蒙特卡罗计算机模拟应用一例

通过圆周率的计算，说明蒙特卡罗计算机模拟方法的应用过程     

相对论高次谐波的蒙特卡罗模拟

根据相对论条件下的经典电磁辐射理论,采用蒙特卡罗计算方法对类氢离子在强激光场下辐射的高次谐波进行了模拟计算,获得了相对论领域内激光参数（频率、脉冲波形、强度等）对高次谐波影响的各种计算结果,并对结果进行了理论分析。     

相对论高次谐波的蒙特卡罗模拟

 根据相对论条件下的经典电磁辐射理论,采用蒙特卡罗计算方法对类氢离子在强激光场下辐射的高次谐波进行了模拟计算,获得了相对论领域内激光参数（频率、脉冲波形、强度等）对高次谐波影响的各种计算结果,并对结果进行了理论分析。     

X射线能谱测量的蒙特卡罗成像模拟

针对高能强流电子束轰击高Z靶产生的X射线的能谱测量问题,采用蒙特卡罗方法进行成像模拟研究。高能X射线能谱通常由对X射线经过衰减体的直穿透射率曲线进行解谱获得。设计了带多准直孔的截锥体模型,在单次模拟成像中获得完整的衰减透射率曲线,有效避免了散射光子对透射率曲线以及X射线能谱重建的影响。成像面采用非均匀划分网格计数,将大部分探测计数点集中于各准直孔出口处,保证关注区域模拟成像的精细度,同时减少总体计算时间。     

接触热阻的蒙特卡罗法模拟

 用蒙特卡罗随机模拟法计算了固体接触界面间的接触热阻。取服从高斯分布的随机数作为粗糙峰高度，分别对每个粗糙峰计算出单点接触热阻，再并联所有单点接触热阻得到总的接触热阻。修正了“积分法”的理论结果且与发表的文献数据相符。     

反应堆蒙特卡罗临界模拟中均匀裂变源算法的改进

在反应堆pin-by-pin精细建模及蒙特卡罗模拟计算研究中, 由于不同栅元的功率密度差异较大, 导致蒙特卡罗方法临界计算的样本在不同栅元之间的分配不均衡, 由此引起栅元内的各种计数的统计误差差异较大. 为使大部分栅元内计数的统计误差降至一个合理的水平, 单纯增加总样本已不是一个高效的解决方法. 通过在特定临界计算迭代算法的基础上改进并实现均匀裂变源算法的思想, 对大亚湾压水堆pin-by-pin模型取得了具有较高效率的数值结果. 本工作为具有自主知识产权的蒙特卡罗粒子输运模拟软件JMCT最终达到反应堆pin-by-pin模型(包括一系列国际基准模型)的模拟性能要求提供了一个有效的工具.     

直接模拟蒙特卡罗法对连续流体传热和流动的模拟

直接模拟蒙特卡罗法在稀薄气体模拟中获得了成功的应用。然而,在处理连续介质传热和流动问题时,模拟的速度极大地制约了该方法的应用。尤其对于大系统的连续流动,该方法的收敛速度几乎无法实现。鉴于此,本文通过引入超粒子,对直接模拟蒙特卡罗方法进行改进,并将其应用于连续介质传热和流动模拟之中。     

基于蒙特卡罗模拟方法的光源用LED封装光学结构设计

为满足正在兴起的LED照明光源的设计优化要求,必须探索适合LED光学结构设计的有效方法。引进蒙特卡罗(Monte Carlo)随机模拟方法对常规形式发光二极管(LED)的光学封装结构进行模拟,得出了LED的光强分布,并进行实际测量,模拟结果与实验所得结果吻合较好,证明蒙特卡罗方法是进行LED光源光学结构设计的一种有效工具,可以以此作为LED光源的设计优化手段。重点探讨了此方法模拟中的随机数构造、优化;LED模型;仿真的计算机实现,仿真结果的验证;结构优化的思路等问题。     

金属凝固与晶体生长过程的蒙特卡罗模拟

采用导热的有限差分计算与图形蒙特卡罗方法相结合的手段,模拟了不同的冷却条件,熔体初温以及杂质含量情况下的金属凝固及其晶体生长过程。模拟得了与典型的边界冷却凝固实验相一致的微观组织结构,同时,通过对模拟结果的分析,总结了不同的冷却条件、熔体初温以及含杂质情况对金属凝固组织的一些影响规律。     

外力驱动作用下高分子链在表面吸附性质的计算机模拟

采用退火法模拟研究受外力F驱动的高分子链在吸引表面的吸附特性.通过高分子链的平均表面接触数〈M〉与温度T之间的关系计算临界吸附温度T_c,并发现T_c随着F的增加而减小;进而通过高分子链的均方回转半径分析外力驱动作用对高分子链构象的影响,并从回转半径极小值或者垂直外力方向的y和z分量的变化交叉校验临界吸附点T_c.模拟计算了处于吸附状态的高分子链随着外力F的增加是否会发生吸附状态到脱附状态的相变以及发生相变所需施加的外力是否由温度所决定.模拟结果表明:两种不同温度下高分子链的吸附性质和构象性质受外力驱动作用而产生不同现象,在温度区间T*_cTT_c时会发生脱附现象,而在TT*_c时不会发生脱附现象.     

Potential-Decomposition Strategy in Markov Chain Monte Carlo Sampling Algorithms

We introduce the potential-decomposition strategy （PDS）, which can be used in Markov chain Monte Carlo sampling algorithms. PDS can be designed to make particles move in a modified potential that favors diffusion in phase space, then, by rejecting some trial samples, the target distributions can be sampled in an unbiased manner. Furthermore, if the accepted trial samples are insumcient, they can be recycled as initial states to form more unbiased samples. This strategy can greatly improve efficiency when the original potential has multiple metastable states separated by large barriers. We apply PDS to the 2d Ising model and a double-well potential model with a large barrier, demonstrating in these two representative examples that convergence is accelerated by orders of magnitude.     

Coupling control variates for Markov chain Monte Carlo

We show that Markov couplings can be used to improve the accuracy of Markov chain Monte Carlo calculations in some situations where the steady-state probability distribution is not explicitly known. The technique generalizes the notion of control variates from classical Monte Carlo integration. We illustrate it using two models of nonequilibrium transport.     

Potential-Decomposition Strategy in Markov Chain Monte Carlo Sampling Algorithms

We introduce the potential-decomposition strategy (PDS), which can be used in Markov chain Monte Carlo sampling algorithms. PDS can be designed to make particles move in a modified potential that favors diffusion in phase space, then, by rejecting some trial samples, the target distributions can be sampled in an unbiased manner. Furthermore, if the accepted trial samples are insufficient, they can be recycled as initial states to form more unbiasedsamples. This strategy can greatly improve efficiency when the original potential has multiple metastable states separated by large barriers. We apply PDS to the 2d Ising model and a double-well potential model with a large barrier, demonstrating in these two representative examples that convergence is accelerated by orders of magnitude.     

The Convergence of Markov Chain Monte Carlo Methods: From the Metropolis Method to Hamiltonian Monte Carlo

From its inception in the 1950s to the modern frontiers of applied statistics, Markov chain Monte Carlo has been one of the most ubiquitous and successful methods in statistical computing. The development of the method in that time has been fueled by not only increasingly difficult problems but also novel techniques adopted from physics. Here, the history of Markov chain Monte Carlo is reviewed from its inception with the Metropolis method to the contemporary state‐of‐the‐art in Hamiltonian Monte Carlo, focusing on the evolving interplay between the statistical and physical perspectives of the method.     

Measuring preferential attachment in growing networks with missing-timelines using Markov chain Monte Carlo

Preferential attachment is widely recognised as the principal driving force behind the evolution of many growing networks, and measuring the extent to which it occurs during the growth of a network is important for explaining its overall structure. Conventional methods require that the timeline of a growing network is known, that is, the order in which the nodes of the network appeared in time is available. But growing network datasets are commonly accompanied by missing-timelines, in which instance the order of the nodes in time cannot be readily ascertained from the data. To address this shortcoming, we propose a Markov chain Monte Carlo algorithm for measuring preferential attachment in growing networks with missing-timelines. Key to our approach is that any growing network model gives rise to a probability distribution over the space of networks. This enables a growing network model to be fitted to a growing network dataset with missing-timeline, allowing not only for the prevalence of preferential attachment to be estimated as a model parameter, but the timeline also. Parameter estimation is achieved by implementing a novel Metropolis–Hastings sampling scheme for updating both the preferential attachment parameter and timeline. A simulation study demonstrates that our method accurately measures the occurrence of preferential attachment in networks generated according to the underlying model. What is more, our approach is illustrated on a small sub-network of the United States patent citation network. Since the timeline for this example is in fact known, we are able to validate our approach against the conventional methods, showing that they give mutually consistent estimates.     

利用Bayesian-MCMC方法从雷达回波反演海洋波导

应用贝叶斯-蒙特卡罗（Bayesian-MCMC）方法将海洋波导参数的先验信息描述为先验概率密度,结合雷达回波资料（电磁波传播损耗）,得到待反演海洋波导参数的后验概率密度,用马尔可夫链蒙特卡罗（MCMC）-Gibbs采样器采样后验概率密度分布,并用样本最大似然估计值作为对海洋波导参数分布的估计.数值实验结果表明,该方法对先验信息进行了有效利用,反演精度高于遗传算法的反演精度.该方法较为充分利用先验信息,得到解的概率分布,即解的不确定性分析,这在实际应用中有一定的参考价值. 关键词： 波导 电磁波传播损耗 贝叶斯-蒙特卡罗 概率分布     

Random walk on distant mesh points Monte Carlo methods

A new technique for obtaining Monte Carlo algorithms based on the Markov chains with a finite number of states is suggested. Instead of the classical random walk on neighboring mesh points, a general way of constructing Monte Carlo algorithms that could be called random walk on distant mesh points is considered. It is applied to solve boundary value problems. The numerical examples indicate that the new methods are less laborious and therefore more efficient.In conclusion, we mention that all Monte Carlo algorithms are parallel and could be easily realized on parallel computers.     

A Markov-property Monte Carlo method: One-dimensional Ising model

In this paper we introduce a new Monte Carlo procedure based on the Markov property. This procedure is particularly well suited to massively parallel computation. We illustrate the method on the critical phenomena of the well known one-dimensional Ising model. In the course of this work we found that the autocorrelation time for the Metropolis Monte Carlo algorithm is closely given by the square of the correlation length. We find speedup factors of the order of 1 million for the method as implemented on the CM2 relative to a serial machine. Our procedure gives error estimates which are quite consistent with the observed deviations from the analytically known exact results.     

平行束缪子透射成像蒙特卡罗模拟

使用蒙特卡罗程序Geant4,模拟平行束缪子垂直于理想探测器平面入射法国试验客体（FTO）模型,在模型的上方和下方各放置三块理想探测器,用以输出缪子位置信息,从而确定入射与出射缪子径迹。通过三种方法统计缪子穿过模型前后的透射比,对模型进行透射成像,得到不同的成像结果。统计方法一和方法二分辨力可达2 mm2 mm,统计方法三可达1 mm1 mm,Cu与W区分较为明显,而且可显示出FTO模型中心的空气球,FTO模型与模型周围空气的边界十分清晰。模拟结果表明,平行束入射缪子可以进行透射成像。     

A new Monte Carlo power method for the eigenvalue problem of transfer matrices

We propose a new Monte Carlo method for calculating eigenvalues of transfer matrices leading to free energies and to correlation lengths of classical and quantum many-body systems. Generally, this method can be applied to the calculation of the maximum eigenvalue of a nonnegative matrix  such that all the matrix elements of Â^k are strictly positive for an integerk. This method is based on a new representation of the maximum eigenvalue of the matrix  as the thermal average of a certain observable of a many-body system. Therefore one can easily calculate the maximum eigenvalue of a transfer matrix leading to the free energy in the standard Monte Carlo simulations, such as the Metropolis algorithm. As test cases, we calculate the free energies of the square-lattice Ising model and of the spin-1/2XY Heisenberg chain. We also prove two useful theorems on the ergodicity in quantum Monte Carlo algorithms, or more generally, on the ergodicity of Monte Carlo algorithms using our new representation of the maximum eigenvalue of the matrixÂ.