An improved high order moment-based saddlepoint approximation method for reliability analysis

Moment-based methods use only statistical moments of random variables for reliability analysis. The cumulative distribution function (CDF) or probability density function (PDF) of a performance function can be constructed from the perspective of the first few statistical moments, and the failure probability can be evaluated accordingly. However, existing moment-based methods may lead to large errors or instability. As such, the present paper focuses on the high order moment method for higher accuracy of reliability estimation by combining the common saddlepoint approximation technique, and an improved high order moment-based saddlepoint approximation (SPA) method for reliability analysis is presented. The approximated cumulant generating function (CGF) and the CDF of the performance function in terms of its first four statistical-moments are constructed. The developed method can be used for reliability evaluation of uncertain structures follow any types of distribution. Several numerical examples are given to demonstrate the efficacy and accuracy of the proposed method. Comparisons of the new method and several existing high order moment methods are also made on the reliability assessment.     

Probabilistic sensitivity of limit states of structures. The Monte Carlo simulation

The main issue of the paper is the probabilistic sensitivity of the limit states of structures with respect to selected input design variables. Attempt to the problem is done by the dedicated Monte Carlo simulation procedure. Basic design variables are random variables of given probability distributions, presented in the form of random numbers. Uni-parametrical increment of the dominant basic variable (basic variables) is done, finally achieving the structural limit state. The simulation procedure restuls in a set of limit multipliers. Statistical analysis leads to the estimate of the probability density function of the limit state. Thus the numerical image is presented of the probabilistic sensitivity of the structural limit state. Reliability or the probability of failure are to be estimated, as statistical parameters of the histogram. Numerical examples of engineering structures illustrate the method introduced in the paper, conclusions are formulated eventually. (© 2009 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

结构的失效可能度及模糊概率计算方法

依据模糊可能性理论,系统地建立含模糊变量时结构的可靠性计算模型。旨在解决模糊结构、模糊-随机结构和模糊状态假设下结构的可靠性计算问题。所建模型可给出模糊结构失效的可能度和模糊-随机结构失效概率的可能性分布。研究表明:对同时含模糊变量和随机变量的混合可靠性计算问题,把失效概率(或可靠度)作为模糊变量,能更客观地反映系统的安全状况。算例分析说明了文中方法的合理性和有效性。     

A computational framework of kinematic accuracy reliability analysis for industrial robots

A new computational method to evaluate comprehensively the positional accuracy reliability for single coordinate, single point, multipoint and trajectory accuracy of industrial robots is proposed using the sparse grid numerical integration method and the saddlepoint approximation method. A kinematic error model of end-effector is constructed in three coordinate directions using the sparse grid numerical integration method considering uncertain parameters. The first-four order moments and the covariance matrix for three coordinates of the end-effector are calculated by extended Gauss–Hermite integration nodes and corresponding weights. The eigen-decomposition is conducted to transform the interdependent coordinates into independent standard normal variables. An equivalent extreme value distribution of response is applied to assess the reliability of kinematic accuracy. The probability density function and probability of failure for extreme value distribution are then derived through the saddlepoint approximation method. Four examples are given to demonstrate the effectiveness of the proposed method.     

A probabilistic theory of random maps

We present a probabilistic theory of random maps with discrete time and continuous state. The forward and backward Kolmogorov equations as well as the FPK equation governing the evolution of the probability density function of the system are derived. The moment equations of arbitrary order are derived, and the reliability and first passage time problem are also studied. Examples are presented to demonstrate the application of the theoretical development. Numerical solutions including the time histories of moment evolution, steady state probability density function, reliability and first passage time probability density function for time discrete random maps are included. The present work compliments the existing theory of continuous time stochastic processes.     

A new method for reliability analysis of structures with mixed random and convex variables

This paper proposes a method combining projection-outline-based active learning strategy with Kriging metamodel for reliability analysis of structures with mixed random and convex variables. In this method, it is determined that the approximation accuracy of projection outlines on the limit-state surface is crucial for estimation of failure probability instead of the whole limit-state surface. To efficiently improve the approximation accuracy of projection outlines, a new projection-outline-based active learning strategy is developed to sequentially obtain update points located around the projection outlines. Taking into account the influence of metamodel uncertainty on the estimation of failure probability, a quantification function of metamodel uncertainty is developed and introduced in the stopping condition of Kriging metamodel update. Finally, Monte Carlo simulation is employed to calculate the failure probability based on the refined Kriging metamodel. Four examples including the Burro Creek Bridge and a piezoelectric energy harvester are tested to validate the performance of the proposed method. Results indicate that the proposed method is accurate and efficient for reliability analysis of structures with mixed random and convex variables.     

A practical non-parametric copula algorithm for system reliability with correlations

System reliability analysis involving correlated random variables is challenging because the failure probability cannot be uniquely determined under the given probability information. This paper proposes a system reliability evaluation method based on non-parametric copulas. The approximated joint probability distribution satisfying the constraints specified by correlations has the maximal relative entropy with respect to the joint probability distribution of independent random variables. Thus the reliability evaluation is unbiased from the perspective of information theory. The estimation of the non-parametric copula parameters from Pearson linear correlation, Spearman rank correlation, and Kendall rank correlation are provided, respectively. The approximated maximum entropy distribution is then integrated with the first and second order system reliability method. Four examples are adopted to illustrate the accuracy and efficiency of the proposed method. It is found that traditional system reliability method encodes excessive dependence information for correlated random variables and the estimated failure probability can be significantly biased.     

Saddlepoint approximation for moments of random variables

In this paper, we introduce a saddlepoint approximation method for higher-order moments like E(S − a)₊ ^m , a>0, where the random variable S in these expectations could be a single random variable as well as the average or sum of some i.i.d random variables, and a > 0 is a constant. Numerical results are given to show the accuracy of this approximation method.     

积分波动率二阶变差估计量分析:鞍点算法

本文利用鞍点逼近方法对Black-Scholes模型的积分波动率的二阶变差估计量的估计误差进行分析,得到了相对于中心极限定理更为精细的结果,并且给出了逼近的鞍点算法。结果表明鞍点逼近是中心极限定理的纠正。模拟结果表明鞍点算法给出的估计误差分布相对于正态逼近更合理。该结果在对积分波动率进行统计假设检验时是有意义的。     

Numerical simulation based on fuzzy stochastic analysis

In this paper mathematical methods for fuzzy stochastic analysis in engineering applications are presented. Fuzzy stochastic analysis maps uncertain input data in the form of fuzzy random variables onto fuzzy random result variables. The operator of the mapping can be any desired deterministic algorithm, e.g. the dynamic analysis of structures. Two different approaches for processing the fuzzy random input data are discussed. For these purposes two types of fuzzy probability distribution functions for describing fuzzy random variables are introduced. On the basis of these two types of fuzzy probability distribution functions two appropriate algorithms for fuzzy stochastic analysis are developed. Both algorithms are demonstrated and compared by way of an example.     

Distribution modeling for reliability analysis: Impact of multiple dependences and probability model selection

Reliability analysis requires modeling of joint probability distribution of uncertain parameters, which can be a challenge since the random variables representing the parameter uncertainties may be correlated. For convenience, a Gaussian data dependence is commonly assumed for correlated random variables. This paper first investigates the effect of multidimensional non-Gaussian data dependences underlying the multivariate probability distribution on reliability results. Using different bivariate copulas in a vine structure, various data dependences can be modeled. The associated copula parameters are identified from available statistical information by moment matching techniques. After the development of the vine copula model for representing the multivariate probability distribution, the reliability involving correlated random variables is evaluated based on the Rosenblatt transformation. The impact of data dependence is significant because a large deviation in failure probability is observed, which emphasizes the need for accurate dependence characterization. A practical method for dependence modeling based on limited data is thus provided. The result demonstrates that the non-Gaussian data dependences can be real in practice, and the reliability can be biased if the Gaussian dependence is used inappropriately. Moreover, the effect of conditioning order on reliability should not be overlooked except that the vine structure contains only one type of copula.     

On the compounded bivariate Poisson distribution: A unified treatment

A unified treatment is presented here of compounding with the bivariate Poisson distribution. Exploiting the exponential nature of its probability generating function, it is shown that the pgf of the compound distribution is the moment generating function of the compounding random variable. This relationship leads to rather interesting general results. Particularly, the development of the conditional distribution is simplified. Four cases are presented in detail.     

Iterated probability distributions and extremes with random sample size

In this paper the possible nondegenerated limit distributions for the n-fold mapping of a given probability distribution are considered. If the mapping used for the iteration procedure is a probability generating function of a positive integer-valued random variable then the results can be applied to the max-stability of distributions of random variables with random sample size.     

Reliability of Elastic‐Plastic Frame Structures

An effective algorithm for estimating the reliability of frame structures made of an elastic perfectly plastic material is presented. It is assumed that the structure is working in uniaxial state of stress, the material parameters and the limit values of the loads are described by random variables, all loads are acting in a static manner, the probability density functions of all random variables, which describe the structure and loads are known. As the structure reliability measure the probability of failure and corresponding reliability index are regarded. The dead (constant) load and the climatic loads are taken into account. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

基于指数分布不同定时截尾数据的可靠度的置信下限

本文基于指数分布不同定时截尾数据,利用鞍点逼近法给出参数估计的概率分布的近似公式,进而给出可靠度的近似置信下限,并通过数值模拟及实例计算说明本文方法的可行性。     

Interval optimization based line sampling method for fuzzy and random reliability analysis

For structural system with fuzzy variables as well as random variables, a novel algorithm for obtaining membership function of fuzzy reliability is presented on interval optimization based Line Sampling (LS) method. In the presented algorithm, the value domain of the fuzzy variables under the given membership level is firstly obtained according to their membership functions. Then, in the value domain of the fuzzy variables, bounds of reliability of the structure are obtained by the nesting analysis of the interval optimization, which is performed by modern heuristic methods, and reliability analysis, which is achieved by the LS method in the reduced space of the random variables. In this way the uncertainties of the input variables are propagated to the safety measurement of the structure, and the membership function of the fuzzy reliability is obtained. The presented algorithm not only inherits the advantage of the direct Monte Carlo method in propagating and distinguishing the fuzzy and random uncertainties, but also can improve the computational efficiency tremendously in case of acceptable precision. Several examples are used to illustrate the advantages of the presented algorithm.     

Saddlepoint Approximation for the Distribution Function Near the Mean

We repair numerical difficulties in applying saddlepoint tail probability approximations when the ordinate at which the approximation is evaluated is near the mean of the distribution approximated. These modifications apply to double saddlepoint approximations to conditional distributions as well.     

Saddlepoint Approximations to the Probability of Ruin in Finite Time for the Compound Poisson Risk Process Perturbed by Diffusion

A large deviations type approximation to the probability of ruin within a finite time for the compound Poisson risk process perturbed by diffusion is derived. This approximation is based on the saddlepoint method and generalizes the approximation for the non-perturbed risk process by Barndorff-Nielsen and Schmidli (Scand Actuar J 1995(2):169–186, 1995). An importance sampling approximation to this probability of ruin is also provided. Numerical illustrations assess the accuracy of the saddlepoint approximation using importance sampling as a benchmark. The relative deviations between saddlepoint approximation and importance sampling are very small, even for extremely small probabilities of ruin. The saddlepoint approximation is however substantially faster to compute.     

A Sparse Grid Stochastic Collocation Discontinuous Galerkin Method for Constrained Optimal Control Problem Governed by Random Convection Dominated Diffusion Equations

A sparse grid stochastic collocation method combined with discontinuous Galerkin method is developed for solving convection dominated diffusion optimal control problem with random coefficients. By the optimal control theory, an optimality system is obtained for the problem, which consists of a state equation, a co-state equation and an inequality. Based on finite dimensional noise assumption of random field, the random coefficients are assumed to have finite term expansions depending on a finite number of mutually independent random variables in the probability space. An approximation scheme is established by using a discontinuous Galerkin method for the physical space and a sparse grid stochastic collocation method based on the Smolyak construction for the probability space, which leads to the solution of uncoupled deterministic problems. A priori error estimates are derived for the state, co-state and control variables. Numerical experiments are presented to illustrate the theoretical results.     

随机结构系统基于可靠性的优化设计

提出了以梁板（薄板）为基体的随机结构系统（即结构元件的面积、长度、弹性模量和强度等均为随机变量）在随机载荷作用下,基于可靠性的优化设计方法．给出了随机结构系统安全余量和系统可靠性指标的敏度表达式;给出最佳矢量型算法．首先是用改进的一次二阶矩和随机有限元法求出安全余量的可靠性指标的表达式,然后用概率网络估算(PNET)法求出系统失效概率的公式,对该式两边求导得出了系统可靠性指标的敏度表达式,进而用最佳矢量型算法进行优化设计．在优化迭代过程中,采用梯度步和最佳矢量步相结合的方法进行计算．最后给出了一个算例,说明该方法计算效率高,收敛稳定,适合工程应用．