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蒙特卡罗模拟 (查看源代码)

2021年8月1日 (日) 14:59的版本

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无编辑摘要

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此词条暂由彩云小译翻译，翻译字数共4901，未经人工整理和审校，带来阅读不便，请见谅。

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|keywords=~~玻尔兹曼方程、玻尔兹曼常数、H定理~~

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|keywords=蒙特卡罗方法、蒙特卡洛实验、蒙特卡洛模拟

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|description=~~奥地利物理学家、哲学家~~

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|description=重复的随机抽样来获得数值结果，利用随机性来解决理论上可能是确定性的问题，最优化，数值积分，依据概率分布生成图像

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尽管其概念和算法简单，但与蒙特卡罗模拟相关的计算成本却是高的惊人。一般情况下，该方法需要大量的样本来获得良好的近似，如果单个样本的处理时间较长，可能会导致总运行时间长度难以控制。<ref>Shonkwiler, R. W.; Mendivil, F. (2009). ''Explorations in Monte Carlo Methods''. Springer.</ref>尽管在非常复杂的问题中，这是一个严重的限制，但该算法令人尴尬的并行性质允许通过本地处理器、集群、云计算、GPU、FPGA等的并行计算策略来降低高昂的成本(或许降低到可以接受的水平上)。<ref>Atanassova, E.; Gurov, T.; Karaivanova, A.; Ivanovska, S.; Durchova, M.; Dimitrov, D. (2016). "On the parallelization approaches for Intel MIC architecture". ''AIP Conference Proceedings''. '''1773''': 070001. doi:10.1063/1.4964983.</ref><ref>Cunha Jr, A.; Nasser, R.; Sampaio, R.; Lopes, H.; Breitman, K. (2014). "Uncertainty quantification through the Monte Carlo method in a cloud computing setting". ''Computer Physics Communications''. '''185'''(5): 1355–1363. doi:10.1016/j.cpc.2014.01.006.</ref><ref>Wei, J.; Kruis, F.E. (2013). "A GPU-based parallelized Monte-Carlo method for particle coagulation using an acceptance–rejection strategy". ''Chemical Engineering Science''. '''104''': 451–459. doi:10.1016/j.ces.2013.08.008.</ref><ref>Lin, Y.; Wang, F.; Liu, B. (2018). "Random number generators for large-scale parallel Monte Carlo simulations on FPGA". ''Journal of Computational Physics''. '''360''': 93–103. doi:10.1016/j.jcp.2018.01.029.</ref>

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== Overview 概述 ==

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==Overview 概述==

Monte Carlo methods vary, but tend to follow a particular pattern:

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蒙特卡罗方法各不相同，但趋于遵循一个特定的模式:

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# Define a domain of possible inputs

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#Define a domain of possible inputs

Define a domain of possible inputs

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1、定义可能输入的域

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# Generate inputs randomly from a [[probability distribution]] over the domain

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#Generate inputs randomly from a [[probability distribution]] over the domain

Generate inputs randomly from a probability distribution over the domain

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2、从域上的概率分布随机生成输入

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# Perform a [[Deterministic algorithm|deterministic]] computation on the inputs

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#Perform a [[Deterministic algorithm|deterministic]] computation on the inputs

Perform a deterministic computation on the inputs

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3、对输入进行确定性计算

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# Aggregate the results

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#Aggregate the results

Aggregate the results

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例如，考虑一个单位正方形内嵌的四分之一圆。考虑到它们的面积比是π/4，π的值可以用蒙特卡罗方法来近似：<ref name=":9" />

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# Draw a square, then [[inscribed figure|inscribe]] a quadrant within it

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#Draw a square, then [[inscribed figure|inscribe]] a quadrant within it

Draw a square, then inscribe a quadrant within it

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1、画一个正方形，然后在其中划出一个四分之一圆

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# [[uniform distribution (continuous)|Uniformly]] scatter a given number of points over the square

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#[[uniform distribution (continuous)|Uniformly]] scatter a given number of points over the square

Uniformly scatter a given number of points over the square

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2、在正方形上均匀散布给定数量的点

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# Count the number of points inside the quadrant, i.e. having a distance from the origin of less than 1

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#Count the number of points inside the quadrant, i.e. having a distance from the origin of less than 1

Count the number of points inside the quadrant, i.e. having a distance from the origin of less than 1

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3、计算四分之一圆内的点数，即满足距离原点小于1的

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# The ratio of the inside-count and the total-sample-count is an estimate of the ratio of the two areas, π/4. Multiply the result by 4 to estimate π.

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#The ratio of the inside-count and the total-sample-count is an estimate of the ratio of the two areas, π/4. Multiply the result by 4 to estimate π.

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The ratio of the inside-count and the total-sample-count is an estimate of the ratio of the two areas, |4}}. Multiply the result by 4 to estimate .

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<nowiki>The ratio of the inside-count and the total-sample-count is an estimate of the ratio of the two areas, |4}}. Multiply the result by 4 to estimate .</nowiki>

4、四分之一圆内部计数与总样本计数之比是两个区域之比的估计值，π/4。把结果乘以4就可以估算出π的值。

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有两个重要的考虑因素:

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# If the points are not uniformly distributed, then the approximation will be poor.

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#If the points are not uniformly distributed, then the approximation will be poor.

If the points are not uniformly distributed, then the approximation will be poor.

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1、如果这些点不是均匀分布的，那么近似效果就会很差。

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# There are many points. The approximation is generally poor if only a few points are randomly placed in the whole square. On average, the approximation improves as more points are placed.

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#There are many points. The approximation is generally poor if only a few points are randomly placed in the whole square. On average, the approximation improves as more points are placed.

There are many points. The approximation is generally poor if only a few points are randomly placed in the whole square. On average, the approximation improves as more points are placed.

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应用蒙特卡罗方法需要大量的随机数，这也就刺激了'''伪随机数生成器 Pseudorandom Number Generators'''的发展，伪随机数生成器比以前用于统计抽样的随机数表要快得多。

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== History 历史 ==

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==History 历史==

Before the Monte Carlo method was developed, simulations tested a previously understood deterministic problem, and statistical sampling was used to estimate uncertainties in the simulations. Monte Carlo simulations invert this approach, solving deterministic problems using probabilistic metaheuristics (see simulated annealing).

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在高级信号处理和'''贝叶斯推断 Bayesian Inference'''中使用'''序列蒙特卡罗方法 Sequential Monte Carlo'''是最近才出现的。1993年，高登等人在他们的开创性工作中发表了蒙特卡罗重采样算法在贝叶斯推论统计学中的首次应用。<ref name=":12" /> 作者将他们的算法命名为“自举过滤器” ，并证明了与其他过滤方法相比，他们的自举过滤算法不需要任何关于系统状态空间或噪声的假设。此外北川源四郎也进行了”蒙特卡洛过滤器”相关的开创性研究。<ref name=":13" /> 在1990年代中期，'''皮埃尔·德尔·莫勒尔 Pierre Del Moral <ref name="dm9622" />''' 和'''希米尔康 · 卡瓦略 Himilcon Carvalho'''以及皮埃尔 · 德尔 · 莫勒尔、'''安德烈 · 莫宁 André Monin'''和'''杰拉德 · 萨鲁特 Gérard Salut <ref name=":14" />''' 发表了关于粒子过滤器的文章。1989-1992年间，在LAAS-CNRS (系统分析和体系结构实验室)，皮埃尔·德尔·莫勒尔、'''J·C·诺亚 J. C. Noyer'''、'''G·里加尔 G. Rigal''' 和'''杰拉德 · 萨鲁特'''开发了粒子滤波器用于信号处理。他们与STCAN（海军建造和武装服务技术部）、IT公司DIGILOG共同完成了一系列关于雷达/声纳和GPS信号处理问题的限制性和机密性研究报告。<ref name=":15" /><ref name=":16" /><ref name=":17" /><ref name=":18" /><ref name=":19" /><ref name=":20" /> 这些序列蒙特卡罗方法可以解释为一个接受拒绝采样器配备了相互作用的回收机制。

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From 1950 to 1996, all the publications on Sequential Monte Carlo methodologies, including the pruning and resample Monte Carlo methods introduced in computational physics and molecular chemistry, present natural and heuristic-like algorithms applied to different situations without a single proof of their consistency, nor a discussion on the bias of the estimates and on genealogical and ancestral tree based algorithms. The mathematical foundations and the first rigorous analysis of these particle algorithms were written by Pierre Del Moral in 1996.<ref name="dm9622"/><ref name=":22">{{cite journal|last1 = Del Moral|first1 = Pierre|title = Measure Valued Processes and Interacting Particle Systems. Application to Non Linear Filtering Problems|journal = Annals of Applied Probability|date = 1998|edition = Publications du Laboratoire de Statistique et Probabilités, 96-15 (1996)|volume = 8|issue = 2|pages = 438–495|url = http://projecteuclid.org/download/pdf_1/euclid.aoap/1028903535|doi = 10.1214/aoap/1028903535|citeseerx = 10.1.1.55.5257}}</ref>

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From 1950 to 1996, all the publications on Sequential Monte Carlo methodologies, including the pruning and resample Monte Carlo methods introduced in computational physics and molecular chemistry, present natural and heuristic-like algorithms applied to different situations without a single proof of their consistency, nor a discussion on the bias of the estimates and on genealogical and ancestral tree based algorithms. The mathematical foundations and the first rigorous analysis of these particle algorithms were written by Pierre Del Moral in 1996.<ref name="dm9622" /><ref name=":22">{{cite journal|last1 = Del Moral|first1 = Pierre|title = Measure Valued Processes and Interacting Particle Systems. Application to Non Linear Filtering Problems|journal = Annals of Applied Probability|date = 1998|edition = Publications du Laboratoire de Statistique et Probabilités, 96-15 (1996)|volume = 8|issue = 2|pages = 438–495|url = http://projecteuclid.org/download/pdf_1/euclid.aoap/1028903535|doi = 10.1214/aoap/1028903535|citeseerx = 10.1.1.55.5257}}</ref>

从1950年到1996年，所有关于顺序蒙特卡罗方法的出版物，包括计算物理和分子化学中引入的删减和重采样蒙特卡罗方法，目前应用于不同的情况的自然和类启发式算法，没有任何一致性证明，也没有讨论估计的偏差和基于谱系和遗传树的算法。皮埃尔 · 德尔 · 莫勒尔在1996年的写作中阐述了关于这些粒子算法的数学基础，并对其第一次进行了严格的分析。<ref name="dm9622" /><ref name=":22" />

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*Monte Carlo method: Pouring out a box of coins on a table, and then computing the ratio of coins that land heads versus tails is a Monte Carlo method of determining the behavior of repeated coin tosses, but it is not a simulation.

*蒙特卡洛方法：将一盒硬币倒在桌子上，然后计算正面与反面落地的硬币比例。这是一种确定重复掷硬币行为的蒙特卡洛方法，但它不是模拟。

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*Monte Carlo simulation: Drawing <nowiki>''</nowiki>a large number<nowiki>''</nowiki> of pseudo-random uniform variables from the interval [0,1] at one time, or once at many different times, and assigning values less than or equal to 0.50 as heads and greater than 0.50 as tails, is a <nowiki>''</nowiki>Monte Carlo simulation<nowiki>''</nowiki> of the behavior of repeatedly tossing a coin.

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* Monte Carlo simulation: Drawing <nowiki>''</nowiki>a large number<nowiki>''</nowiki> of pseudo-random uniform variables from the interval [0,1] at one time, or once at many different times, and assigning values less than or equal to 0.50 as heads and greater than 0.50 as tails, is a <nowiki>''</nowiki>Monte Carlo simulation<nowiki>''</nowiki> of the behavior of repeatedly tossing a coin.

*蒙特卡罗模拟法：一次或多次从区间[0,1]中绘制“大量”伪随机均匀变量，赋值小于或等于0.50作为正面，大于0.50为反面。这是一个多次掷硬币的“蒙特卡罗模拟”行为。

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为了评估随机数质量对蒙特卡罗模拟结果的影响，天体物理学研究人员测试了通过英特尔的RDRAND指令集生成的加密安全伪随机数，并将其与'''梅森旋转算法 Mersenne Twister'''生成的伪随机数进行了比较，在蒙特卡洛模拟褐矮星射电耀斑的过程中。RDRAND是最接近真实随机数生成器的伪随机数生成器。对于生成107个随机数的试验，典型伪随机数生成器生成的模型与RDRAND之间没有统计差异。<ref name=":24" />

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=== Monte Carlo simulation versus "what if" scenarios 蒙特卡罗模拟与“假设”情景 ===

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===Monte Carlo simulation versus "what if" scenarios 蒙特卡罗模拟与“假设”情景===

There are ways of using probabilities that are definitely not Monte Carlo simulations – for example, deterministic modeling using single-point estimates. Each uncertain variable within a model is assigned a "best guess" estimate. Scenarios (such as best, worst, or most likely case) for each input variable are chosen and the results recorded.

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蒙特卡罗方法被广泛应用于工程设计中的敏感度分析和工艺设计中的定量概率分析。这种需求来源于典型过程模拟的交互性、共线性和非线性行为。比如说,

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* In [[microelectronics|microelectronics engineering]], Monte Carlo methods are applied to analyze correlated and uncorrelated variations in [[Analog signal|analog]] and [[Digital data|digital]] [[integrated circuits]].

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*In [[microelectronics|microelectronics engineering]], Monte Carlo methods are applied to analyze correlated and uncorrelated variations in [[Analog signal|analog]] and [[Digital data|digital]] [[integrated circuits]].

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* 在'''微电子工程 Microelectronics Engineering'''中，蒙特卡罗方法被用于分析模拟和数字集成电路中相关和不相关的变化。

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*在'''微电子工程 Microelectronics Engineering'''中，蒙特卡罗方法被用于分析模拟和数字集成电路中相关和不相关的变化。

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* In [[geostatistics]] and [[geometallurgy]], Monte Carlo methods underpin the design of [[mineral processing]] [[process flow diagram|flowsheets]] and contribute to [[quantitative risk analysis]].<ref name="mbv01">{{Cite book | last1 =Mazhdrakov | first1 =Metodi | last2 =Benov | first2 =Dobriyan |last3=Valkanov|first3=Nikolai | year =2018 | title =The Monte Carlo Method. Engineering Applications | publisher =ACMO Academic Press | volume = | pages = 250| isbn =978-619-90684-3-4 | doi = |url=https://books.google.com/books?id=t0BqDwAAQBAJ&q=the+monte+carlo+method+engineering+applications+mazhdrakov}}</ref>

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*In [[geostatistics]] and [[geometallurgy]], Monte Carlo methods underpin the design of [[mineral processing]] [[process flow diagram|flowsheets]] and contribute to [[quantitative risk analysis]].<ref name="mbv01">{{Cite book | last1 =Mazhdrakov | first1 =Metodi | last2 =Benov | first2 =Dobriyan |last3=Valkanov|first3=Nikolai | year =2018 | title =The Monte Carlo Method. Engineering Applications | publisher =ACMO Academic Press | volume = | pages = 250| isbn =978-619-90684-3-4 | doi = |url=https://books.google.com/books?id=t0BqDwAAQBAJ&q=the+monte+carlo+method+engineering+applications+mazhdrakov}}</ref>

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* 在'''地质统计学 Geostatistics'''和'''地质冶金学 Geometallurgy'''中，蒙特卡罗方法是矿物处理流程设计的基础，并有助于定量风险分析。<ref name="mbv01" />

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*在'''地质统计学 Geostatistics'''和'''地质冶金学 Geometallurgy'''中，蒙特卡罗方法是矿物处理流程设计的基础，并有助于定量风险分析。<ref name="mbv01" />

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* In [[wind energy]] yield analysis, the predicted energy output of a wind farm during its lifetime is calculated giving different levels of uncertainty ([[Percentile|P90]], P50, etc.)

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*In [[wind energy]] yield analysis, the predicted energy output of a wind farm during its lifetime is calculated giving different levels of uncertainty ([[Percentile|P90]], P50, etc.)

* 在风能产量分析中，考虑不同的不确定性(P90、P50等)，计算风电场在其生命周期内的预测发电量。

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* Impacts of pollution are simulated<ref name="IntPanis1">Int Panis et al. 2001</ref> and diesel compared with petrol.<ref name="IntPanis2">Int Panis et al. 2002</ref>

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*Impacts of pollution are simulated<ref name="IntPanis1">Int Panis et al. 2001</ref> and diesel compared with petrol.<ref name="IntPanis2">Int Panis et al. 2002</ref>

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* 模拟了污染产生的影响，<ref name="IntPanis1" /> 并将柴油和汽油进行了比较。<ref name="IntPanis2" />

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*模拟了污染产生的影响，<ref name="IntPanis1" /> 并将柴油和汽油进行了比较。<ref name="IntPanis2" />

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* In [[fluid dynamics]], in particular [[gas dynamics|rarefied gas dynamics]], where the Boltzmann equation is solved for finite [[Knudsen number]] fluid flows using the [[direct simulation Monte Carlo]]<ref name=":33">G. A. Bird, Molecular Gas Dynamics, Clarendon, Oxford (1976)</ref> method in combination with highly efficient computational algorithms.<ref name=":34">{{cite journal | last1 = Dietrich | first1 = S. | last2 = Boyd | first2 = I. | year = 1996 | title = A Scalar optimized parallel implementation of the DSMC technique | url = | journal = Journal of Computational Physics | volume = 126 | issue = 2| pages = 328–42 | doi=10.1006/jcph.1996.0141|bibcode = 1996JCoPh.126..328D }}</ref>

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*In [[fluid dynamics]], in particular [[gas dynamics|rarefied gas dynamics]], where the Boltzmann equation is solved for finite [[Knudsen number]] fluid flows using the [[direct simulation Monte Carlo]]<ref name=":33">G. A. Bird, Molecular Gas Dynamics, Clarendon, Oxford (1976)</ref> method in combination with highly efficient computational algorithms.<ref name=":34">{{cite journal | last1 = Dietrich | first1 = S. | last2 = Boyd | first2 = I. | year = 1996 | title = A Scalar optimized parallel implementation of the DSMC technique | url = | journal = Journal of Computational Physics | volume = 126 | issue = 2| pages = 328–42 | doi=10.1006/jcph.1996.0141|bibcode = 1996JCoPh.126..328D }}</ref>

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* 在流体动力学，特别是'''稀薄气体动力学 Rarefied Gas Dynamics'''中，采用'''直接模拟蒙特卡罗方法 Direct Simulation Monte Carlo'''<ref name=":33" /> 结合高效计算算法求解有限'''努森数 Knudsen Number'''流体的玻尔兹曼方程。<ref name=":34" />

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*在流体动力学，特别是'''稀薄气体动力学 Rarefied Gas Dynamics'''中，采用'''直接模拟蒙特卡罗方法 Direct Simulation Monte Carlo'''<ref name=":33" /> 结合高效计算算法求解有限'''努森数 Knudsen Number'''流体的玻尔兹曼方程。<ref name=":34" />

* In [[autonomous robotics]], [[Monte Carlo localization]] can determine the position of a robot. It is often applied to stochastic filters such as the [[Kalman filter]] or [[particle filter]] that forms the heart of the [[Simultaneous localization and mapping|SLAM]] (simultaneous localization and mapping) algorithm.

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* 在自主机器人中，'''蒙特卡洛定位 Monte Carlo Localization'''可以确定机器人的位置。它通常应用于随机滤波器，如'''卡尔曼滤波器 Kalman Filter'''或'''粒子滤波器 Particle Filter'''，构成'''同步定位和映射算法 SLAM (Simultaneous Localization and Mapping)'''的核心。

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*在自主机器人中，'''蒙特卡洛定位 Monte Carlo Localization'''可以确定机器人的位置。它通常应用于随机滤波器，如'''卡尔曼滤波器 Kalman Filter'''或'''粒子滤波器 Particle Filter'''，构成'''同步定位和映射算法 SLAM (Simultaneous Localization and Mapping)'''的核心。

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* In [[telecommunications]], when planning a wireless network, design must be proved to work for a wide variety of scenarios that depend mainly on the number of users, their locations and the services they want to use. Monte Carlo methods are typically used to generate these users and their states. The network performance is then evaluated and, if results are not satisfactory, the network design goes through an optimization process.

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*In [[telecommunications]], when planning a wireless network, design must be proved to work for a wide variety of scenarios that depend mainly on the number of users, their locations and the services they want to use. Monte Carlo methods are typically used to generate these users and their states. The network performance is then evaluated and, if results are not satisfactory, the network design goes through an optimization process.

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* 在电信行业，在规划无线网络时，必须证明设计适用于各种不同用户数量、用户位置和他们想使用的服务的场景。蒙特卡罗方法通常用于生成这些用户及其状态。然后对网络性能进行评估，如果结果不能令人满意，则进行网络设计优化。

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*在电信行业，在规划无线网络时，必须证明设计适用于各种不同用户数量、用户位置和他们想使用的服务的场景。蒙特卡罗方法通常用于生成这些用户及其状态。然后对网络性能进行评估，如果结果不能令人满意，则进行网络设计优化。

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* In [[reliability engineering]], Monte Carlo simulation is used to compute system-level response given the component-level response. For example, for a transportation network subject to an earthquake event, Monte Carlo simulation can be used to assess the ''k''-terminal reliability of the network given the failure probability of its components, e.g. bridges, roadways, etc.<ref name=":35">Nabian, Mohammad Amin; Meidani, Hadi (2017-08-28). "Deep Learning for Accelerated Reliability Analysis of Infrastructure Networks". ''Computer-Aided Civil and Infrastructure Engineering''. '''33''' (6): 443–458. arXiv:1708.08551. Bibcode:2017arXiv170808551N. doi:10.1111/mice.12359. S2CID 36661983.</ref><ref name=":36">{{Cite journal|last1=Nabian|first1=Mohammad Amin|last2=Meidani|first2=Hadi|date=2018|title=Accelerating Stochastic Assessment of Post-Earthquake Transportation Network Connectivity via Machine-Learning-Based Surrogates|url=https://trid.trb.org/view/1496617|journal=Transportation Research Board 97th Annual Meeting|volume=|pages=|via=}}</ref><ref name=":37">{{Cite journal|last1=Nabian|first1=Mohammad Amin|last2=Meidani|first2=Hadi|date=2017|title=Uncertainty Quantification and PCA-Based Model Reduction for Parallel Monte Carlo Analysis of Infrastructure System Reliability|url=https://trid.trb.org/view/1439614|journal=Transportation Research Board 96th Annual Meeting|volume=|pages=|via=}}</ref> Another profound example is the application of the Monte Carlo method to solve the G-Renewal equation of the generalized renewal process.<ref name=":38">Krivtsov, V. V. (2000). ''Modeling and estimation of the generalized renewal process in repairable system reliability analysis'' (PhD). University of Maryland, College Park, ISBN/ISSN: 0599725877.</ref>

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*In [[reliability engineering]], Monte Carlo simulation is used to compute system-level response given the component-level response. For example, for a transportation network subject to an earthquake event, Monte Carlo simulation can be used to assess the ''k''-terminal reliability of the network given the failure probability of its components, e.g. bridges, roadways, etc.<ref name=":35">Nabian, Mohammad Amin; Meidani, Hadi (2017-08-28). "Deep Learning for Accelerated Reliability Analysis of Infrastructure Networks". ''Computer-Aided Civil and Infrastructure Engineering''. '''33''' (6): 443–458. arXiv:1708.08551. Bibcode:2017arXiv170808551N. doi:10.1111/mice.12359. S2CID 36661983.</ref><ref name=":36">{{Cite journal|last1=Nabian|first1=Mohammad Amin|last2=Meidani|first2=Hadi|date=2018|title=Accelerating Stochastic Assessment of Post-Earthquake Transportation Network Connectivity via Machine-Learning-Based Surrogates|url=https://trid.trb.org/view/1496617|journal=Transportation Research Board 97th Annual Meeting|volume=|pages=|via=}}</ref><ref name=":37">{{Cite journal|last1=Nabian|first1=Mohammad Amin|last2=Meidani|first2=Hadi|date=2017|title=Uncertainty Quantification and PCA-Based Model Reduction for Parallel Monte Carlo Analysis of Infrastructure System Reliability|url=https://trid.trb.org/view/1439614|journal=Transportation Research Board 96th Annual Meeting|volume=|pages=|via=}}</ref> Another profound example is the application of the Monte Carlo method to solve the G-Renewal equation of the generalized renewal process.<ref name=":38">Krivtsov, V. V. (2000). ''Modeling and estimation of the generalized renewal process in repairable system reliability analysis'' (PhD). University of Maryland, College Park, ISBN/ISSN: 0599725877.</ref>

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* 在可靠性工程中，给定部件级响应，蒙特卡罗仿真用来计算系统级响应。例如，对于一个受地震事件影响的交通网络，给定其组件，如桥梁、道路等的失效概率，蒙特卡洛模拟可以用来评估网络的k-终端可靠性。<ref name=":35" /><ref name=":36" /><ref name=":37" /> 另一个意义深远的例子是应用蒙特卡罗方法求解'''广义更新过程 Generalized Renewal Process'''的G-更新方程。<ref name=":38" />

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*在可靠性工程中，给定部件级响应，蒙特卡罗仿真用来计算系统级响应。例如，对于一个受地震事件影响的交通网络，给定其组件，如桥梁、道路等的失效概率，蒙特卡洛模拟可以用来评估网络的k-终端可靠性。<ref name=":35" /><ref name=":36" /><ref name=":37" /> 另一个意义深远的例子是应用蒙特卡罗方法求解'''广义更新过程 Generalized Renewal Process'''的G-更新方程。<ref name=":38" />

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* In [[signal processing]] and [[Bayesian inference]], [[particle filter]]s and [[Sequential Monte Carlo method|sequential Monte Carlo techniques]] are a class of [[mean field particle methods]] for sampling and computing the posterior distribution of a signal process given some noisy and partial observations using interacting [[empirical measure]]s.

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*In [[signal processing]] and [[Bayesian inference]], [[particle filter]]s and [[Sequential Monte Carlo method|sequential Monte Carlo techniques]] are a class of [[mean field particle methods]] for sampling and computing the posterior distribution of a signal process given some noisy and partial observations using interacting [[empirical measure]]s.

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* 在信号处理和贝叶斯推断中，粒子滤波器和序列蒙特卡罗技术是一类平均场粒子方法，用于对给定噪声和局部观测的信号过程进行采样和计算后验分布，使用相互作用的经验测度。

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*在信号处理和贝叶斯推断中，粒子滤波器和序列蒙特卡罗技术是一类平均场粒子方法，用于对给定噪声和局部观测的信号过程进行采样和计算后验分布，使用相互作用的经验测度。

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* In groundwater modeling, Monte Carlo methods are utilized to generate a large number of realizations of heterogeneous parameter field for model uncertainty quantification or parameter inversion.

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*In groundwater modeling, Monte Carlo methods are utilized to generate a large number of realizations of heterogeneous parameter field for model uncertainty quantification or parameter inversion.

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* 在地下水模拟中，利用蒙特卡罗方法产生了大量的非均质参数场实现，用于模型不确定性量化或参数反演。<ref>Chen, Shang-Ying; Hsu, Kuo-Chin; Fan, Chia-Ming (15 March 2021). "Improvement of generalized finite difference method for stochastic subsurface flow modeling". ''Journal of Computational Physics''. '''429''': 110002. doi:10.1016/J.JCP.2020.110002.</ref>

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*在地下水模拟中，利用蒙特卡罗方法产生了大量的非均质参数场实现，用于模型不确定性量化或参数反演。<ref>Chen, Shang-Ying; Hsu, Kuo-Chin; Fan, Chia-Ming (15 March 2021). "Improvement of generalized finite difference method for stochastic subsurface flow modeling". ''Journal of Computational Physics''. '''429''': 110002. doi:10.1016/J.JCP.2020.110002.</ref>

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===Climate change and radiative forcing 气候变化与辐射强迫 ===

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===Climate change and radiative forcing 气候变化与辐射强迫===

The [[IPCC|Intergovernmental Panel on Climate Change]] relies on Monte Carlo methods in [[probability density function]] analysis of [[radiative forcing]].

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蒙特卡罗方法的统计标准是由萨维罗斯基制定的。<ref name=":41" /> 在应用统计学中，蒙特卡罗方法至少可用于四种目的:

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#To compare competing statistics for small samples under realistic data conditions. Although [[type I error]] and power properties of statistics can be calculated for data drawn from classical theoretical distributions (''e.g.'', [[normal curve]], [[Cauchy distribution]]) for [[asymptotic]] conditions (''i. e'', infinite sample size and infinitesimally small treatment effect), real data often do not have such distributions.<ref name=":43">Sawilowsky, Shlomo S.; Fahoome, Gail C. (2003). ''Statistics via Monte Carlo Simulation with Fortran''. Rochester Hills, MI: JMASM. ISBN <bdi>978-0-9740236-0-1</bdi>.</ref>

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# To compare competing statistics for small samples under realistic data conditions. Although [[type I error]] and power properties of statistics can be calculated for data drawn from classical theoretical distributions (''e.g.'', [[normal curve]], [[Cauchy distribution]]) for [[asymptotic]] conditions (''i. e'', infinite sample size and infinitesimally small treatment effect), real data often do not have such distributions.<ref name=":43">Sawilowsky, Shlomo S.; Fahoome, Gail C. (2003). ''Statistics via Monte Carlo Simulation with Fortran''. Rochester Hills, MI: JMASM. ISBN <bdi>978-0-9740236-0-1</bdi>.</ref>

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#To compare competing statistics for small samples under realistic data conditions. Although type I error and power properties of statistics can be calculated for data drawn from classical theoretical distributions (e.g., normal curve, Cauchy distribution) for asymptotic conditions (i. e, infinite sample size and infinitesimally small treatment effect), real data often do not have such distributions. 比较在现实数据条件下小样本的竞争统计。虽然根据经典理论分布(例如，正态曲线，'''柯西分布 Cauchy distribution''')数据的渐近条件(即，无限大的样本量和无限小的处理效果) ， i 型误差和统计的幂次特性可以进行计算，但是实际数据往往没有这样的分布。<ref name=":43" />

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# To compare competing statistics for small samples under realistic data conditions. Although type I error and power properties of statistics can be calculated for data drawn from classical theoretical distributions (e.g., normal curve, Cauchy distribution) for asymptotic conditions (i. e, infinite sample size and infinitesimally small treatment effect), real data often do not have such distributions. 比较在现实数据条件下小样本的竞争统计。虽然根据经典理论分布(例如，正态曲线，'''柯西分布 Cauchy distribution''')数据的渐近条件(即，无限大的样本量和无限小的处理效果) ， i 型误差和统计的幂次特性可以进行计算，但是实际数据往往没有这样的分布。<ref name=":43" />

#To provide implementations of [[Statistical hypothesis testing|hypothesis tests]] that are more efficient than exact tests such as [[permutation tests]] (which are often impossible to compute) while being more accurate than critical values for [[asymptotic distribution]]s.

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#To provide a random sample from the posterior distribution in Bayesian inference. This sample then approximates and summarizes all the essential features of the posterior. 提供一份来自后验概率贝叶斯推断的随机样本。然后基于这个样本进行近似和总结后验的所有基本特征。

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#To provide efficient random estimates of the Hessian matrix of the negative log-likelihood function that may be averaged to form an estimate of the [[Fisher information]] matrix.<ref name=":44">Spall, James C. (2005). "Monte Carlo Computation of the Fisher Information Matrix in Nonstandard Settings". ''Journal of Computational and Graphical Statistics''. '''14''' (4): 889–909. CiteSeerX 10.1.1.142.738. doi:10.1198/106186005X78800. S2CID 16090098.</ref><ref name=":45">{{Cite journal |doi = 10.1016/j.csda.2009.09.018|title = Efficient Monte Carlo computation of Fisher information matrix using prior information|journal = Computational Statistics & Data Analysis|volume = 54|issue = 2|pages = 272–289|year = 2010|last1 = Das|first1 = Sonjoy|last2 = Spall|first2 = James C.|last3 = Ghanem|first3 = Roger}}</ref>

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# To provide efficient random estimates of the Hessian matrix of the negative log-likelihood function that may be averaged to form an estimate of the [[Fisher information]] matrix.<ref name=":44">Spall, James C. (2005). "Monte Carlo Computation of the Fisher Information Matrix in Nonstandard Settings". ''Journal of Computational and Graphical Statistics''. '''14''' (4): 889–909. CiteSeerX 10.1.1.142.738. doi:10.1198/106186005X78800. S2CID 16090098.</ref><ref name=":45">{{Cite journal |doi = 10.1016/j.csda.2009.09.018|title = Efficient Monte Carlo computation of Fisher information matrix using prior information|journal = Computational Statistics & Data Analysis|volume = 54|issue = 2|pages = 272–289|year = 2010|last1 = Das|first1 = Sonjoy|last2 = Spall|first2 = James C.|last3 = Ghanem|first3 = Roger}}</ref>

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#To provide efficient random estimates of the Hessian matrix of the negative log-likelihood function that may be averaged to form an estimate of the Fisher information matrix. 提供负对数似然函数的海赛矩阵的有效随机估计，这些估计的平均值可以形成'''费雪信息量 Fisher Information'''矩阵的估计。<ref name=":44" /><ref name=":45" />

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# To provide efficient random estimates of the Hessian matrix of the negative log-likelihood function that may be averaged to form an estimate of the Fisher information matrix. 提供负对数似然函数的海赛矩阵的有效随机估计，这些估计的平均值可以形成'''费雪信息量 Fisher Information'''矩阵的估计。<ref name=":44" /><ref name=":45" />

Monte Carlo methods are also a compromise between approximate randomization and permutation tests. An approximate [[randomization test]] is based on a specified subset of all permutations (which entails potentially enormous housekeeping of which permutations have been considered). The Monte Carlo approach is based on a specified number of randomly drawn permutations (exchanging a minor loss in precision if a permutation is drawn twice—or more frequently—for the efficiency of not having to track which permutations have already been selected).

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#Play a simulated game starting with that node. 以该节点开始玩一个模拟游戏。

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#Use the results of that simulated game to update the node and its ancestors.

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# Use the results of that simulated game to update the node and its ancestors.

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#Use the results of that simulated game to update the node and its ancestors. 使用模拟游戏的结果来更新节点及其祖先。

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# Use the results of that simulated game to update the node and its ancestors. 使用模拟游戏的结果来更新节点及其祖先。

The net effect, over the course of many simulated games, is that the value of a node representing a move will go up or down, hopefully corresponding to whether or not that node represents a good move.

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美国海岸警卫队在其计算机建模软件—'''搜救最优规划系统 Search and Rescue Optimal Planning System (SAROPS)''' 中使用蒙特卡罗方法，以便在搜索和救援行动中计算可能的船只位置。每个模拟可以生成多达一万个数据点，这些数据点是根据提供的变量随机分布的。<ref name=":54" /> 然后根据这些数据推断生成搜索模式，以优化包容概率(POC)和检测概率(POD) ，这两者合起来等于总体成功概率(POS)。最终，作为概率分布的一个实际应用，以最迅速和最便捷的救援方法，拯救生命和资源。<ref name=":55" />

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===Finance and business 金融与商业===

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===Finance and business 金融与商业 ===

Monte Carlo simulation is commonly used to evaluate the risk and uncertainty that would affect the outcome of different decision options. Monte Carlo simulation allows the business risk analyst to incorporate the total effects of uncertainty in variables like sales volume, commodity and labour prices, interest and exchange rates, as well as the effect of distinct risk events like the cancellation of a contract or the change of a tax law.

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在金融领域中，蒙特卡罗方法经常被用于评估业务单位或公司层面的项目投资，或其他金融估值。它们可以用来模拟项目进度，其中模拟汇总了对最坏情况、最好情况和每个任务最可能持续时间的估计，以确定整个项目的结果[https://risk.octigo.pl/]蒙特卡罗方法也用于期权定价，违约风险分析。<ref name=":56" /><ref name=":57" /><ref name="kr11" /> 此外，它们还可以用来估计医疗干预的财务影响。<ref name=":58" />

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===Law 法律===

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=== Law 法律===

A Monte Carlo approach was used for evaluating the potential value of a proposed program to help female petitioners in Wisconsin be successful in their applications for [[Harassment Restraining Order|harassment]] and [[Domestic Abuse Restraining Order|domestic abuse restraining orders]]. It was proposed to help women succeed in their petitions by providing them with greater advocacy thereby potentially reducing the risk of [[rape]] and [[physical assault]]. However, there were many variables in play that could not be estimated perfectly, including the effectiveness of restraining orders, the success rate of petitioners both with and without advocacy, and many others. The study ran trials that varied these variables to come up with an overall estimate of the success level of the proposed program as a whole.<ref name="montecarloanalysis">Elwart, Liz; Emerson, Nina; Enders, Christina; Fumia, Dani; Murphy, Kevin (December 2006). "Increasing Access to Restraining Orders for Low Income Victims of Domestic Violence: A Cost-Benefit Analysis of the Proposed Domestic Abuse Grant Program" (PDF). State Bar of Wisconsin. Archived from the original (PDF) on 6 November 2018. Retrieved 2016-12-12.</ref>

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蒙特卡洛方法被用来评估一个拟议的方案的潜在价值，以帮助威斯康星州的女性请愿者成功地申请骚扰和家庭虐待限制令。提议帮助妇女成功地提出请愿，向她们提供更多的宣传，从而有可能减少强奸和人身攻击的风险。然而，还有许多变量无法完全估计，包括限制令的有效性，上访者的成功率，无论有没有主张，以及许多其他因素。这项研究通过改变这些变量进行了试验，得出了对整个计划成功程度的总体评估。<ref name="montecarloanalysis" />

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==Use in mathematics 数学应用==

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== Use in mathematics 数学应用==

In general, the Monte Carlo methods are used in mathematics to solve various problems by generating suitable random numbers (see also [[Random number generation]]) and observing that fraction of the numbers that obeys some property or properties. The method is useful for obtaining numerical solutions to problems too complicated to solve analytically. The most common application of the Monte Carlo method is Monte Carlo integration.

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一般来说，蒙特卡罗方法在数学中通过产生合适的随机数(也见随机数产生)和观察符合某些性质的数字分数来解决各种问题。这种方法对于求解解析求解过于复杂的问题的数值解是有用的。蒙特卡罗方法最常用的应用是蒙地卡罗积分。

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=== Integration 积分 ===

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=== Integration 积分===

{{Main|Monte Carlo integration}}[[File:Monte-carlo2.gif|thumb|Monte-Carlo integration works by comparing random points with the value of the functionMonte-Carlo integration works by comparing random points with the value of the function

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另一类方法是模拟体积上的随机游动(马尔科夫蒙特卡洛)。这些方法包括 Metropolis-Hastings 算法、 Gibbs 抽样、 Wang 和 Landau 算法以及交互式 MCMC 方法，如序贯蒙特卡罗抽样。<ref name=":61" />

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=== Simulation and optimization 模拟与优化 ===

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===Simulation and optimization 模拟与优化===

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最著名的重要性抽样方法，Metropolis–Hastings 演算法，可以推广，这提供了一种方法，允许分析(可能是高度非线性)与复杂的先验信息和数据与任意噪声分布的反问题。<ref name=":63" /><ref name=":64" />

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===Philosophy 哲学===

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=== Philosophy 哲学===

Popular exposition of the Monte Carlo Method was conducted by McCracken<ref name=":65">McCracken, D. D., (1955) The Monte Carlo Method, Scientific American, 192(5), pp. 90-97</ref>. Method's general philosophy was discussed by [[Elishakoff]]<ref name=":66">Elishakoff, I., (2003) Notes on Philosophy of the Monte Carlo Method, International Applied Mechanics, 39(7), pp.753-762</ref> and Grüne-Yanoff and Weirich<ref name=":67">Grüne-Yanoff, T., & Weirich, P. (2010). The philosophy and epistemology of simulation: A review, Simulation & Gaming, 41(1), pp. 20-50</ref>.

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由 McCracken 主持的蒙特卡罗方法博览会的普及展览。<ref name=":65" />方法的一般哲学由 Elishakoff、<ref name=":66" /> Grüne-Yanoff 和 weurich 讨论。<ref name=":67" />

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== See also 另见 ==

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==See also 另见==

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== References 参考文献 ==

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==References 参考文献 ==

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=== Citations ===

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===Citations===

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=== Sources ===

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===Sources===

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