Quasi-Monte Carlo methods for lattice systems: A first look
Quasi-Monte Carlo methods for lattice systems: A first look
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格子系统的拟蒙特卡罗方法：初步了解
DOI：
10.1016/j.cpc.2013.10.011
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发表时间：
2014
期刊：
影响因子：
--
通讯作者：
M. Müller-Preussker
中科院分区：
文献类型：
--
作者：
K. Jansen;H. Leövey;A. Ammon;A. Griewank;M. Müller-Preussker
我们研究了拟蒙特卡罗方法对量子力学欧几里德格子系统的适用性，以改善此类理论的可观测量的渐近误差行为。在大多数情况下，通过对由普通马尔科夫链蒙特卡罗模拟产生的随机观测值进行平均来计算可观测值的误差表现为N−1/2，其中N是观测值的数量。借助于拟蒙特卡罗方法，可以将某些问题的这种行为改进到N−1，甚至在问题足够规则的情况下进一步改进。我们将这种方法应用于量子谐振子和非谐振子等简单系统，并验证了改进的误差比例。节目摘要节目标题：QAR-0.1目录标识：AERJ_v1_0节目摘要网址：http://cpc.CS.QUB。交流电。UK/摘要/AERJ_v1_0。超文本标记语言程序可从：CPC程序图书馆，女王大学，贝尔法斯特，北爱尔兰许可条款：GNU通用公共许可证版本3，编号。分布式程序中的行数，包括测试数据等：67759第分布式程序的字节数，包括测试数据等：2165365分布格式：TAR。GZ编程语言：C和C++。电脑：个人电脑。操作系统：在GNU/Linux上测试，应该可以很轻松地移植到其他操作系统上。代码是否被矢量化或并行化了？注意：没有RAM：内存使用量直接随样本和维度的数量而变化：使用的字节数=“样本数”ד维度数”×8字节(双精度)。分类：4.13、11.5、23。外部例程：FFTW3库(http://www.FFTW。问题的性质：某些通过费曼路径积分被表述为量子场论的物理模型，例如量子色动力学，需要对路径积分进行非微扰处理。唯一已知的实现这一点的方法是格子正则化。在该公式中，路径积分被离散化为有限但非常高维的积分。到目前为止，只有蒙特卡罗，特别是马尔可夫链-蒙特卡罗方法，如Metropolis或混合蒙特卡罗算法被用来计算路径积分的近似解。这些算法经常在可观测样本中导致不希望看到的自相关效应，并且在任何情况下都会受到与N−1/2成正比的慢渐近误差行为的影响，如果N是样本数。解的方法：本程序应用准蒙特卡罗方法和重加权技术(分别是加权均匀采样法)来产生非简谐振子的不相关样本，其渐近误差行为得到了改善。不同寻常的特点：准蒙特卡罗方法的应用在格子场理论领域是相当革命性的。运行时间：运行时间直接取决于样本数量N和维度d。在现代计算机上，运行N=2 16=65536(包括9个副本运行)和d=100的运行时间不会超过1分钟。
We investigate the applicability of quasi-Monte Carlo methods to Euclidean lattice systems for quantum mechanics in order to improve the asymptotic error behavior of observables for such theories. In most cases the error of an observable calculated by averaging over random observations generated from an ordinary Markov chain Monte Carlo simulation behaves like N− 1/2, where N is the number of observations. By means of quasi-Monte Carlo methods it is possible to improve this behavior for certain problems to N− 1, or even further if the problems are regular enough. We adapted and applied this approach to simple systems like the quantum harmonic and anharmonic oscillator and verified an improved error scaling. Program summary Program title: qar-0.1 Catalogue identifier: AERJ_v1_0 Program summary URL: http://cpc. cs. qub. ac. uk/summaries/AERJ_v1_0. html Program obtainable from: CPC Program Library, Queen’s University, Belfast, N. Ireland Licensing provisions: GNU General Public Licence version 3 No. of lines in distributed program, including test data, etc.: 67759 No. of bytes in distributed program, including test data, etc.: 2165365 Distribution format: tar. gz Programming language: C and C++. Computer: PC. Operating system: Tested on GNU/Linux, should be portable to other operating systems with minimal efforts. Has the code been vectorized or parallelized?: No RAM: The memory usage directly scales with the number of samples and dimensions: Bytes used=“number of samples”דnumber of dimensions”× 8 Bytes (double precision). Classification: 4.13, 11.5, 23. External routines: FFTW 3 library (http://www. fftw. org) Nature of problem: Certain physical models formulated as a quantum field theory through the Feynman path integral, such as quantum chromodynamics, require a non-perturbative treatment of the path integral. The only known approach that achieves this is the lattice regularization. In this formulation the path integral is discretized to a finite, but very high dimensional integral. So far only Monte Carlo, and especially Markov chain-Monte Carlo methods like the Metropolis or the hybrid Monte Carlo algorithm have been used to calculate approximate solutions of the path integral. These algorithms often lead to the undesired effect of autocorrelation in the samples of observables and suffer in any case from the slow asymptotic error behavior proportional to N− 1/2, if N is the number of samples. Solution method: This program applies the quasi-Monte Carlo approach and the reweighting technique (respectively the weighted uniform sampling method) to generate uncorrelated samples of observables of the anharmonic oscillator with an improved asymptotic error behavior. Unusual features: The application of the quasi-Monte Carlo approach is quite revolutionary in the field of lattice field theories. Running time: The running time depends directly on the number of samples N and dimensions d. On modern computers a run with up to N= 2 16= 65536 (including 9 replica runs) and d= 100 should not take much longer than one minute.
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DOI：
10.1088/1742-6596/454/1/012043
发表时间：
2013
期刊：
Journal of Physics: Conference Series
影响因子：
--
作者：
K. Jansen;H. Leövey;A. Nube;A. Griewank;M. Müller-Preussker
通讯作者：
M. Müller-Preussker