Application of finitized power series distributions to accelerated variate generation. Part II: the case of the logarithmic distribution

The method of aliasing vastly expands the palette of discrete random variate generation methodologies while providing excellent speed. However, its application is limited to finitely supported distributions. We demonstrate that an application of moment preserving finitization called the Negative Taylor Series Finitization (NTSF) method for the power series family of discrete distributions, when coupled with the method of aliasing, can greatly improve infinitely supported discrete random variate generation speed with certain limitations. We illustrate this with the logarithmic power series distribution, and we compare four published algorithms designed to generate random variates from a logarithmic distribution to the aliasing method of random variate generation from an NTSF version of the same distribution. We compare the accuracy and speed (user‐time) of these various methods for generating variates from a logarithmic distribution.     

New results using an integrated model and recursive algorithm for image motion estimation

A linear equation in the affine parameters used to model image motion may be derived by Taylor series expansion and truncation, and windowed spatial integration. Two methods for reducing errors in the Taylor approximation are discussed and results are presented.     

The development of a high-order Taylor expansion solution to the chemical rate equation for the simulation of complex biochemical systems

A numerical method for evaluating chemical rate equations is presented. This method was developed by expressing the system of coupled, first-degree, ordinary differential chemical rate equations as a single tensor equation. The tensorial rate equation is invariant in form for all reversible and irreversible reaction schemes that can be expressed as first- and second-order reaction steps, and can accommodate any number of reactive components. The tensor rate equation was manipulated to obtain a simple formula (in terms of rate constants and initial concentrations) for the power coefficients of the Taylor expansion of the chemical rate equation. The Taylor expansion formula was used to develop a FORTRAN algorithm for analysing the time development of chemical systems. A computational experiment was performed with a Michaelis-Menten scheme in which step size and expansion order (to the 100th term) were varied; the inclusion of high-order terms of the Taylor expansion was shown to reduce truncation and round-off errors associated with Runge-Kutta methods and lead to increased computational efficiency.     

A class of explicit two-step superstable methods for second-order linear initial value problems

A class of explicit two-step superstable methods of fourth algebraic order for the numerical solution of second-order linear initial value problems is presented in this article. We need Taylor expansion at an internal grid point and collocation formulae for the derivatives of the solution to derive a method and then modify it into a class of methods having the desired stability properties. Computational results are presented to demonstrate the applicability of the methods to some standard problems.     

Fast Approximation of the Discrete Gauss Transform in Higher Dimensions

We present a novel approach for the fast approximation of the discrete Gauss transform in higher dimensions. The algorithm is based on the dual-tree technique and introduces a new Taylor series expansion. It compares favorably to existing methods especially when it comes to higher dimensions and a broad range of bandwidths. Numerical results with different datasets in up to 62 dimensions demonstrate its performance.     

A novel approach to uncertainty analysis using methods of hybrid dimension reduction and improved maximum entropy

Structural and Multidisciplinary Optimization - Methods of uncertainty analysis based on statistical moments are more convenient than methods that use a Taylor series expansion because the moments...     

An observability problem for nonlinear discrete-data systems

A series expansion method and an iterative scheme are presented to estimate states or initial conditions of nonlinear discrete data systems from output measurements when the system equations are known. The methods are based on the Taylor series expansion of the trajectory about some nominal trajectory. The schemes when applied to linear systems result in dead beat observers.     

Multi-order Arnoldi-based model order reduction of second-order time-delay systems

In this paper, we discuss the Krylov subspace-based model order reduction methods of second-order systems with time delays, and present two structure-preserving methods for model order reduction of these second-order systems, which avoid to convert the second-order systems into first-order ones. One method is based on a Krylov subspace by using the Taylor series expansion, the other method is based on the Laguerre series expansion. These two methods are used in the multi-order Arnoldi algorithm to construct the projection matrices. The resulting reduced models can not only preserve the structure of the original systems, but also can match a certain number of approximate moments or Laguerre expansion coefficients. The effectiveness of the proposed methods is demonstrated by two numerical examples.     

On the Butcher group and general multi-value methods

This paper proves a theorem (“Theorem 6”) on the composition of, what we call, Butcher series. This Theorem is shown to be fundamental for the theory of Runge-Kutta methods: the formulas for the Taylor expansion of RK-methods and multiderivative RK-methods as well as formulas for the operation of the “Butcher group” (which describes the composition of RK-methods) are easy consequences. We do not attempt to realize the series as (generalized) Runge-Kutta methods, so we are not forced to restrict ourselves to the finite dimensional case. The theory extends to the multiderivative case as well, and the formulas remain valid for series which are not realizable as Runge-Kutta methods at all. Finally we extend the multi-value methods of J. Butcher [2] to the multiderivative case, which leads to a big class of integration methods for ordinary differential equations, including the methods of Nordsieck and Gear [3]. The defintions and notations of [4] are used throughout this paper, many of the results are proved here again.     

三类保辛摄动及其数值比较

摄动法近似应当保辛．本文指出,有限元位移法自动保辛,有限元混合能表示也保辛．摄动法的刚度阵Taylor级数展开能证明保辛;混合能的Taylor级数展开摄动也证明了保辛．但传递辛矩阵的Taylor级数展开摄动却不能保辛．辛矩阵只能在乘法群下保辛,故传递辛矩阵的保辛摄动必须采用正则变换的乘法．虽然刚度阵加法摄动、混合能矩阵加法摄动与传递辛矩阵正则变换乘法摄动都保辛,但这3种摄动近似并不相同．最后通过数值例题给出了对比．     

Application of the modified Taylor approximation

In this paper we introduce and describe a new scheme for the numerical integration of smooth functions. The scheme is based on the modified Taylor expansion and is suitable for functions that exhibit near-sinusoidal or repetitive behaviour. We discuss the method and its rate of convergence, then implement it for the approximation of certain integrals. Examples include integrands involving Airy wave, Bessel, Gamma, and elliptic functions. The results, from the data in the tables, demonstrate that the method converges rapidly and approximates the integral as well as some well-known numerical integration methods used with sufficiently small step sizes.     

Explicit Numerov type methods with reduced number of stages

We present in this paper a new approach for the derivation of hybrid explicit Numerov type methods. The new methodology does not require the intermediate use of high accuracy interpolatory nodes, since we only need the Taylor expansion of the internal points. As a consequence, a sixth-order method is produced at a cost of only four stages per step instead of six stages needed for the methods which have appeared in the literature until now. Numerical results over some well-known problems in physics and mechanics indicate the superiority of the new method.     

A Spectral Method for the Time Evolution in Parabolic Problems

Numerical time propagation of linear parabolic problems is commonly performed by Taylor expansion based schemes, such as Runge–Kutta. However, explicit schemes of this type impose a stringent stability restriction on the time step when the space discretization matrix is poorly conditioned. Thus the computational work required for integration over a long and fixed time interval is controlled by stability rather than by accuracy of the scheme. We develop an improved time evolution scheme based on a new Chebyshev series expansion for solving time-dependent inhomogeneous parabolic initial-boundary value problems in which the stability condition is relaxed. Spectral accuracy of the time evolution scheme is achieved. Additionally, the approximation derived here can be useful for solving quasi-linear parabolic evolution problems by exponential time differencing methods     

任意梯度折射率介质中光线追迹的仿真与分析

针对研究光线传输问题,光线在任意梯度折射率介质中的传输路径难以用解析式给出精确解,通常采用数值方法求解,而欧拉法、龙格库塔法和泰勒级数展开法,正是针对介质中光线传输进行追迹的数值方法。在对折射率离散分布介质中的光线追迹过程中,所需空阊点的折射率及其梯度采用距离加权插值和Barron梯度算子进行求解。通过对任意梯度折射率介质中的光线传输进行仿真,并将仿真结果与解析解进行比较和分析,结果表明龙格库塔法的追迹精度最高,泰勒级数展开法次之,而欧拉法的相对最低;此外,光线追迹精度还受到追迹步长和插值方法精度的影响。     

Controlling the Wrapping Effect in the Solution of ODEs for Asteroids

During the last decade, substantial progress has been made in fighting the wrapping effect in self-validated integrations of linear systems. However, it is still the main problem limiting the applicability of such methods to the long-term integration of non-linear systems. Here we show how high-order self-validated methods can successfully overcome this obstacle.We study and compare the validated integration of a Kepler problem with conventional and high-order methods represented by AWA and Taylor models, respectively. We show that this simple model problem exhibits significant wrapping that is particularly difficult to control for conventional first-order methods. It will become clear that utilizing high-order methods with shrink wrapping allows the system to be analyzed in a fully validated context over large integration times. By comparing high-order Taylor model integrations with Taylor model methods subjected to an artificial wrapping effect, we show that utilizing high-order methods to propagate initial conditions is indeed the foremost reason for the successful suppression of the wrapping effect.To further demonstrate that high-order Taylor model methods can be used for the integration of complicated non-linear systems, we summarize results obtained from a fully verified and self-validated orbit integration of the near earth asteroid 1997 XF11. Since this asteroid will have several close encounters with Earth, its analysis is an important application of reliable computations.     

An iterative method for the computation of the solutions of nonlinear equations

In this paper, we present new iterative methods for the computation of zeros of C ¹ functions. The idea is mainly based on a new asymptotic expansion (the Bernoulli expansion) for regular functions. Just as the Newton method is derived from the linear part of the Taylor polynomial, the new methods are analogously derived from the quadratic part of the Bernoulli expansion. We prove that the proposed procedures combine the assured convergence of bisection-like algorithms with a superlinear convergence speed which characterizes Newton-like methods. We show that the order of this new procedure is p= 2 and that the cost per iteration is completely equivalent to that of the Newton method. Finally some numerical experiments are performed. The related results seem to indicate that at least one of the proposed techniques works better than the Newton method. Moreover, the given method used in connection with an enclosing-interval procedure [2], is competitive with the ones recently proposed by Alefeld and Potra [2]. Received: July 1997 / Accepted: January 1998     

Optimal buffer allocation of serial production lines with quality inspection machines

Using Taylor series expansion and probability generating function technique, we present an approximation method for the analysis of the average steady state throughput of serial production lines with unreliable machines, finite buffers and quality inspection machines. Employing the approximation method, we propose an analytic method for the optimal buffer allocation to achieve a desired throughput. The proposed methods are validated by computer simulations.     

The layer-wise method and the backpropagation hybrid approach tolearning a feedforward neural network

Feedforward neural networks (FNNs) have been proposed to solve complex problems in pattern recognition and classification and function approximation. Despite the general success of learning methods for FNNs, such as the backpropagation (BP) algorithm, second-order optimization algorithms and layer-wise learning algorithms, several drawbacks remain to be overcome. In particular, two major drawbacks are convergence to a local minima and long learning time. We propose an efficient learning method for a FNN that combines the BP strategy and optimization layer by layer. More precisely, we construct the layer-wise optimization method using the Taylor series expansion of nonlinear operators describing a FNN and propose to update weights of each layer by the BP-based Kaczmarz iterative procedure. The experimental results show that the new learning algorithm is stable, it reduces the learning time and demonstrates improvement of generalization results in comparison with other well-known methods.     

A Higher-Order Taylor Expansion Approach to Simulation of Stochastic Forward-Looking Models with an Application to a Nonlinear Phillips Curve Model

We propose to apply to the simulation of general nonlinearrational-expectation models a method where the expectation functions areapproximated through a higher-order Taylor expansion. This method has beenadvocated by Judd (1998) and others for the simulation of stochasticoptimal-control problems and we extend its application to more general cases.The coefficients for the first-order approximation of the expectation functionare obtained using a generalized eigenvalue decomposition as it is usual forthe simulation of linear rational-expectation models. Coefficients forhigher-order terms in the Taylor expansion are then obtained by solving asuccession of linear systems. In addition, we provide a method to reduce abias in the computation of the stochastic equilibrium of such models. Theseprocedures are made available in DYNARE, a MATLAB and GAUSS based simulationprogram.This method is then applied to the simulation of a macroeconomic modelembodying a nonlinear Phillips curve. We show that in this case a quadraticapproximation is sufficient, but different in important ways from thesimulation of a linearized version of the model.     

Quartic parameters for acoustic applications of lattice Boltzmann scheme

Using the Taylor expansion method, we show that it is possible to improve the lattice Boltzmann method for acoustic applications. We derive a formal expansion of the eigenvalues of the discrete approximation and fit the parameters of the scheme to enforce fourth order accuracy. The corresponding discrete equations are solved with the help of symbolic manipulation. The solutions are obtained in the case of D3Q27 lattice Boltzmann scheme. Various numerical tests support the coherence of this approach.