Error Estimation in the Histogram Monte Carlo Method

We examine the sources of error in the histogram reweighting method for Monte Carlo data analysis. We demonstrate that, in addition to the standard statistical error which has been studied elsewhere, there are two other sources of error, one arising through correlations in the reweighted samples, and one arising from the finite range of energies sampled by a simulation of finite length. We demonstrate that while the former correction is usually negligible by comparison with statistical fluctuations, the latter may not be, and give criteria for judging the range of validity of histogram extrapolations based on the size of this latter correction.     

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.     

Histogram Monte Carlo position-space renormalization group: Applications to the site percolation

We study site percolation on the square lattice and show that, when augmented with histogram Monte Carlo simulations for large lattices, the cell-to-cell renormalization group approach can be used to determine the critical probability accurately. Unlike the cell-to-site method and an alternate renormalization group approach proposed recently by Sahimi and Rassamdana, both of which rely onab initio numerical inputs, the cell-to-cell scheme is free of prior knowledge and thus can be applied more widely.     

A boundary-based net-exchange Monte Carlo method for absorbing and scattering thick media

A boundary-based net-exchange Monte Carlo method was introduced (JQSRT 74 (2001) 563) that allows to bypass the difficulties encountered by standard Monte Carlo algorithms in the limit of optically thick absorption (and/or for quasi-isothermal configurations). With the present paper, this method is extended to scattering media. Developments are fully 3D, but illustrations are presented for plane parallel configuration. Compared to standard Monte Carlo algorithms, convergence qualities have been improved over a wide range of absorption and scattering optical thicknesses. The proposed algorithm still encounters a convergence difficulty in the case of optically thick, highly scattering media.     

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.     

Monte Carlo方法在B样条函数分频理论中的应用

针对弹道参数的B样条表示问题,运用Monte Carlo方法对B样条函数分频理论进行仿真论证,得到B样条延拓节点的补充方法,并验证B样条函数逼近空间相容性定理的正确性.比较Monte Carlo方法和B样条分频算法的计算结果,说明分频算法合理.将该算法应用于外弹道跟踪测量数据的处理中,可以确定出表示弹道所需要的样条节点个数及样条节点序列的排布,有效减少待估参数个数.     

采用加权直方图分析和蒙特卡洛重采样法对自由能级的校正研究

本文采用加权直方图分析方法和蒙特卡洛重采样方法进行自由基校正研究. 生成的自由能表面几乎可以收敛到具有足够采样情况的精准表面,并且比常规的加权直方图分析校正方法能更稳定地处理采样不足的情况,为检测能级校正表面的不确定性提供指南,并且定义了明确的标准用以确定能改善其视觉效果的自由能表面平滑程度. 本文通过水中的丙氨酸二肽和KillerRed蛋白质子转移的自由能图证明该方法的优势,说明蒙特卡洛重采样法可以作为在产生自由能表面实际的系统中的实用工具.     

Monte Carlo法模拟油滴的理论分布

在密立根油滴实验中,应用Monte Carlo法数值地模拟实验室条件下可能的油滴带电量Q、平衡电压U和匀速运动时间t的理论分布,分别从三维分布、时间分布及数量方面比较3种选择油滴的标准,给出较合理的结论.     

Monte Carlo method and sensitivity estimations

It is shown that, starting from any existing Monte Carlo algorithm for estimation of a physical quantity A, it is possible to implement a simple additional procedure that simultaneously estimates the sensitivity of A to any problem parameter. The corresponding supplementary cost is very low as no additional random sampling is required. The principle is presented on a formal basis and simple radiative transfer examples are used for illustration.     

Exact Monte Carlo for few-fermion systems

We have reconsidered the fundamental difficulties of fermion Monte Carlo as applied to few-body systems. We conclude that necessary ingredients of successful algorithms include the following: There must be equal populations of random walkers that carry positive and negative weights. The positions of positive walkers should be selected from a distribution that uses Green's functions to couple all walkers. The positions of negative walkers should be generated from those of positive walkers by means of odd permutations. The correct importance functions that take into account the global interactions of the populations are different for positive and negative walkers. Use of such importance functions breaks the symmetry that otherwise would exist between configurations (of the entire population) and configurations derived by interchanging positive and negative walkers. Based upon these observations, we have constructed a stable and accurate algorithm that solves a fully-polarized, three-dimensional, three-body model problem.     

求解中子动力学方程的加权蒙特卡罗方法

为了实现基于蒙特卡罗方法的中子动力学计算,在传统的直接蒙特卡罗动力学方法的基础上,提出了一种加权蒙特卡罗动力学方法。该方法通过引入粒子权重的概念,隐式考虑中子俘获反应和裂变反应过程中中子数目的变化,避免了模拟粒子的数目随时间的变化,降低了统计偏差,消除了程序计算过程中粒子的存库操作,提高了计算精度。基于单能点堆模型,开发了中子动力学计算程序NECP-Dandi,进行了大量数值验证与分析,包括无缓发中子、单组缓发中子、六组缓发中子、正阶跃反应性引入、负阶跃反应性引入、正脉冲反应性、负脉冲反应性和正线性反应性引入等情况。数值结果表明,相比于直接蒙特卡罗动力学方法,加权蒙特卡罗动力学方法在计算结果的精度和计算效率上有较为明显的改进,程序结构更为简洁。     

蒙特卡洛法模拟G型辐冷器抛物面屏辐射的新算法

提出了筛选法模拟辐射能束随机发射位置的新算法 ,解决了模拟 G型辐冷器抛物面屏这种立体曲面辐射能束随机发射位置的难题 ,并以切平面为发射基础 ,推导出辐射能束随机发射方向的求解算法。通过上述算法 ,模拟了辐射过程 ,计算了相应的耦合因子 ,并对结果进行了误差分析。计算结果表明 ,基于新算法的蒙特卡洛法准确模拟了 G型辐冷器辐射过程 ,计算精度高、速度快     

Flat Histogram Monte Carlo Simulations of Triangulated Fixed-Connectivity Surface Models

Using the Wang-Landau flat histogram Monte Carlo (FHMC) simulation technique, we were able to study two types of triangulated spherical surface models in which the two-dimensional extrinsic curvature energy is assumed in the Hamiltonian. The Gaussian bond potential is also included in the Hamiltonian of the first model, but it is replaced by a hard-wall potential in the second model. The results presented in this paper are in good agreement with the results previously reported by our group. The transition of surface fluctuations and collapsing transition were studied using the canonical Metropolis Monte Carlo simulation technique and were found to be of the first-order. The results obtained in this paper also show that the FHMC technique can be successfully applied to triangulated surface models. It is non-trivial whether the technique is applicable or not to surface models because the simulations are performed on relatively large surfaces.     

纳米晶粒生长的动力学蒙特卡罗模型

给出应用于纳米晶粒生长的动力学蒙特卡罗模型,并对模拟方法做了细致的讨论.作为一个应用实例,对薄膜生长做了分类模拟研究.结果显示,改变三维Ehrlich-Schwoebel势垒将导致薄膜生长过程中多层原子岛的结构相变.     

Monte Carlo simulation using the Fourier transform of the interatomic potential

When doing Monte Carlo simulations using continuous potentials, the evaluation of the configurational potential energy ink-space by Fourier transformation is shown to be a computationally attractive scheme for systems where the long-range interatomic interaction spans a dimension comparable to the size of the simulated system.     

Note on Monte Carlo methods without necessary detailed balance

Simple criteria for convergence of Monte Carlo algorithms not necessarily requiring detailed balance for any specified transition probability are derived and it is shown that it is possible to view the algorithm as a superimposition of a Brownian motion on configurational space coupled to the transition probabilities. As such, the error contributions due to a particular Monte Carlo algorithm and the integration limits in configuration space must be distinguished from those due to the nonuniform sampling of the Brownian motion, and criteria related to the number of steps required to distinguish these errors are provided for the simplest cases involving one dimension and symmetrical probability distributions.     

A generalized optimization of Monte Carlo particle transport schemes

Formulations and algorithms for optimizing biased random walk problems in radiation transport are described. A matrix-integral equation is constructed by coupling the second moment and its derivative with respect to the biasing parameter. The optimization is based on estimation of the second moment around the score and the Monte Carlo perturbation algorithm to treat the variation in the biasing parameter.     

Auxiliary-field Monte Carlo methods

I discuss Monte Carlo algorithms for quantum many-body systems that employ an auxiliary field to linearize a two-body interaction. These reduce the evaluation of the partition function to sampling many one-body evolutions in a fluctuating field. Fermions and bosons are treated on an equal footing. Applications to potential models and to quantum spin systems are discussed. This work was supported in part by the National Science Foundation, grants PHY82-07332 and PHY85-05682. The potential-model studies were done in collaboration with G. Sugiyama, while A. Khan and T. Troudet were responsible for the work on the quantum spin systems.     

Monte Carlo Shell Model Mass Predictions

The nuclear mass calculation is discussed in terms of large-scale shell model calculations. First, the development and limitations of the conventional shell model calculations are mentioned. In order to overcome the limitations, the Quantum Monte Carlo Diagonalization (QMCD) method has been proposed. The basic formulation and features of the QMCD method are presented as well as its application to the nuclear shell model, referred to as Monte Carlo Shell Model (MCSM). The MCSM provides us with a breakthrough in shell model calculations: the structure of low-lying states can be studied with realistic interactions for a nearly unlimited variety of nuclei. Thus, the MCSM can contribute significantly to the study of nuclear masses. An application to N∼20 unstable nuclei far from the β-stability line is mentioned. This revised version was published online in July 2006 with corrections to the Cover Date.