【作者】 李强；

【导师】 洪志敏；

【作者基本信息】 内蒙古工业大学 ， 统计学， 2017， 硕士

【摘要】 蒙特卡罗(MC)方法作为一种统计方法,起初主要作为确定论方法的补充使用.但是随着计算机的快速发展以及对计算机的普及与应用打破了这一局面,使得原本耗时的模拟过程变得更加快捷,这极大的促进了MC方法的发展.MC方法的应用范围也从起初的核领域延伸到其它领域,现今已成为解决许多物理、工程、金融等领域中问题的重要计算工具,在参数估计和可靠性设计投资风险和投标报价生物医学统计物理学等方面也有极为广泛的应用,对于确定性问题,也可用MC方法进行解决,以及求解各类方程(组)、计算多重积分、无穷级数等.但是由于MC方法经过随机模拟得到的近似值难免会与所估计的真值存在偏差,本文针对该问题展开研究.首先,论文对MC方法的收敛性与误差估计进行了分析,总结了常用的六种方差缩减的方法:重要抽样法、控制变量法、对偶变量法、条件期望法、分层抽样法、相关抽样法.其次,论文讨论了常见的方程组、定积分、级数问题的蒙特卡罗对偶变量法,通过建立概率统计模型、抽样产生随机样本进而获得确定性问题的估计值,结果显示该技巧可有效降低模拟精度且缩短计算机运行时间,充分展示了对偶变量法的高效性.最后,讨论线性代数系统的自适应蒙特卡罗求解算法.该方法包括自适应重要性抽样蒙特卡罗和自适应相关抽样蒙特卡罗,并通过求解具体算例比较蒙特卡罗方法和确定性方法的求解时间和算法效率,从而得出自适应蒙特卡罗方法可以获得更快的收敛速度。更多还原

【Abstract】 The Monte Carlo(MC)method as a statistical method was initially used primarily to supplement the deterministic method.The popularity and application of computers breaking the situation with the rapid development of computers,the original simulation process that is time-consuming has become more efficient,this chance greatly contributed to the development of MC methods.The range of application of the MC method from the beginning of the nuclear field also extended to other areas.Now the MC method has solved many problems in the field of physics,engineering,finance and other important computing tools,the investment risk in the parameter estimation and reliability design and bid quotation biomedical statistical physics and MC method has the extremely widespread application.The certainty problems also can use the MC method and all kinds of equations(group)and calculate multiple integral,infinite series,etc.However,since the approximation of MC method with random simulation is inevitably to deviate the estimated true value,following this problem we study in this paper.Firstly,the convergence of the MC method and error estimates have been analyzed in this paper,and six kinds of variance reduction methods which were commonly used have been summarized.Six kinds of variance have been presented,such as important sampling method,the control variable method,antithetic variables method,conditioning expectation method,stratified sampling method,related sampling method.Secondly,in this paper,MC antithetic variables method is discussed with system of equations,definite integral and series.Estimated values based on related sampling method can be achieved by constructing probability and statistics models and sampling for random samples.The result shows that this technique can reduce the simulation precision and shorten the computer running time effectively.Finally,we discuss the adaptive Monte Carlo solution algorithm of linear algebraic system.The adaptive Monte Carlo method includes adaptive importance sampling Monte Carlo and adaptive correlation sampling Monte Carlo,comparing Monte Carlo method anddeterministic method by solving concrete examples,the efficiency of the algorithm is obtained,and the adaptive Monte Carlo method can obtain a faster convergence rate.更多还原