浅谈蒙特卡罗法在概率统计教学中的应用

在概率统计教学中利用蒙特卡罗法设计适当的模拟实验,结合多媒体在课堂教学中实时演示,能很好地加深学生对概率统计概念、思想的理解,进而培养学生的统计思维能力.     

统计视角下的结构突变检验设计及其应用

在结构突变检验中,针对参数统计检验需要依赖于严格假定的局限,本文创新性地提出了一套切实可行的非参数统计检验方法和检验程序。蒙特卡罗模拟表明,这套非参数统计组合设计具有较高的检验效力;结构突变前后样本量大小并不影响对截距与斜率双突变以及截距不变、斜率突变进行检验的检验效力,而对截距突变、斜率不变以及无结构突变检验影响较大。     

Weighted Monte Carlo Methods for Approximate Solution of a Nonlinear Boltzmann Equation

We construct weighted modifications of statistical modeling of an ensemble of interacting particles which is connected with approximate solution of a nonlinear Boltzmann equation.     

吸波材料与微波暗室问题的数学模型

为分析尖劈形材料的吸波性能,建立了半圆镜像模型将辐射波在尖劈空间的传播等效为射线对等分半圆的切割,确定了反射波线和吸波材料几何参数之间的解析关系.对于微波暗室问题,假设在余弦辐射体上随机选取一个方向的能量波束集中了其所有能量,分别建立了加权能量分布模型和Monte Carlo模型,仿真验证了两模型结论的一致性.     

Corrections to the Article “Weighted Monte Carlo Methods for Approximate Solution of a Nonlinear Boltzmann Equation”

Corrections are given to the above-mentioned article.     

Multi-Level Monte Carlo Path Integral Molecular Dynamics for Thermal Average Calculation in the Nonadiabatic Regime

With the path integral approach, the thermal average in multi-electronic-state quantum systems can be approximated by the ring polymer representation on an extended configuration space, where the additional degrees of freedom are associated with the surface index of each bead. The primary goal of this work is to propose a more efficient sampling algorithm for the calculation of such thermal averages. We reformulate the extended ring polymer approximation according to the configurations of the surface indexes, and by introducing a proper reference measure, the reformulation is recast as a ratio of two expectations of function expansions. By quantitatively estimating the sub-estimators, and minimizing the total variance of the sampled average, we propose a multi-level Monte Carlo path integral molecular dynamics method (MLMC-PIMD) to achieve an optimal balance of computational cost and accuracy.     

Antithetic Methods for Gibbs Samplers

In this article we propose a modification to the output from Metropolis-within-Gibbs samplers that can lead to substantial reductions in the variance over standard estimates. The idea is simple: at each time step of the algorithm, introduce an extra sample into the estimate that is negatively correlated with the current sample, the rationale being that this provides a two-sample numerical approximation to a Rao–Blackwellized estimate. As the conditional sampling distribution at each step has already been constructed, the generation of the antithetic sample often requires negligible computational effort. Our method is implementable whenever one subvector of the state can be sampled from its full conditional and the corresponding distribution function may be inverted, or the full conditional has a symmetric density. We demonstrate our approach in the context of logistic regression and hierarchical Poisson models. The data and computer code used in this article are available online.     

两部雷达最佳部署问题研究

本文研究两部雷达联合发现目标概率,采用Monte Carlo法解决了最佳部署问题。     

基于蒙特卡罗方法的中国中长期煤炭需求预测

为避免传统预测方法的参数取值主观性问题,采用参数随机产生的蒙特卡罗方法预测中国中长期煤炭需求.首先分析了经济增长、能源结构和产业结构三个主要煤炭需求影响因素,并基于1980~2015年间各影响因素及煤炭消费的历史数据和最小二乘法的多元线性回归拟合煤炭需求方程.在此基础上,构建各影响因素的概率分布,采用蒙特卡罗方法模拟1981~2015年的煤炭需求,发现仿真结果可以较好拟合现实,可作为仿真预测的有效工具.结合经济新常态和能源结构调整的现状,控制参数取值范围进行蒙特卡罗仿真预测,结果显示,2016~2025年的煤炭需求呈先上升后下降趋势,并于2020年达到需求的峰值40.25亿吨,这些结果对于煤炭产业的科学决策有重要的作用.     

A TWO-STAGE ALGORITHM OF NUMERICAL EVALUATION OF INTEGRALS IN NUMBER-THEORETIC METHODS

1.IntroductionTheNumber-TheoreticMethod(NTM)isaspecialmethodwhichrepresentsacombinationofnumbertheoryandnumericalanalysis.Thewidestrangeofapplicationsandindeedthehistoricaloriginofthismethodisfoundinnumericalintegration.Alsorelatedproblemssuchajsinte...     

Multilevel sequential Monte Carlo: Mean square error bounds under verifiable conditions

In this article, we consider the multilevel sequential Monte Carlo (MLSMC) method of Beskos et al. (Stoch. Proc. Appl. [to appear]). This is a technique designed to approximate expectations w.r.t. probability laws associated to a discretization. For instance, in the context of inverse problems, where one discretizes the solution of a partial differential equation. The MLSMC approach is especially useful when independent, coupled sampling is not possible. Beskos et al. show that for MLSMC the computational effort to achieve a given error, can be less than independent sampling. In this article we significantly weaken the assumptions of Beskos et al., extending the proofs to non-compact state-spaces. The assumptions are based upon multiplicative drift conditions as in Kontoyiannis and Meyn (Electron. J. Probab. 10 [2005]: 61–123). The assumptions are verified for an example.     

定积分计算方法及其数值试验

以定积分概念所蕴含的数学思想和哲学思想为指导,给出定积分近似计算的两种方法,即定义法和蒙特卡罗方法,并借助Matlab软件编程对其加以验证,数值试验结果显示所给方法可行且有效．     

Statistical mechanics and ocean circulation

We apply equilibrium statistical mechanics based upon the conservation of energy and potential enstrophy to the mass-density distribution within the ocean, using a Monte Carlo method that conserves the buoyancy of each fluid particle. The equilibrium state resembles the buoyancy structure actually observed.     

Weighted Monte Carlo method for an approximate solution of the nonlinear coagulation equation

New weighted modifications of direct statistical simulation methods designed for the approximate solution of the nonlinear Smoluchowski equation are developed on the basis of stratification of the interaction distribution in a multiparticle system according to the index of a pair of interacting particles. The weighted algorithms are validated for a model problem with a known solution. It is shown that they effectively estimate variations in the functionals with varying parameters, in particular, with the initial number N ₀ of particles in the simulating ensemble. The computations performed for the problem with a known solution confirm the semiheuristic hypothesis that the model error is O(N ₀ ^?1 ). Estimates are derived for the derivatives of the approximate solution with respect to the coagulation coefficient.     

Dominant eigenvalue problem for positive integral operators and its solution by Monte Carlo method

In this paper, a method of numerical solution to the dominant eigenvalue problem for positive integral operators is presented. This method is based on results of the theory of positive operators developed by Krein and Rutman. The problem is solved by Monte Carlo method constructing random variables in such a way that differences between results obtained and the exact ones would be arbitrarily small. Some numerical results are shown.     

Consensus formation of two-level opinion dynamics

Opinion dynamics have received significant attention in recent years. This paper proposes a bounded confidence opinion model for a group of agents with two different confidence levels. Each agent in the population is endowed with a confidence interval around her opinion with radius αd or (1-α)d, where α ∈ (0,1/2] represents the differentiation of confidence levels. We analytically derived the critical confidence bound d_c = 1/(4α) for the two-level opinion dynamics on ?. A single opinion cluster is formed with probability 1 above this critical value regardless of the ratio p of agents with high/low confidence. Extensive numerical simulations are performed to illustrate our theoretical results. Noticed is a clear impact of p on the collective behavior: more agents with high confidence lead to harder agreement. It is also experimentally revealed that the sharpness of the threshold d_c increases with α but does not depend on p.