蒙特卡洛方法在仿真中的应用

本文介绍了蒙特卡洛方法,一种利用随机数(或伪随机数)来解决许多类型计算问题的通用算法。首先描述了蒙特卡洛方法的基本原理,并且通过两个典型应用案例,讨论了蒙特卡洛方法的适用范围和使用条件,展示了该方法的优点,体现了该方法在解决高自由度问题方面的优势。     

Solving Systems of Linear Equations with Relaxed Monte Carlo Method

The problem of solving systems of linear algebraic equations by parallel Monte Carlo numerical methods is considered. A parallel Monte Carlo method with relaxation is presented. This is a report of a research in progress, showing the effectiveness of this algorithm. Theoretical justification of this algorithm and numerical experiments are presented. The algorithms were implemented on a cluster of workstations using MPI.     

基于蒙特卡罗最小二乘的实验数据拟合方法

采用最小二乘法拟合化工实验数据,相关系数接近于1,精度高,但所得的结果与经验关联式大相径庭。蒙特卡罗方法是一种基于概率模型的非确定性数值方法。蒙特卡罗最小二乘拟合方法处理化工实验数据,应用中更为灵活,适用范围更广。在Excel电子表格中,利用工作表中的数据与VBA混合编程很容易完成蒙特卡罗最小二乘数据拟合,VBA实现与Excel电子表格的数据通讯及并行处理实验数据,读取工作表中的实验数据,计算随机点的大致搜索范围,进行最小二乘统计分析,将结果输出到工作表中。蒙特卡罗最小二乘拟合方法采用与最小二乘法相同的精度标准,在符合大数定理的基础上,精度大幅度提高。蒙特卡罗方法在随机搜索点较小时,误差很大,当随机搜索点达到10000时,其精度与最小二乘法相差无几,却得到与经验关联式十分接近的准数关系方程,取得了实践与理论统一的实验效果。     

基于蒙特卡罗模拟修正的随机矩阵去噪方法

针对蕴含噪声信息较少的小组合股票市场,提出使用蒙特卡罗模拟修正的随机矩阵去噪方法。首先通过数据模拟生成随机矩阵,然后利用大量的模拟数据来同时修正噪声下界和上界,最终对噪声范围进行精确测定。运用道琼斯中国88指数和香港恒生50指数的数据进行实证分析,结果表明,与LCPB法、PG+法和KR法相比,在特征值、特征向量和反比参率方面, 蒙特卡罗模拟去噪方法修正后噪声范围的合理性及有效性得到很大的提升;对去噪前后的相关矩阵进行投资组合,得知在相同的期望收益率下,蒙特卡罗模拟去噪方法具有最小的风险值,能够为资产组合选择和风险管理等金融应用提供一定的参考。     

基于RMC的蒙特卡罗程序性能优化

蒙特卡罗MC方法是核反应堆设计和分析中重要的粒子输运模拟方法.MC方法能够模拟复杂几何形状且计算结果精度高,缺点是需要耗费大量时间进行上亿规模粒子模拟.如何提高蒙特卡罗程序的性能成为大规模蒙特卡罗数值模拟的挑战.基于堆用蒙特卡罗分析程序RM C,先后开展了基于TCMalloc动态内存分配优化、OpenMP线程调度策略优...     

基于Monte Carlo method的均衡度确定模型

蒙特卡罗方法(Monte Carlo method),也称统计模拟方法,是一种以概率统计理论为指导的一类非常重要的数值计算方法,是指使用随机数(或更常见的伪随机数)来解决很多计算问题的方法,本文尝试建立警察服务平台的均衡度模型并用蒙特卡罗方法求解,实验结果可以满足一般的应用需求。     

Computing the principal eigenvalue of the Laplace operator by a stochastic method

We describe a Monte Carlo method for the numerical computation of the principal eigenvalue of the Laplace operator in a bounded domain with Dirichlet conditions. It is based on the estimation of the speed of absorption of the Brownian motion by the boundary of the domain. Various tools of statistical estimation and different simulation schemes are developed to optimize the method. Numerical examples are studied to check the accuracy and the robustness of our approach.     

Large-scale reliability-based structural optimization

A robust and efficient methodology is presented for treating large-scale reliability-based structural optimization problems. The optimization is performed with evolution strategies, while the reliability analysis is carried out with the Monte Carlo simulation method incorporating the importance sampling technique to reduce the sample size. Efficient hybrid methods are implemented to solve the reanalysis-type problems that arise in the optimization phase with evolution strategies and in the reliability analysis with Monte Carlo simulations. These hybrid solution methods are based on the preconditioned conjugate gradient algorithm using efficient preconditioning schemes. The numerical tests presented demonstrate the computational advantages of the proposed methods, which become more pronounced for large-scale optimization problems.     

Formulation and computational issues for stability of two-layer inelastic fluids

A numerical solution technique based on the Chebyshev pseudospectral method is presented for solving boundary value and generalized complex eigenvalue problems which are valid over connected domains coupled through interfacial boundary conditions. As an example, the eigenvalue problem that describes the linear stability of two superposed inelastic Carreau–Yasuda fluids in plane Poiseuille flow is considered. Collocation points are formed by following two different approaches and it is shown that the accuracy of the results are highly dependent on the choice of collocation points. Therefore, in the success of pseudospectral method, a proper selection of collocation points for boundary value and eigenvalue problems is very crucial.     

Monte Carlo方法与计算机模拟应用研究

介绍了Monte Carlo方法,提出其在模拟Buffer问题时存在的一个问题,并给出改进的方法;提出了用Monte Carlo方法产生任意分布随机变量的原理及方法,并对Beta分布和标准正态分布随机变量进行了计算机模拟和效果检验。     

Monte Carlo methods for derivatives of options with discontinuous payoffs

Various Monte Carlo methods have been proposed to estimate the derivatives of contingent claims prices. The Monte Carlo approximate likelihood ratio estimator is studied. Recent convergence results are extended in order to show that the Monte Carlo approximate likelihood ratio derivative estimator is asymptotically equivalent, up to a second-order bias component, to an estimator based on a covariation approximation, the Monte Carlo Covariation estimator. Both converge slower than the Monte Carlo Malliavin derivative estimators. Theoretical convergence results are illustrated in a numerical experiment dealing with the risk management of digital options in a CEV model.     

Reliability-based topology optimization of double layer grids using a two-stage optimization method

This paper presents a two-stage optimization method for reliability-based topology optimization (RBTO) of double layer grids that considers uncertainties in applied loads. The optimization is performed using a two-stage optimization by employing the method of moving asymptotes (MMA) and ant colony optimization (ACO), which is called MMA-ACO method. For implementation of MMA-ACO, the structural stiffness is maximized using MMA, first. Then, the results of MMA are used to enhance ACO through the following four modifications: (I) finding the structural importance rate of elements or joints and using this to achieve a better topology, (II) determining the number of compressive and tensile element types, (III) changing the lower limit of available cross-sectional areas for the elements of each group and (IV) modifying the generation of random stable structures. In reliability analysis, multiple criteria i.e. stiffness and eigenvalue are considered where the probability of failure in each mode is calculated by Monte Carlo simulation (MCS). To reduce the computational time, the eigenvalues are evaluated using the third order approximation (TOA). Through numerical examples, reliability-based topology designs of typical double layer grids are obtained by ACO and MMA-ACO methods. The numerical results reveal the computational advantages and effectiveness of the proposed MMA-ACO method for the RBTO of double layer grids. Also, the importance of considering uncertainties is then demonstrated by comparing the results obtained by those of other failure probabilities.     

Analysis of complex survey data using SAS

Commonly used statistical methods and software packages typically assume that observations are independent and identically distributed and fail to account for complex sampling designs when present. I suggest an approach to analyzing complex survey data in SAS, using weighted generalized estimating equations. Limited Monte Carlo simulations support the method. An example demonstrates application of the method and compares results to those from software commonly used in the analysis of complex survey data.     

Numerical integration using Wang-Landau sampling

We report a new application of Wang-Landau sampling to numerical integration that is straightforward to implement. It is applicable to a wide variety of integrals without restrictions and is readily generalized to higher-dimensional problems. The feasibility of the method results from a reinterpretation of the density of states in statistical physics to an appropriate measure for numerical integration. The properties of this algorithm as a new kind of Monte Carlo integration scheme are investigated with some simple integrals, and a potential application of the method is illustrated by the evaluation of integrals arising in perturbation theory of quantum many-body systems.     

基于蒙特卡罗方法的目标跟踪

为了更鲁棒和快速地进行目标跟踪,在基于粒子滤波的目标跟踪方法的启发下,提出了一种新的基于蒙特卡罗方法的目标跟踪方法。该方法首先运用蒙特卡罗技术对下一帧目标可能出现的位置和尺度进行抽样;然后计算各抽样与参考目标的相似度;最后通过估计目标状态来获得跟踪目标。实验表明,该方法无需目标运动信息,特别适用于目标灵活运动时的跟踪,与现有的算法相比,不仅算法实现简单,同时有较好的鲁棒性和通用性。     

Discontinuous Galerkin methods for elliptic partial differential equations with random coefficients

This paper proposes and analyses two numerical methods for solving elliptic partial differential equations with random coefficients, under the finite noise assumption. First, the stochastic discontinuous Galerkin method represents the stochastic solution in a Galerkin framework. Second, the Monte Carlo discontinuous Galerkin method samples the coefficients by a Monte Carlo approach. Both methods discretize the differential operators by the class of interior penalty discontinuous Galerkin methods. Error analysis is obtained. Numerical results show the sensitivity of the expected value and variance with respect to the penalty parameter of the spatial discretization.     

MCMC方法在生物逆问题求解中的应用

提出用马尔科夫链蒙特卡罗（MCMC）方法来求解生物逆问题。导出待求参数分布规律的后验概率密度函数;采用自适应Metropolis算法构造Markov链;然后截取收敛的链序列计算数学期望,成功估计出未知参数。数值实验结果表明,该方法具有很高的估计精度和较好的抗噪声性能。     

Automatic synthesis of uncertain models for linear circuit simulation: A polynomial chaos theory approach

A generalized and automated process for the evaluation of system uncertainty using computer simulation is presented. Wiener–Askey polynomial chaos and generalized polynomial chaos expansions along with Galerkin projections, are used to project a resistive companion system representation onto a stochastic space. Modifications to the resistive companion modeling method that allow for individual models to be produced independently from one another are presented. The results of the polynomial chaos system simulation are compared to Monte Carlo simulation results from PSPICE and C++. The comparison of the simulation results from the various methods demonstrates that polynomial chaos circuit simulation is accurate and advantageous. The algorithms and processes presented in this paper are the basis for the creation of a computer-aided design (CAD) simulator for linear networks containing uncertain parameters.     

含指数积分型阻尼和周期激励系统的稳态随机响应分析

运用基于短时高斯逼近的广义胞映射方法,研究了含指数积分型非粘性阻尼和周期激励系统在高斯白噪声作用下的稳态响应.首先介绍了方法的实施过程,并推导了系统的矩方程.然后给出了系统的稳态概率密度函数,分析了阻尼系数和松弛参数对稳态响应的影响,并通过直接Monte Carlo模拟的结果验证了广义胞映射方法的有效性.     

A fast approximate method for parameter sensitivity estimation in Monte Carlo structural reliability

In estimating the effect of a change in a random variable parameter on the (time-invariant) probability of structural failure estimated through Monte Carlo methods the usual approach is to carry out a duplicate simulation run for each parameter being varied. The associated computational cost may become prohibitive when many random variables are involved. Herein a procedure is proposed in which the numerical results from a Monte Carlo reliability estimation procedure are converted to a form that will allow the basic ideas of the first order reliability method to be employed. Using these allows sensitivity estimates of low computational cost to be made. Illustrative examples with sensitivities computed both by conventional Monte Carlo and the proposed procedure show good agreement over a range of probability distributions for the input random variables and for various complexities of the limit state function.