Variance reduction method based on sensitivity derivatives,Part 2

A previous paper introduced a sampling method (SDES) based on sensitivity derivatives to construct statistical moment estimates that are more efficient than standard Monte Carlo estimates. In this paper we sharpen previous theoretical results and introduce a criterion to guarantee that the variance of SDES estimates is smaller than the variance of the Monte Carlo estimate. Previous numerical experiments demonstrated, and here we prove analytically, that the first-order SDES and Monte Carlo estimates converge at the same rate. We illustrate the efficiency of the SDES method of order n, where n is fixed, to estimate statistical moments with a Korteweg–de Vries equation with uncertain initial conditions.     

An efficient sampling method for stochastic inverse problems

A general framework is developed to treat inverse problems with parameters that are random fields. It involves a sampling method that exploits the sensitivity derivatives of the control variable with respect to the random parameters. As the sensitivity derivatives are computed only at the mean values of the relevant parameters, the related extra cost of the present method is a fraction of the total cost of the Monte Carlo method. The effectiveness of the method is demonstrated on an example problem governed by the Burgers equation with random viscosity. It is specifically shown that this method is two orders of magnitude more efficient compared to the conventional Monte Carlo method. In other words, for a given number of samples, the present method yields two orders of magnitude higher accuracy than its conventional counterpart.     

An efficient statistically equivalent reduced method on stochastic model updating

The demand for computational efficiency and reduced cost presents a big challenge for the development of more applicable and practical approaches in the field of uncertainty model updating. In this article, a computationally efficient approach, which is a combination of Stochastic Response Surface Method (SRSM) and Monte Carlo inverse error propagation, for stochastic model updating is developed based on a surrogate model. This stochastic surrogate model is determined using the Hermite polynomial chaos expansion and regression-based efficient collocation method. This paper addresses the critical issue of effectiveness and efficiency of the presented method. The efficiency of this method is demonstrated as a large number of computationally demanding full model simulations are no longer essential, and instead, the updating of parameter mean values and variances is implemented on the stochastic surrogate model expressed as an explicit mathematical expression. A three degree-of-freedom numerical model and a double-hat structure formed by a number of bolted joints are employed to illustrate the implementation of the method. Using the Monte Carlo-based method as the benchmark, the effectiveness and efficiency of the proposed method is verified.     

An interval algorithm for sensitivity analysis of coupled vibro-acoustic systems

Sensitivity analysis is a vital part in the optimization design of coupled vibro-acoustic systems. A new interval sensitivity-analysis method for vibro-acoustic systems is proposed in this paper. This method relies on only interval perturbation analysis instead of partial derivatives and difference operations. For strongly nonlinear systems, in particular, this methodology requires parameter variation over narrower ranges in comparison with other methods. To implement sensitivity analysis based on this method, the interval ranges of the responses of the vibro-acoustic system with interval parameters should first be obtained. Therefore, an interval perturbation-analysis method is presented for obtaining the interval bounds of the sound-pressure responses of a coupled vibro-acoustic system with interval parameters. The interval perturbation method is then compared with the Monte Carlo method, which can be taken as the benchmark for comparative accuracy. Two numerical examples involving sensitivity analysis of vibro-acoustic systems illustrate the feasibility and effectiveness of the proposed interval-based method.     

Estimation of derivatives with respect to parameters of a functional of a diffusion process moving in a domain with absorbing boundary

A statistical method is proposed for estimating derivatives with respect to parameters of a functional of a diffusion process moving in a domain with absorbing boundary. The functional considered defines the probability representation of the solution of a corresponding parabolic first boundary-value problem. The problem posed is tackled by numerically solving stochastic differential equations (SDE) using the Euler method. An error of the proposed method is evaluated, and estimates of the variance of the resultant parametric derivatives are given. Some numerical results are presented.     

MC方差减小技术在算术平均亚式外汇期权定价中的应用

建立了利率和汇率波动率均为随机情形下算术平均亚式外汇期权的定价模型.由于其定价问题求解十分困难,运用蒙特卡罗(Monte Carlo)方法并结合控制变量方差减小技术进行模拟,有效地减小了模拟方差,得到了期权定价问题的数值结果.     

Numerical approximation of conditional asymptotic variances using Monte Carlo simulation

We consider in this paper the use of Monte Carlo simulation to numerically approximate the asymptotic variance of an estimator of a population parameter. When the variance of an estimator does not exist in finite samples, the variance of its limiting distribution is often used for inferences. However, in this case, the numerical approximation of asymptotic variances is less straightforward, unless their analytical derivation is mathematically tractable. The method proposed does not assume the existence of variance in finite samples. If finite sample variance does exist, it provides a more efficient approximation than the one based on the convergence of finite sample variances. Furthermore, the results obtained will be potentially useful in evaluating and comparing different estimation procedures based on their asymptotic variances for various types of distributions. The method is also applicable in surveys where the sample size required to achieve a fixed margin of error is based on the asymptotic variance of the estimator. The proposed method can be routinely applied and alleviates the complex theoretical treatment usually associated with the analytical derivation of the asymptotic variance of an estimator which is often managed on a case by case basis. This is particularly appealing in view of the advance of modern computing technology. The proposed numerical approximation is based on the variances of a certain truncated statistic for two selected sample sizes, using a Richardson extrapolation type formulation. The variances of the truncated statistic for the two sample sizes are computed based on Monte Carlo simulations, and the theory for optimizing the computing resources is also given. The accuracy of the proposed method is numerically demonstrated in a classical errors-in-variables model where analytical results are available for the purpose of comparisons.     

A new method for parameter sensitivity estimation in structural reliability analysis

For the parameter sensitivity estimation with implicit limit state functions in the time-invariant reliability analysis, the common Monte Carlo simulation based approach involves multiple trials for each parameter being varied, which will increase associated computational cost and the cost may become inevitably high especially when many random variables are involved. Another effective approach for this problem is featured as constructing the equivalent limit state function (usually called response surface) and performing the estimation in FORM/SORM. However, as the equivalent limit state function is polynomial in the traditional response surface method, it is not a good approximation especially for some highly non-linear limit state functions. To solve the above two problems, a new method, support vector regression based response surface method, is therefore presented in this paper. The support vector regression algorithm is employed to construct the equivalent limit state function and FORM/SORM is used in the parameter sensitivity estimation, and then two illustrative examples are given. It is shown that the computational cost of the sensitivity estimation can be greatly reduced and the accuracy can be retained, and results of the sensitivity estimation obtained by the proposed method are in satisfactory agreement with those computed by the conventional Monte Carlo methods.     

On the parametric uncertainty quantification of the Rothermel's rate of spread model

Parametric uncertainty quantification of the Rothermel's fire spread model is presented using the Polynomial Chaos expansion method under a Non-Intrusive Spectral Projection (NISP) approach. Several Rothermel's model input parameters have been considered random with an associated prescribed probability density function. Two different vegetation fire scenarios are considered and NISP method results and performance are compared with four other stochastic methodologies: Sensitivity Derivative Enhance Sampling; two Monte Carlo techniques; and Global Sensitivity Analysis. The stochastic analysis includes a sensitivity analysis study to quantify the direct influence of each random parameter on the solution. The NISP approach achieved performance three orders of magnitude faster than the traditional Monte Carlo method. The NISP capability to perform uncertainty quantification associated with fast convergence makes it well suited to be applied for stochastic prediction of fire spread.     

变系数部分线性模型的加权随机约束s-K估计

研究随机约束条件下半参数变系数部分线性模型的参数估计问题,当回归模型线性部分变量存在多重共线性时,基于Profile最小二乘方法、s-K估计和加权混合估计构造参数向量的加权随机约束s-K估计,并在均方误差矩阵准则下给出新估计量优于s-K估计和加权混合估计的充要条件,最后通过蒙特卡洛数值模拟验证所提出估计量的有限样本性质.