general validity and the great potential of the IHPTM for solving strongly nonlinear problems.     

On perturbation analysis of queueing networks with finitelysupported service time distributions

Infinitesimal perturbation analysis (IPA) has emerged as an efficient tool for estimating the gradient of a function defined on the steady state of a queuing network. Differentiability of such functions is often assumed due to difficulties in proving it. It is pointed out that such functions may be nondifferentiable at an infinite set of points, dense in a given interval, for a large class of realistic system models. This phenomenon is largely due to correlation of traffic patterns on the links of a network, and to the presence of atoms in the distributions of their service times. The issue of nondifferentiability goes beyond IPA, to any method for estimating the gradients of functions in steady state     

一种新的多普勒中心频率实时估计算法

针对可编程器件实现合成孔径雷达多普勒中心频率实时估计问题,提出了复符号估计算法.新算法利用合成孔径雷达回波先验零均值圆对称复高斯分布统计特性,运用复反正弦定律非线性估计回波相关值,降低了多普勒中心频率估计复杂度,适合硬件实现.分析和仿真结果表明,与已有的符号多普勒中心频率估计算法相比,复符号多普勒中心频率估计算法不但大大降低了运算量和相应硬件的实现面积,提高了整个多普勒中心频率估计运算部件的运行效率,而且具有与其相当的估计性能.     

Sensitivity analysis and optimization of throughput in a production line with blocking

Production lines with limited storage capacities can be modeled as cyclic queueing networks with finite buffers and general service times. A new technique, called perturbation analysis of discrete event dynamic systems, is applied to these queueing networks. An estimate of the gradient of the system throughput is obtained by perturbation analysis based on only one sample trajectory of the system. We show that the estimate is strongly consistent. Using this perturbation analysis estimate of gradient, we can apply the Robbins-Monro stochastic procedure in optimizing the system throughput. Compared to the conventional Kiefer-Wolfowitz optimization procedure, this approach saves a large amount of computation. For a real system, the gradient estimate can be obtained without changing any parameters in the system. The results also hold for systems with general routing but in which no server can block more than one other server simultaneously.     

Approximation of an arbitrary filter and its recursive implementation

Filtering is one of the important techniques in computer vision. It has been widely used in edge detection, image restoration, range image segmentation, etc. However, the efficient implementation of an arbitrary filter has been a challenging problem until now. In this paper, a novel method is proposed to implement an arbitrary filter. Firstly, an efficient recursive structure is proposed to implement any (polynomial) × (exponential)-type (PET) filter. The computational complexity and structure are independent of its filter mask size or its bandwidth. Secondly, a new method is proposed—Lagurre spectrum decomposition method—to obtain the PET approximation of any filters. As an example, the above method is applied to the approximation and implementation of Gaussian filters and experiments have shown that a perfect approximation can be obtained with only third-order Lagurre bases, and therefore only a fourth-order recursive filter is needed to implement Gaussian filters. Finally, the comparison of the present method with the known ones shows that (1) Lagurre polynomial bases are orthogonal with each other, so the filter approximation is simple, (2) the bases are complete and the completeness guarantees the approximation error can be reduced to zero, (3) the method can be used to design both Gaussian and any other filters.     

On derivative estimation of single-server queues via structural infinitesimal perturbation analysis

In Dai and Ho (1994) we developed a method, referred to asstructural infinitesimal perturbation analysis (SIPA), to address the need for derivative estimation with respect to a special type of parameter. However, it was not clear how much computational effort is required to implement this method. Derivative estimation via SIPA can be complicated in implementation. Such computational problems, also arise in several other derivative estimation methods. In this paper we take SIPA as a typical method and apply it to a special class of DEDS-several variations of single-server queues, focusing on the issue of implementation. We demonstrate that SIPA can be efficiently implemented. In some cases, it can be as simple as theinfinitesimal perturbation analysis (IPA), method which is considered to be the most efficient method available so far. The main approach we take is to combine SIPA with finite perturbation analysis and cut-and-paste techniques. Explicit formulae are given to various problems, some being impossible to solve using the traditional IPA method. Numerical examples are employed to illustrate the results.     

Mesh Statistics for Robust Curvature Estimation

While it is usually not difficult to compute principal curvatures of a smooth surface of sufficient differentiability, it is a rather difficult task when only a polygonal approximation of the surface is available, because of the inherent ambiguity of such representation. A number of different approaches has been proposed in the past that tackle this problem using various techniques. Most papers tend to focus on a particular method, while an comprehensive comparison of the different approaches is usually missing. We present results of a large experiment, involving both common and recently proposed curvature estimation techniques, applied to triangle meshes of varying properties. It turns out that none of the approaches provides reliable results under all circumstances. Motivated by this observation, we investigate mesh statistics, which can be computed from vertex positions and mesh connectivity information only, and which can help in deciding which estimator will work best for a particular case. Finally, we propose a meta‐estimator, which makes a choice between existing algorithms based on the value of the mesh statistics, and we demonstrate that such meta‐estimator, despite its simplicity, provides considerably more robust results than any existing approach.     

A posteriori error analysis of a fully-mixed formulation for the Brinkman–Darcy problem

We develop the a posteriori error analysis for a mixed finite element method applied to the coupling of Brinkman and Darcy equations in 3D, modelling the interaction of viscous and non-viscous flow effects across a given interface. The system is formulated in terms of velocity and pressure within the Darcy subdomain, together with vorticity, velocity and pressure of the fluid in the Brinkman region, and a Lagrange multiplier enforcing pressure continuity across the interface. The solvability of a fully-mixed formulation along with a priori error bounds for a finite element method have been recently established in Álvarez et al. ( Comput Methods Appl Mech Eng 307:68–95, 2016). Here we derive a residual-based a posteriori error estimator for such a scheme, and prove its reliability exploiting a global inf-sup condition in combination with suitable Helmholtz decompositions, and interpolation properties of Clément and Raviart–Thomas operators. The estimator is also shown to be efficient, following a localisation strategy and appropriate inverse inequalities. We present numerical tests to confirm the features of the estimator and to illustrate the performance of the method in academic and application-oriented problems.