Path integral solution for non-linear system enforced by Poisson White Noise

In this paper the response in terms of probability density function of non-linear systems under Poisson White Noise is considered. The problem is handled via path integral (PI) solution that may be considered as a step-by-step solution technique in terms of probability density function. First the extension of the PI to the case of Poisson White Noise is derived, then it is shown that at the limit when the time step becomes an infinitesimal quantity the Kolmogorov–Feller (K–F) equation is fully restored enforcing the validity of the approximations made in obtaining the conditional probability appearing in the Chapman Kolmogorov equation (starting point of the PI). Spectral counterpart of the PI, ruling the evolution of the characteristic function is also derived. It is also shown that using appropriately the PI for Poisson White Noise also the case of Normal White Noise be easily derived.     

The Folded Normal Distribution: Accuracy of Estimation By Maximum Likelihood

This paper is a sequel to two previous papers on the subject of the Folded Normal Distribution appearing in Technometrics, S, pp. 543–550 and 551–562, (1961). In these earlier papers methods for estimating the parameters of the Folded Normal Distribution were proposed. Thii paper gives approximations for the standard errors for the maximum liklihood estimates of these parameters.     

Improving the modified XFEM for optimal high-order approximation

This paper investigates the accuracy of high-order extended finite element methods (XFEMs) for the solution of discontinuous problems with both straight and curved weak discontinuities in two dimensions. The modified XFEM, a specific form of the stable generalised finite element method, is found to offer advantages in cost and complexity over other approaches, but suffers from suboptimal rates of convergence due to spurious higher-order contributions to the approximation space. An improved modified XFEM is presented, with basis functions “corrected” by projecting out higher-order contributions that cannot be represented by the standard finite element basis. The resulting corrections are independent of the equations being solved and need be pre-computed only once for geometric elements of a given order. An accurate numerical integration scheme that correctly integrates functions with curved discontinuities is also presented. Optimal rates of convergence are then recovered for Poisson problems with both straight and quadratically curved discontinuities for approximations up to order p ≤ 4. These are the first truly optimal convergence results achieved using the XFEM for a curved weak discontinuity and are also the first optimally convergent results achieved using the modified XFEM for any problem with approximations of order p>1. Almost optimal rates of convergence are recovered for an elastic problem with a circular weak discontinuity for approximations up to order p ≤ 4.     

A shock process with a non-cumulative damage

Two types of non-cumulative damage shock models are considered. Based on the distribution of damage, caused by a shock effecting a system, the intervals with small, intermediate and large damage are introduced. The initial homogeneous Poisson shock process is split into three homogeneous Poisson processes and studied independently. Several criteria of failure are considered, based on the assumption that shocks with a small level of damage are harmless for a system, shocks with a large level of damage results in the system's failure and shocks with an intermediate level of damage can result in the system's failure only with some probability. The second model is based on an assumption that shocks with a small level of damage are harmless to a system, if they are not too close to each other. The probability of the system's failure-free performance in [0,t) is derived explicitly. Simple asymptotic exponential approximations are obtained The accuracy of this method is analyzed. Possible generalizations are discussed.     

A unified treatment of superconvergent recovered gradient functions for piecewise linear finite element approximations

Piecewise linear finite element approximations to two-dimensional Poisson problems are treated. For simplicity, consideration is restricted to problems having Dirichlet boundary conditions and defined on rectangular domains Ω which are partitioned by a uniform triangular mesh. It is also required that the solutions u ∈ H³ (Ω). A method is proposed for recovering the gradients of the finite element approximations to a root mean square accuracy of O(h²), both at element edge mid-points and element vertices, using simple averaging schemes over adjacent elements. Piecewise linear interpolants (respectively discontinuous and continuous) are then fitted to these recovered gradients, and are shown to be O(h²) estimates for ?u in the L₂-norm, and thus superconvergent. A discussion is given of the extension of the results to problems with more general region and mesh geometries, boundary conditions and with solutions of lower regularity, and also to other second-order elliptic boundary value problems, e.g. the problem of planar linear elasticity.     

First passage time of filtered Poisson process with exponential shape function

Solving some integro-differential equation we find the Laplace transform of the first passage time for filtered Poisson process generated by pulses with uniform or exponential distributions. Also, the martingale technique is applied for approximations of expectations and distributions of the first passage times. The approximations accuracy is verified with the help of Monte-Carlo simulations.     

基于非Gauss风载的高耸结构风振可靠性分析

基于风载非Gauss模型推导了Davenport谱下结构脉动风载的五阶统计矩表达。通过响应概率特征函数和Fourier变换,求得Gram-Charlier级数形式的振形位移和速度响应联合概率统计分布。利用Rice公式和Poisson假设,研究了不同风载模型对高耸结构风振可靠性分析结果的影响。算例分析表明:高耸结构风振可靠性分析需要采用风载非Gauss模型;模态位移和速度响应的联合概率统计分布在截断于五阶Hermite多项式时具有较好精度;可以引入模态位移与速度独立性假设以简化高耸结构风振可靠性分析,计算结果是偏于安全的。     

The Determination of Single Sampling Attribute Plans with Given Producer's and Consumer's Risk

The paper gives a survey of solutions to the problem of determining a single sampling plan (n, c) so that P(p ₁) > 1 – α, P(p ₂) ≤ β, and c is as small as possible, where P(p) denotes the operating characteristic, p ₁ < p ₂ and 1 – α > β. Solutions corresponding to Poisson, binomial, and hypergeometric operating characteristics are discussed and compared. Both exact and approximate formulas are given, and the accuracy of the approximations is evaluated by numerical investigations.     

A Computationally viable higher-order theory for laminated composite plates

A variational higher-order theory involving all transverse strain and stress components is proposed for the analysis of laminated composite plates. Derived from three-dimensional elasticity with emphasis on developing a viable computational methodology, the theory is well suited for finite element approximations as it incorporates both C⁰ and C^?1 continuous kinematic fields and Poisson boundary conditions. From the theory, a simple three-node stretching-bending finite element is developed and applied to the problem of cylindrical bending of a symmetric carbon/epoxy laminate for which an exact solution is available. Both the analytic and finite element results were found to be in excellent agreement with the exact solution for a wide range of the length-to-thickness ratio. The proposed higher-order theory has the same computational advantages as first-order shear-deformable theories. The present methodology, however, provides greater predictive capabilities, especially, for thick-section composites.     

Simple cylindrical and spherical finite elements

Finite element approximations are developed for three‐dimensional domains naturally represented in either cylindrical or spherical coordinates. Lines of constant radius, axial length, or angle are used to represent the domain and cast approximations that are natural for these geometries. As opposed to general isoparametric three‐dimensional elements generated in conventional parent space, these elements can be evaluated analytically and do not generate geometric discretization error. They also allow for anisotropic material coefficients that are frequently aligned in either cylindrical or spherical coordinates. Several examples are provided that show convergence properties and comparison with analytical solutions of the Poisson equation.     

Level‐cut inhomogeneous filtered Poisson field for two‐phase microstructures

Level‐cut homogeneous filtered Poisson fields developed in (J. Appl. Phys. 2003; 94 (6):3762–3770) to model two‐phase microstructures are defined, and their properties are briefly reviewed. Filtered Poisson fields are sums of randomly scaled and oriented kernels that are centered at the points of homogeneous Poisson fields. The cuts of these fields above specified thresholds are called level‐cut homogeneous filtered Poisson fields. It is shown that an arbitrary inhomogeneous Poisson field becomes homogeneous if observed in new coordinates, and that the mapping relating inhomogeneous and homogeneous Poisson fields can be constructed in a simple manner. This mapping and the model in (J. Appl. Phys. 2003; 94 (6): 3762–3770) provide an efficient algorithm for generating arbitrary inhomogeneous two‐phase microstructures. Developments in (Int. J. Numer. Meth. Engng 2008; DOI: 10.1002/nme.2340 ), using arguments essentially identical to those in (J. Appl. Phys. 2003; 94 (6):3762–3770) to define and generate inhomogeneous Poisson fields, overlook the natural extension of results in (J. Appl. Phys. 2003; 94 (6): 3762–3770) to these fields provided by the mapping constructed in this paper. Copyright © 2008 John Wiley & Sons, Ltd.     

Backorder minimization in multiproduct assemble-to-order systems

We consider a multiproduct assemble-to-order system. Components are built to stock with inventory controlled by base-stock rules, but the final products are assembled to order. Customer orders of each product follow a batch Poisson process. The leadtimes for replenishing component inventory are stochastic. We study the optimal allocation of a given budget among component inventories so as to minimize a weighted average of backorders over product types. We derive easy-to-compute bounds and approximations for the expected number of backorders and use them to formulate surrogate optimization problems. Efficient algorithms are developed to solve these problems, and numerical examples illustrate the effectiveness of the bounds and approximations.     

Three dimensional eddy current calculation using a network method

First order finite element or finite difference approximations to the electric and magnetic scalar potential Poisson equations are represented by network models. The resulting linked-circuit problem is solved using nodal variables for the larger, magnetic part of the problem and mesh variables for the smaller, conducting region. This method of solution is shown to compare favourably, in terms of number of variables, with conventional finite element techniques. A surface impedance model is also presented for the solution of saturated eddy-current problems. This model uses actual iron B/H curves rather than the approximations usually employed. Both calculation methods are verified using experimental results.     

The Use of Probability Limits of COM–Poisson Charts and their Applications

The conventional c and u charts are based on the Poisson distribution assumption for the monitoring of count data. In practice, this assumption is not often satisfied, which requires a generalized control chart to monitor both over‐dispersed as well as under‐dispersed count data. The Conway–Maxwell–Poisson (COM–Poisson) distribution is a general count distribution that relaxes the equi‐dispersion assumption of the Poisson distribution and in fact encompasses the special cases of the Poisson, geometric, and Bernoulli distributions. In this study, the exact k‐sigma limits and true probability limits for COM–Poisson distribution chart have been proposed. The comparison between the 3‐sigma limits, the exact k‐sigma limits, and the true probability limits has been investigated, and it was found that the probability limits are more efficient than the 3‐sigma and the k‐sigma limits in terms of (i) low probability of false alarm, (ii) existence of lower control limits, and (iii) high discriminatory power of detecting a shift in the parameter (particularly downward shift). Finally, a real data set has been presented to illustrate the application of the probability limits in practice. Copyright © 2012 John Wiley & Sons, Ltd.     

Extreme value of typhoon-induced non-stationary buffeting response of long-span bridges

This paper concerns the extreme value of typhoon-induced non-stationary buffeting response of long-span bridges. The framework of non-stationary buffeting analysis is briefly introduced first, in which the non-stationary buffeting response is regarded as the summation of a time-varying mean response and a dynamic response that can be represented by a zero-mean evolutionary Gaussian process characterized by an evolutionary power spectral density (EPSD) function. The formulas for determining approximate probabilistic characteristics of extreme non-stationary responses are then derived by extending the currently-used Poisson and Vanmarcke approximations. By comparing with the Monte Carlo solution, the extended approximations for extreme value of non-stationary responses are found reliable and accurate enough. Particularly, the extended Vanmarcke approximation can give closer results to the Monte Carlo solution than the extended Poisson approximation. The extended Vanmarcke approximation is finally applied to the Stonecutters Bridge to find the extreme value of non-stationary buffeting response of the bridge to a strong typhoon. The results show that the extreme displacement responses of the bridge from the non-stationary buffeting analysis are larger than those predicted by the conventional stationary buffeting analysis, and therefore the non-stationary buffeting analysis is necessary.     

Some Approximations for the Noncentral-F Distribution

Among the various approximations for the noncentral-F distribution. the best known are the computationally simpler approximation due to Severo and Zelen [18] and Laubscher [9], which requires only the normal distribution, and a more accurate approximation due to Tiku [21], which requires the F distribution. Thcsr approximations are systematically evaluated. Some variations of these approximations are considered and found to he of comparable accuracy. An Edgeworth-acries approximation, which requires only the normal distribution, is developed and sren to be superior in accuracy. A FORTRAN-subroutine for the Edgeworth-series approximation is given.     

Indirect T‐Trefftz and F‐Trefftz methods for solving boundary value problem of Poisson equation

Abstract

The Poisson equation can be solved by first finding a particular solution and then solving the resulting Laplace equation. In this paper, a computational procedure based on the Trefftz method is developed to solve the Poisson equation for two‐dimensional domains. The radial basis function approach is used to find an approximate particular solution for the Poisson equation. Then, two kinds of Trefftz methods, the T‐Trefftz method and F‐Trefftz method, are adopted to solve the resulting Laplace equation. In order to deal with the possible ill‐posed behaviors existing in the Trefftz methods, the truncated singular value decomposition method and L‐curve concept are both employed. The Poisson equation of the type, ?² u = f(x, u), in which x is the position and u is the dependent variable, is solved by the iterative procedure. Numerical examples are provided to show the validity of the proposed numerical methods and some interesting phenomena are carefully discussed while solving the Helmholtz equation as a Poisson equation. It is concluded that the F‐Trefftz method can deal with a multiply connected domain with genus p(p > 1) while the T‐Trefftz method can only deal with a multiply connected domain with genus 1 if the domain partition technique is not adopted.     

Modeling animal-vehicle collisions using diagonal inflated bivariate Poisson regression

Two types of animal-vehicle collision (AVC) data are commonly adopted for AVC-related risk analysis research: reported AVC data and carcass removal data. One issue with these two data sets is that they were found to have significant discrepancies by previous studies. In order to model these two types of data together and provide a better understanding of highway AVCs, this study adopts a diagonal inflated bivariate Poisson regression method, an inflated version of bivariate Poisson regression model, to fit the reported AVC and carcass removal data sets collected in Washington State during 2002–2006. The diagonal inflated bivariate Poisson model not only can model paired data with correlation, but also handle under- or over-dispersed data sets as well.Compared with three other types of models, double Poisson, bivariate Poisson, and zero-inflated double Poisson, the diagonal inflated bivariate Poisson model demonstrates its capability of fitting two data sets with remarkable overlapping portions resulting from the same stochastic process. Therefore, the diagonal inflated bivariate Poisson model provides researchers a new approach to investigating AVCs from a different perspective involving the three distribution parameters (λ₁, λ₂ and λ₃). The modeling results show the impacts of traffic elements, geometric design and geographic characteristics on the occurrences of both reported AVC and carcass removal data. It is found that the increase of some associated factors, such as speed limit, annual average daily traffic, and shoulder width, will increase the numbers of reported AVCs and carcass removals. Conversely, the presence of some geometric factors, such as rolling and mountainous terrain, will decrease the number of reported AVCs.     

Multiples,poisson distributions,and chance: An analysis of the Brannigan-Wanner model

Brannigan andWanner argue that the empirical distribution of multiple grades can be more adequately explained in terms of a negative contagious poisson model. This alternative is based on a Zeitgeist theory which places emphasis on the role of communication in scientific discovery. Nonetheless, a detailed analysis indicates the following: (a) mathematically, the simple Poisson is the limiting case of the contagious Poisson when the contagion parameter approaches zero; (b) empirically, the mean and variance are so nearly equal (i. e., the contagion effect is very small) that predictions from the contagious Poisson are virtually equivalent to those of the simple Poisson; (c) in particular, both distributions predict that multiples are less common than singletons and even nulltons, the latter occurring with a probability of over one third (thereby implying that chance plays a much bigger part than Zeitgeist or maturational theories would suggest); (d) estimates from theSimonton, Merton, andOgburn-Thomas data sets all concur that the contagion effect is not only small, but positive besides, yielding a modest positive contagious Poisson that contradicts the principal tenet of the communication interpretation.     

Convergence,accuracy and stability of finite element approximations of a class of non-linear hyperbolic equations

Numerical stability criteria and rates of convergence are derived for finite element approximations of the non- linear wave equation u_tt?F(u_x) = f(x, t), where F(u_x) possesses properties generally encountered in non-linear elasticity. Piecewise linear finite element approximations in x and central difference approximations in t are studied.