更改

'''卡洛斯 Kalos'''和'''惠特洛克 Whitlock'''<ref>Kalos, Malvin H.; Whitlock, Paula A. (2008). ''Monte Carlo Methods''. Wiley-VCH. ISBN <bdi>978-3-527-40760-6</bdi>.</ref>指出，这几种方法的区别并不总是容易分辨。例如，来自原子的辐射是一种自然的随机过程。它可以直接模拟，也可以用随机方程描述其平均行为，这些随机方程本身可以用蒙特卡罗方法求解。“实际上，同样的计算机代码可以同时被看作是‘自然模拟’或者通过自然抽样解方程。”

'''卡洛斯 Kalos'''和'''惠特洛克 Whitlock'''<ref>Kalos, Malvin H.; Whitlock, Paula A. (2008). ''Monte Carlo Methods''. Wiley-VCH. ISBN <bdi>978-3-527-40760-6</bdi>.</ref>指出，这几种方法的区别并不总是容易分辨。例如，来自原子的辐射是一种自然的随机过程。它可以直接模拟，也可以用随机方程描述其平均行为，这些随机方程本身可以用蒙特卡罗方法求解。“实际上，同样的计算机代码可以同时被看作是‘自然模拟’或者通过自然抽样解方程。”

'''Monte Carlo and random numbers'''

'''Monte Carlo and random numbers 蒙特卡罗和随机数'''

 

The main idea behind this method is that the results are computed based on repeated random sampling and statistical analysis. The Monte Carlo simulation is, in fact, random experimentations, in the case that, the results of these experiments are not well known. Monte Carlo simulations are typically characterized by many unknown parameters, many of which are difficult to obtain experimentally. Monte Carlo simulation methods do not always require truly random numbers to be useful (although, for some applications such as primality testing, unpredictability is vital). Many of the most useful techniques use deterministic, pseudorandom sequences, making it easy to test and re-run simulations. The only quality usually necessary to make good simulations is for the pseudo-random sequence to appear "random enough" in a certain sense.

The main idea behind this method is that the results are computed based on repeated random sampling and statistical analysis. The Monte Carlo simulation is, in fact, random experimentations, in the case that, the results of these experiments are not well known. Monte Carlo simulations are typically characterized by many unknown parameters, many of which are difficult to obtain experimentally. Monte Carlo simulation methods do not always require truly random numbers to be useful (although, for some applications such as primality testing, unpredictability is vital). Many of the most useful techniques use deterministic, pseudorandom sequences, making it easy to test and re-run simulations. The only quality usually necessary to make good simulations is for the pseudo-random sequence to appear "random enough" in a certain sense.

这种方法的主要思想是基于重复随机抽样和统计分析来计算结果。蒙特卡洛模拟实际上是一种随机实验，在这种情况下，这些实验的结果并不为人所知。蒙特卡罗模拟的典型特征是有许多未知参数，其中许多参数很难通过实验获得。<ref>Shojaeefard, MH; Khalkhali, A; Yarmohammadisatri, Sadegh (2017). "An efficient sensitivity analysis method for modified geometry of Macpherson suspension based on Pearson Correlation Coefficient". ''Vehicle System Dynamics''. '''55''' (6): 827–852. Bibcode:2017VSD....55..827S. doi:10.1080/00423114.2017.1283046. S2CID 114260173.</ref>蒙特卡罗模拟方法并不总是要求真正的随机数是有用的(尽管对于一些应用程序，如质数测试，不可预测性是至关重要的)。<ref>Davenport, J. H. (1992). "Primality testing revisited". ''Papers from the international symposium on Symbolic and algebraic computation - ISSAC '92''. ''Proceeding ISSAC '92 Papers from the International Symposium on Symbolic and Algebraic Computation''. pp. 123–129. CiteSeerX 10.1.1.43.9296. doi:10.1145/143242.143290. ISBN <bdi>978-0-89791-489-5</bdi>. S2CID 17322272.</ref>许多最有用的技术是使用确定性的伪随机序列，使测试和重新运行模拟变得很容易。伪随机序列在某种意义上表现地“足够随机”，这是进行良好模拟唯一必需的性质。

这种方法的主要思想是基于重复随机抽样和统计分析来计算结果。蒙特卡洛模拟实际上是一种随机实验，在这种情况下，这些实验的结果并不为人所知。蒙特卡罗模拟的典型特征是有许多未知参数，其中许多参数很难通过实验获得。<ref>Shojaeefard, MH; Khalkhali, A; Yarmohammadisatri, Sadegh (2017). "An efficient sensitivity analysis method for modified geometry of Macpherson suspension based on Pearson Correlation Coefficient". ''Vehicle System Dynamics''. '''55''' (6): 827–852. Bibcode:2017VSD....55..827S. doi:10.1080/00423114.2017.1283046. S2CID 114260173.</ref> 蒙特卡罗模拟方法并不总是要求真正的随机数是有用的(尽管对于一些应用程序，如质数测试，不可预测性是至关重要的)。<ref>Davenport, J. H. (1992). "Primality testing revisited". ''Papers from the international symposium on Symbolic and algebraic computation - ISSAC '92''. ''Proceeding ISSAC '92 Papers from the International Symposium on Symbolic and Algebraic Computation''. pp. 123–129. CiteSeerX 10.1.1.43.9296. doi:10.1145/143242.143290. ISBN <bdi>978-0-89791-489-5</bdi>. S2CID 17322272.</ref> 许多最有用的技术是使用确定性的伪随机序列，使测试和重新运行模拟变得很容易。伪随机序列在某种意义上表现地“足够随机”，这是进行良好模拟唯一必需的性质。

What this means depends on the application, but typically they should pass a series of statistical tests. Testing that the numbers are uniformly distributed or follow another desired distribution when a large enough number of elements of the sequence are considered is one of the simplest and most common ones. Weak correlations between successive samples are also often desirable/necessary.

What this means depends on the application, but typically they should pass a series of statistical tests. Testing that the numbers are uniformly distributed or follow another desired distribution when a large enough number of elements of the sequence are considered is one of the simplest and most common ones. Weak correlations between successive samples are also often desirable/necessary.

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" />

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" />

=== Monte Carlo simulation versus "what if" scenarios ===

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

 

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.<ref name=":25">Vose 2000, p. 13</ref>

 

By contrast, Monte Carlo simulations sample from a [[probability distribution]] for each variable to produce hundreds or thousands of possible outcomes. The results are analyzed to get probabilities of different outcomes occurring.<ref>{{harvnb|Vose|2000|page=16}}</ref> For example, a comparison of a spreadsheet cost construction model run using traditional "what if" scenarios, and then running the comparison again with Monte Carlo simulation and [[triangular distribution|triangular probability distribution]]s shows that the Monte Carlo analysis has a narrower range than the "what if" analysis.{{Examples|date=May 2012}}  This is because the "what if" analysis gives equal weight to all scenarios (see [[Corporate finance#Quantifying uncertainty|quantifying uncertainty in corporate finance]]), while the Monte Carlo method hardly samples in the very low probability regions. The samples in such regions are called "rare events".

 

 

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.

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.

使用概率的方法不一定都是蒙特卡洛模拟——例如，使用单点估计的确定性建模。模型中的每个不确定变量都被赋予一个“最佳猜测”估计。为每个输入变量选择场景(如最佳、最差或最可能的情况)并记录结果。<ref name=":25" />

使用概率的方法不一定都是蒙特卡洛模拟——例如，使用单点估计的确定性建模。模型中的每个不确定变量都被赋予一个“最佳猜测”估计。为每个输入变量选择场景(如最佳、最差或最可能的情况)并记录结果。<ref name=":25">Vose 2000, p. 13</ref>

By contrast, Monte Carlo simulations sample from a probability distribution for each variable to produce hundreds or thousands of possible outcomes. The results are analyzed to get probabilities of different outcomes occurring. For example, a comparison of a spreadsheet cost construction model run using traditional "what if" scenarios, and then running the comparison again with Monte Carlo simulation and triangular probability distributions shows that the Monte Carlo analysis has a narrower range than the "what if" analysis.  This is because the "what if" analysis gives equal weight to all scenarios (see quantifying uncertainty in corporate finance), while the Monte Carlo method hardly samples in the very low probability regions. The samples in such regions are called "rare events".

By contrast, Monte Carlo simulations sample from a probability distribution for each variable to produce hundreds or thousands of possible outcomes. The results are analyzed to get probabilities of different outcomes occurring. For example, a comparison of a spreadsheet cost construction model run using traditional "what if" scenarios, and then running the comparison again with Monte Carlo simulation and triangular probability distributions shows that the Monte Carlo analysis has a narrower range than the "what if" analysis.  This is because the "what if" analysis gives equal weight to all scenarios (see quantifying uncertainty in corporate finance), while the Monte Carlo method hardly samples in the very low probability regions. The samples in such regions are called "rare events".

{{See also|Monte Carlo method in statistical physics}}

{{See also|Monte Carlo method in statistical physics}}

Monte Carlo methods are very important in [[computational physics]], [[physical chemistry]], and related applied fields, and have diverse applications from complicated [[quantum chromodynamics]] calculations to designing [[heat shield]]s and [[aerodynamics|aerodynamic]] forms as well as in modeling radiation transport for radiation dosimetry calculations.<ref>{{cite journal | doi = 10.1088/0031-9155/59/4/R151 | pmid=24486639 | volume=59 | issue=4 | title=GPU-based high-performance computing for radiation therapy | journal=Physics in Medicine and Biology | pages=R151–R182|bibcode = 2014PMB....59R.151J | year=2014 | last1=Jia | first1=Xun | last2=Ziegenhein | first2=Peter | last3=Jiang | first3=Steve B | pmc=4003902 }}</ref><ref>{{cite journal | doi = 10.1088/0031-9155/59/6/R183 | volume=59 | issue=6 | title=Advances in kilovoltage x-ray beam dosimetry | journal=Physics in Medicine and Biology | pages=R183–R231|bibcode = 2014PMB....59R.183H | pmid=24584183 | date=Mar 2014| last1=Hill | first1=R | last2=Healy | first2=B | last3=Holloway | first3=L | last4=Kuncic | first4=Z | last5=Thwaites | first5=D | last6=Baldock | first6=C | s2cid=18082594 | url=https://semanticscholar.org/paper/fb231c3d9ade811d793b85623fd32c6ea126d5ff }}</ref><ref>{{cite journal | doi = 10.1088/0031-9155/51/13/R17 | pmid=16790908 | volume=51 | issue=13 | title=Fifty years of Monte Carlo simulations for medical physics | journal=Physics in Medicine and Biology | pages=R287–R301|bibcode = 2006PMB....51R.287R | year=2006 | last1=Rogers | first1=D W O | s2cid=12066026 | url=https://semanticscholar.org/paper/b6d08efc5f0818a01dc60637a4a6f8115482483e }}</ref>  In [[statistical physics]] [[Monte Carlo molecular modeling]] is an alternative to computational [[molecular dynamics]], and Monte Carlo methods are used to compute [[statistical field theory|statistical field theories]] of simple particle and polymer systems.<ref name=":0" /><ref>{{harvnb|Baeurle|2009}}</ref> [[Quantum Monte Carlo]] methods solve the [[many-body problem]] for quantum systems.<ref name="kol10" /><ref name="dp13" /><ref name="dp04" /> In [[Radiation material science|radiation materials science]], the [[binary collision approximation]] for simulating [[ion implantation]] is usually based on a Monte Carlo approach to select the next colliding atom.<ref>{{Cite journal|last1=Möller|first1=W.|last2=Eckstein|first2=W.|date=1984-03-01|title=Tridyn — A TRIM simulation code including dynamic composition changes|journal=Nuclear Instruments and Methods in Physics Research Section B: Beam Interactions with Materials and Atoms|volume=2|issue=1|pages=814–818|doi=10.1016/0168-583X(84)90321-5|bibcode=1984NIMPB...2..814M}}</ref> In experimental [[particle physics]], Monte Carlo methods are used for designing [[particle detector|detectors]], understanding their behavior and comparing experimental data to theory. In [[astrophysics]], they are used in such diverse manners as to model both [[galaxy]] evolution<ref>{{harvnb|MacGillivray|Dodd|1982}}</ref> and microwave radiation transmission through a rough planetary surface.<ref>{{harvnb|Golden|1979}}</ref> Monte Carlo methods are also used in the [[Ensemble forecasting|ensemble models]] that form the basis of modern [[Numerical weather prediction|weather forecasting]].

Monte Carlo methods are very important in [[computational physics]], [[physical chemistry]], and related applied fields, and have diverse applications from complicated [[quantum chromodynamics]] calculations to designing [[heat shield]]s and [[aerodynamics|aerodynamic]] forms as well as in modeling radiation transport for radiation dosimetry calculations.<ref name=":26">{{cite journal | doi = 10.1088/0031-9155/59/4/R151 | pmid=24486639 | volume=59 | issue=4 | title=GPU-based high-performance computing for radiation therapy | journal=Physics in Medicine and Biology | pages=R151–R182|bibcode = 2014PMB....59R.151J | year=2014 | last1=Jia | first1=Xun | last2=Ziegenhein | first2=Peter | last3=Jiang | first3=Steve B | pmc=4003902 }}</ref><ref name=":27">Hill, R; Healy, B; Holloway, L; Kuncic, Z; Thwaites, D; Baldock, C (Mar 2014). "Advances in kilovoltage x-ray beam dosimetry". ''Physics in Medicine and Biology''. '''59''' (6): R183–R231. Bibcode:2014PMB....59R.183H. doi:10.1088/0031-9155/59/6/R183. <nowiki>PMID 24584183</nowiki>. S2CID 18082594.</ref><ref name=":28">Rogers, D W O (2006). "Fifty years of Monte Carlo simulations for medical physics". ''Physics in Medicine and Biology''. '''51''' (13): R287–R301. Bibcode:2006PMB....51R.287R. doi:10.1088/0031-9155/51/13/R17. <nowiki>PMID 16790908</nowiki>. S2CID 12066026.</ref>  In [[statistical physics]] [[Monte Carlo molecular modeling]] is an alternative to computational [[molecular dynamics]], and Monte Carlo methods are used to compute [[statistical field theory|statistical field theories]] of simple particle and polymer systems.<ref name=":0" /><ref name=":29">Baeurle, Stephan A. (2009). "Multiscale modeling of polymer materials using field-theoretic methodologies: A survey about recent developments". ''Journal of Mathematical Chemistry''. '''46''' (2): 363–426. doi:10.1007/s10910-008-9467-3. S2CID 117867762.</ref> [[Quantum Monte Carlo]] methods solve the [[many-body problem]] for quantum systems.<ref name="kol10" /><ref name="dp13" /><ref name="dp04" /> In [[Radiation material science|radiation materials science]], the [[binary collision approximation]] for simulating [[ion implantation]] is usually based on a Monte Carlo approach to select the next colliding atom.<ref name=":30">{{Cite journal|last1=Möller|first1=W.|last2=Eckstein|first2=W.|date=1984-03-01|title=Tridyn — A TRIM simulation code including dynamic composition changes|journal=Nuclear Instruments and Methods in Physics Research Section B: Beam Interactions with Materials and Atoms|volume=2|issue=1|pages=814–818|doi=10.1016/0168-583X(84)90321-5|bibcode=1984NIMPB...2..814M}}</ref> In experimental [[particle physics]], Monte Carlo methods are used for designing [[particle detector|detectors]], understanding their behavior and comparing experimental data to theory. In [[astrophysics]], they are used in such diverse manners as to model both [[galaxy]] evolution<ref name=":31">MacGillivray, H. T.; Dodd, R. J. (1982). "Monte-Carlo simulations of galaxy systems". ''Astrophysics and Space Science''. '''86''' (2): 419–435. doi:10.1007/BF00683346. S2CID 189849365.</ref> and microwave radiation transmission through a rough planetary surface.<ref name=":32">Golden, Leslie M. (1979). "The Effect of Surface Roughness on the Transmission of Microwave Radiation Through a Planetary Surface". ''Icarus''. '''38''' (3): 451–455. Bibcode:1979Icar...38..451G. doi:10.1016/0019-1035(79)90199-4.</ref> Monte Carlo methods are also used in the [[Ensemble forecasting|ensemble models]] that form the basis of modern [[Numerical weather prediction|weather forecasting]].

Monte Carlo methods are very important in computational physics, physical chemistry, and related applied fields, and have diverse applications from complicated quantum chromodynamics calculations to designing heat shields and aerodynamic forms as well as in modeling radiation transport for radiation dosimetry calculations.  In statistical physics Monte Carlo molecular modeling is an alternative to computational molecular dynamics, and Monte Carlo methods are used to compute statistical field theories of simple particle and polymer systems. Quantum Monte Carlo methods solve the many-body problem for quantum systems. In experimental particle physics, Monte Carlo methods are used for designing detectors, understanding their behavior and comparing experimental data to theory. In astrophysics, they are used in such diverse manners as to model both galaxy evolution and microwave radiation transmission through a rough planetary surface. Monte Carlo methods are also used in the ensemble models that form the basis of modern weather forecasting.

Monte Carlo methods are very important in computational physics, physical chemistry, and related applied fields, and have diverse applications from complicated quantum chromodynamics calculations to designing heat shields and aerodynamic forms as well as in modeling radiation transport for radiation dosimetry calculations.  In statistical physics Monte Carlo molecular modeling is an alternative to computational molecular dynamics, and Monte Carlo methods are used to compute statistical field theories of simple particle and polymer systems. Quantum Monte Carlo methods solve the many-body problem for quantum systems.  In radiation materials science, the binary collision approximation for simulating ion implantation is usually based on a Monte Carlo approach to select the next colliding atom. In experimental particle physics, Monte Carlo methods are used for designing detectors, understanding their behavior and comparing experimental data to theory. In astrophysics, they are used in such diverse manners as to model both galaxy evolution and microwave radiation transmission through a rough planetary surface. Monte Carlo methods are also used in the ensemble models that form the basis of modern weather forecasting.

===Engineering 工程学===

===Engineering 工程学===

* 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>

* 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>

* 在地质统计学和地质冶金学中，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。

* 在地质统计学和地质冶金学中，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" />

* 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.)

* 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。

* 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。

* Impacts of pollution are simulated<ref name="IntPanis1">{{harvnb|Int Panis|De Nocker|De Vlieger|Torfs|2001}}</ref> and diesel compared with petrol.<ref name="IntPanis2">{{harvnb|Int Panis|Rabl|De Nocker|Torfs|2002}}</ref>

* 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>

* 模拟了污染产生的影响，并将柴油和汽油进行了比较。

* 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" />

* 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>G. A. Bird, Molecular Gas Dynamics, Clarendon, Oxford (1976)</ref> method in combination with highly efficient computational algorithms.<ref>{{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>

* 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>

* 在流体动力学，特别是稀薄气体动力学中，采用直接模拟蒙特卡罗方法结合高效计算算法求解有限克努森数流体的玻尔兹曼方程。

* 在流体动力学，特别是稀薄气体动力学中，采用直接模拟蒙特卡罗方法<ref name=":33" /> 结合高效计算算法求解有限努森数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.

* 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.

* 在电信行业，在规划无线网络时，必须证明设计适用于各种主要取决于用户数量、他们的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，如果结果不令人满意，则进行网络设计优化。

* 在电信行业，在规划无线网络时，必须证明设计适用于各种主要取决于用户数量、他们的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，如果结果不令人满意，则进行网络设计优化。

* 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>{{cite journal|last1=Nabian|first1=Mohammad Amin|last2=Meidani|first2=Hadi|date=2017-08-28|title=Deep Learning for Accelerated Reliability Analysis of Infrastructure Networks|journal=Computer-Aided Civil and Infrastructure Engineering|volume=33|issue=6|pages=443–458|arxiv=1708.08551|doi=10.1111/mice.12359|bibcode=2017arXiv170808551N|s2cid=36661983}}</ref><ref>{{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>{{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>

* 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>

* 在可靠性工程中，蒙特卡罗仿真被用来计算系统级响应给定的部件级响应。例如，对于一个受地震事件影响的交通网络，蒙特卡洛模拟可以用来评估网络的k终端可靠性，给定其组件，如桥梁、道路等的失效概率。

* 在可靠性工程中，蒙特卡罗仿真被用来计算系统级响应给定的部件级响应。例如，对于一个受地震事件影响的交通网络，蒙特卡洛模拟可以用来评估网络的k终端可靠性，给定其组件，如桥梁、道路等的失效概率。<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的G-更新方程。<ref name=":38" />

* 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.

* 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.

* 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。

* 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。

* 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.

* 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.

* 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。

* 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>

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

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

Probability density function (PDF) of ERF due to total GHG, aerosol forcing and total anthropogenic forcing. The GHG consists of WMGHG, ozone and stratospheric water vapour. The PDFs are generated based on uncertainties provided in Table 8.6. The combination of the individual RF agents to derive total forcing over the Industrial Era are done by Monte Carlo simulations and based on the method in Boucher and Haywood (2001). PDF of the ERF from surface albedo changes and combined contrails and contrail-induced cirrus are included in the total anthropogenic forcing, but not shown as a separate PDF. We currently do not have ERF estimates for some forcing mechanisms: ozone, land use, solar, etc.

Probability density function (PDF) of ERF due to total GHG, aerosol forcing and total anthropogenic forcing. The GHG consists of WMGHG, ozone and stratospheric water vapour. The PDFs are generated based on uncertainties provided in Table 8.6. The combination of the individual RF agents to derive total forcing over the Industrial Era are done by Monte Carlo simulations and based on the method in Boucher and Haywood (2001). PDF of the ERF from surface albedo changes and combined contrails and contrail-induced cirrus are included in the total anthropogenic forcing, but not shown as a separate PDF. We currently do not have ERF estimates for some forcing mechanisms: ozone, land use, solar, etc.

基于总温室气体、气溶胶强迫和总人为强迫的ERF概率密度函数(PDF)。温室气体由WMGHG、臭氧和平流层水蒸汽组成。pdf文件是根据表8.6提供的不确定性生成的。基于Boucher和Haywood(2001)的方法，通过蒙特卡洛模拟，将单个射频agent组合起来，得出工业时代的总强迫。从地面反照率变化和混合尾迹和尾迹诱导的卷云的ERF的PDF包含在总人为强迫中，但没有单独显示为一个PDF。我们目前还没有一些强迫机制的ERF估计:臭氧、土地利用、太阳能等。

基于总温室气体、气溶胶强迫和总人为强迫的ERF概率密度函数(PDF)。温室气体由WMGHG、臭氧和平流层水蒸汽组成。pdf文件是根据表8.6提供的不确定性生成的。基于Boucher和Haywood(2001)的方法，通过蒙特卡洛模拟，将单个射频agent组合起来，得出工业时代的总强迫。从地面反照率变化和混合尾迹和尾迹诱导的卷云的ERF的PDF包含在总人为强迫中，但没有单独显示为一个PDF。我们目前还没有一些强迫机制的ERF估计:臭氧、土地利用、太阳能等。<ref>''Climate Change 2013 The Physical Science Basis'' (PDF). Cambridge University Press. 2013. p. 697. ISBN <bdi>978-1-107-66182-0</bdi>. Retrieved 2 March 2016.</ref>

===Computational biology 计算生物学===

===Computational biology 计算生物学===

Monte Carlo methods are used in various fields of [[computational biology]], for example for [[Bayesian inference in phylogeny]], or for studying biological systems such as genomes, proteins,<ref>{{harvnb|Ojeda|et al.|2009}},</ref> or membranes.<ref>{{harvnb|Milik|Skolnick|1993}}</ref>The systems can be studied in the coarse-grained or ''ab initio'' frameworks depending on the desired accuracy. Computer simulations allow us to monitor the local environment of a particular [[biomolecule|molecule]] to see if some [[chemical reaction]] is happening for instance. In cases where it is not feasible to conduct a physical experiment, [[thought experiment]]s can be conducted (for instance: breaking bonds, introducing impurities at specific sites, changing the local/global structure, or introducing external fields).  

Monte Carlo methods are used in various fields of [[computational biology]], for example for [[Bayesian inference in phylogeny]], or for studying biological systems such as genomes, proteins,<ref name=":39">Ojeda & et al. 2009,</ref> or membranes.<ref name=":40">Milik, M.; Skolnick, J. (Jan 1993). "Insertion of peptide chains into lipid membranes: an off-lattice Monte Carlo dynamics model". ''Proteins''. '''15''' (1): 10–25. doi:10.1002/prot.340150104. <nowiki>PMID 8451235</nowiki>. S2CID 7450512.</ref>The systems can be studied in the coarse-grained or ''ab initio'' frameworks depending on the desired accuracy. Computer simulations allow us to monitor the local environment of a particular [[biomolecule|molecule]] to see if some [[chemical reaction]] is happening for instance. In cases where it is not feasible to conduct a physical experiment, [[thought experiment]]s can be conducted (for instance: breaking bonds, introducing impurities at specific sites, changing the local/global structure, or introducing external fields).  

Monte Carlo methods are used in various fields of computational biology, for example for Bayesian inference in phylogeny, or for studying biological systems such as genomes, proteins, or membranes. The systems can be studied in the coarse-grained or ''ab initio'' frameworks depending on the desired accuracy. Computer simulations allow us to monitor the local environment of a particular molecule to see if some chemical reaction is happening for instance. In cases where it is not feasible to conduct a physical experiment, thought experiments can be conducted (for instance: breaking bonds, introducing impurities at specific sites, changing the local/global structure, or introducing external fields).

Monte Carlo methods are used in various fields of computational biology, for example for Bayesian inference in phylogeny, or for studying biological systems such as genomes, proteins, or membranes. The systems can be studied in the coarse-grained or ''ab initio'' frameworks depending on the desired accuracy. Computer simulations allow us to monitor the local environment of a particular molecule to see if some chemical reaction is happening for instance. In cases where it is not feasible to conduct a physical experiment, thought experiments can be conducted (for instance: breaking bonds, introducing impurities at specific sites, changing the local/global structure, or introducing external fields).

蒙特卡罗方法被用于计算生物学的各个领域，例如在系统发育学中的贝叶斯推断，或者用于研究生物系统，例如基因组、蛋白质或膜。该系统可以在粗粒度或从头开始框架中研究，这取决于所需的准确性。计算机模拟使我们能够监测特定分子的局部环境，看看是否正在发生某种化学反应，例如。在无法进行物理实验的情况下，可以进行思维实验(例如: 打破键，在特定位置引入杂质，改变局部/全球结构，或引入外部场)。

蒙特卡罗方法被用于计算生物学的各个领域，例如在系统发育学中的贝叶斯推断，或者用于研究生物系统，例如基因组、蛋白质<ref name=":39" /> 或膜<ref name=":40" />。该系统可以在粗粒度或从头开始框架中研究，这取决于所需的准确性。计算机模拟使我们能够监测特定分子的局部环境，看看是否正在发生某种化学反应，例如。在无法进行物理实验的情况下，可以进行思维实验(例如: 打破键，在特定位置引入杂质，改变局部/全球结构，或引入外部场)。

===Computer graphics 计算机图形学===

===Computer graphics 计算机图形学===

===Applied statistics 应用统计学===

===Applied statistics 应用统计学===

The standards for Monte Carlo experiments in statistics were set by Sawilowsky.<ref>{{cite journal | last1 = Cassey | last2 = Smith | year = 2014 | title = Simulating confidence for the Ellison-Glaeser Index | url = | journal = Journal of Urban Economics | volume = 81 | issue = | page = 93 | doi =  10.1016/j.jue.2014.02.005}}</ref> In applied statistics, Monte Carlo methods may be used for at least four purposes:

The standards for Monte Carlo experiments in statistics were set by Sawilowsky.<ref name=":41">{{cite journal | last1 = Cassey | last2 = Smith | year = 2014 | title = Simulating confidence for the Ellison-Glaeser Index | url = | journal = Journal of Urban Economics | volume = 81 | issue = | page = 93 | doi =  10.1016/j.jue.2014.02.005}}</ref> In applied statistics, Monte Carlo methods may be used for at least four purposes:

The standards for Monte Carlo experiments in statistics were set by Sawilowsky. In applied statistics, Monte Carlo methods may be used for at least four purposes:

The standards for Monte Carlo experiments in statistics were set by Sawilowsky. In applied statistics, Monte Carlo methods may be used for at least four purposes:

蒙特卡罗方法的统计标准是由 Sawilowsky 制定的。在应用统计学中，蒙特卡罗方法至少可用于四种目的:

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:

#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>{{harvnb|Sawilowsky|Fahoome|2003}}</ref>

#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>

#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 i 型误差和统计的幂次特性可以计算从经典的理论分布(例如，正态曲线，柯西分布)的数据的渐近条件(即，无限大的样本量和无限小的处理效果) ，实际数据往往没有这样的分布。

#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 i 型误差和统计的幂次特性可以计算从经典的理论分布(例如，正态曲线，柯西分布)的数据的渐近条件(即，无限大的样本量和无限小的处理效果) ，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.

#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.

#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。这个样本然后估计和总结所有的基本特征后。

#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。这个样本然后估计和总结所有的基本特征后。

#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>{{Cite journal |doi = 10.1198/106186005X78800|title = Monte Carlo Computation of the Fisher Information Matrix in Nonstandard Settings|journal = Journal of Computational and Graphical Statistics|volume = 14|issue = 4|pages = 889–909|year = 2005|last1 = Spall|first1 = James C.|citeseerx = 10.1.1.142.738|s2cid = 16090098}}</ref><ref>{{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>

#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>

#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 Hessian 矩阵的有效的随机估计，这些估计的平均值可以形成费雪资讯矩阵的估计。

#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 Hessian 矩阵的有效的随机估计，这些估计的平均值可以形成费雪资讯矩阵的估计。<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).

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).

Monte Carlo methods have been developed into a technique called Monte-Carlo tree search that is useful for searching for the best move in a game. Possible moves are organized in a search tree and many random simulations are used to estimate the long-term potential of each move. A black box simulator represents the opponent's moves.

Monte Carlo methods have been developed into a technique called Monte-Carlo tree search that is useful for searching for the best move in a game. Possible moves are organized in a search tree and many random simulations are used to estimate the long-term potential of each move. A black box simulator represents the opponent's moves.

蒙特卡罗方法已经发展成为一种叫做蒙特卡洛树搜索的技术，它可以用来搜索游戏中的最佳移动。可能的移动被组织在一个搜索树和许多随机模拟被用来估计每个移动的长期潜力。一个黑盒模拟器代表对手的动作。

蒙特卡罗方法已经发展成为一种叫做蒙特卡洛树搜索的技术，它可以用来搜索游戏中的最佳移动。可能的移动被组织在一个搜索树和许多随机模拟被用来估计每个移动的长期潜力。一个黑盒模拟器代表对手的动作。<ref name=":45" />

The Monte Carlo tree search (MCTS) method has four steps:<ref>{{cite web|url=http://mcts.ai/about/index.html|title=Monte Carlo Tree Search - About|access-date=2013-05-15|archive-url=https://web.archive.org/web/20151129023043/http://mcts.ai/about/index.html|archive-date=2015-11-29|url-status=dead}}</ref>

The Monte Carlo tree search (MCTS) method has four steps:<ref name=":46">{{cite web|url=http://mcts.ai/about/index.html|title=Monte Carlo Tree Search - About|access-date=2013-05-15|archive-url=https://web.archive.org/web/20151129023043/http://mcts.ai/about/index.html|archive-date=2015-11-29|url-status=dead}}</ref>

The Monte Carlo tree search (MCTS) method has four steps:

The Monte Carlo tree search (MCTS) method has four steps:

蒙特卡罗树搜索(MCTS)方法有四个步骤:

蒙特卡罗树搜索(MCTS)方法有四个步骤：<ref name=":46" />

#Starting at root node of the tree, select optimal child nodes until a leaf node is reached.

#Starting at root node of the tree, select optimal child nodes until a leaf node is reached.

在许多模拟游戏过程中，净效应是代表移动的一个节点的值将上升或下降，希望与该节点是否代表一个好的移动相对应。

在许多模拟游戏过程中，净效应是代表移动的一个节点的值将上升或下降，希望与该节点是否代表一个好的移动相对应。

Monte Carlo Tree Search has been used successfully to play games such as [[Go (game)|Go]],<ref>{{cite book|title=Parallel Monte-Carlo Tree Search| doi=10.1007/978-3-540-87608-3_6|volume=5131|pages=60–71|series=Lecture Notes in Computer Science|year=2008|last1=Chaslot|first1=Guillaume M. J. -B|last2=Winands|first2=Mark H. M|last3=Van Den Herik|first3=H. Jaap|isbn=978-3-540-87607-6|citeseerx = 10.1.1.159.4373}}</ref> [[Tantrix]],<ref>{{cite report|url=https://www.tantrix.com/Tantrix/TRobot/MCTS%20Final%20Report.pdf|title=Monte-Carlo Tree Search in the game of Tantrix: Cosc490 Final Report|last=Bruns|first=Pete}}</ref> [[Battleship (game)|Battleship]],<ref>{{cite web|url=http://www0.cs.ucl.ac.uk/staff/D.Silver/web/Publications_files/pomcp.pdf|title=Monte-Carlo Planning in Large POMDPs|author1=David Silver|author2=Joel Veness|website=0.cs.ucl.ac.uk|accessdate=28 October 2017}}</ref> [[Havannah]],<ref>{{cite book|chapter=Improving Monte–Carlo Tree Search in Havannah| doi=10.1007/978-3-642-17928-0_10|volume=6515|pages=105–115|bibcode=2011LNCS.6515..105L|series=Lecture Notes in Computer Science|year=2011|last1=Lorentz|first1=Richard J|title=Computers and Games|isbn=978-3-642-17927-3}}</ref> and [[Arimaa]].<ref>{{cite web|url=http://www.arimaa.com/arimaa/papers/ThomasJakl/bc-thesis.pdf|author=Tomas Jakl|title=Arimaa challenge – comparison study of MCTS versus alpha-beta methods|website=Arimaa.com|accessdate=28 October 2017}}</ref>

Monte Carlo Tree Search has been used successfully to play games such as [[Go (game)|Go]],<ref name=":47">Chaslot, Guillaume M. J. -B; Winands, Mark H. M; Van Den Herik, H. Jaap (2008). ''Parallel Monte-Carlo Tree Search''. Lecture Notes in Computer Science. '''5131'''. pp. 60–71. CiteSeerX 10.1.1.159.4373. doi:10.1007/978-3-540-87608-3_6. ISBN <bdi>978-3-540-87607-6</bdi>.</ref> [[Tantrix]],<ref name=":48">Bruns, Pete. Monte-Carlo Tree Search in the game of Tantrix: Cosc490 Final Report (PDF) (Report).</ref> [[Battleship (game)|Battleship]],<ref name=":49">David Silver; Joel Veness. "Monte-Carlo Planning in Large POMDPs" (PDF). ''0.cs.ucl.ac.uk''. Retrieved 28 October 2017.</ref> [[Havannah]],<ref name=":50">Lorentz, Richard J (2011). "Improving Monte–Carlo Tree Search in Havannah". ''Computers and Games''. Lecture Notes in Computer Science. '''6515'''. pp. 105–115. Bibcode:2011LNCS.6515..105L. doi:10.1007/978-3-642-17928-0_10. ISBN <bdi>978-3-642-17927-3</bdi>.</ref> and [[Arimaa]].<ref name=":51">Tomas Jakl. "Arimaa challenge – comparison study of MCTS versus alpha-beta methods" (PDF). ''Arimaa.com''. Retrieved 28 October 2017.</ref>

Monte Carlo Tree Search has been used successfully to play games such as Go, Tantrix, Battleship, Havannah, and Arimaa.

Monte Carlo Tree Search has been used successfully to play games such as Go, Tantrix, Battleship, Havannah, and Arimaa.

蒙特卡洛树搜索已成功地用于游戏，如围棋，Tantrix，战舰，Havannah，和 Arimaa。

蒙特卡洛树搜索已成功地用于游戏，如围棋，<ref name=":47" /> Tantrix，<ref name=":48" /> 战舰，<ref name=":49" /> Havannah，<ref name=":50" />和 Arimaa。<ref name=":51" />

{{See also|Computer Go}}

{{See also|Computer Go}}

===Design and visuals 设计与视觉效果===

===Design and visuals 设计与视觉效果===

Monte Carlo methods are also efficient in solving coupled integral differential equations of radiation fields and energy transport, and thus these methods have been used in [[global illumination]] computations that produce photo-realistic images of virtual 3D models, with applications in [[video game]]s, [[architecture]], [[design]], computer generated [[film]]s, and cinematic special effects.<ref>{{harvnb|Szirmay–Kalos|2008}}</ref>

Monte Carlo methods are also efficient in solving coupled integral differential equations of radiation fields and energy transport, and thus these methods have been used in [[global illumination]] computations that produce photo-realistic images of virtual 3D models, with applications in [[video game]]s, [[architecture]], [[design]], computer generated [[film]]s, and cinematic special effects.<ref name=":53">Szirmay–Kalos 2008</ref>

Monte Carlo methods are also efficient in solving coupled integral differential equations of radiation fields and energy transport, and thus these methods have been used in global illumination computations that produce photo-realistic images of virtual 3D models, with applications in video games, architecture, design, computer generated films, and cinematic special effects.

Monte Carlo methods are also efficient in solving coupled integral differential equations of radiation fields and energy transport, and thus these methods have been used in global illumination computations that produce photo-realistic images of virtual 3D models, with applications in video games, architecture, design, computer generated films, and cinematic special effects.

蒙特卡罗方法在解决辐射场和能量传输的耦合积分微分方程方面也很有效，因此这些方法已经被用于全局光源计算，产生虚拟3 d 模型的照片般逼真的图像，应用于视频游戏、建筑、设计、计算机生成的电影和电影特效。

蒙特卡罗方法在解决辐射场和能量传输的耦合积分微分方程方面也很有效，因此这些方法已经被用于全局光源计算，产生虚拟3 d 模型的照片般逼真的图像，应用于视频游戏、建筑、设计、计算机生成的电影和电影特效。<ref name=":53" />

===Search and rescue 搜寻与救援===

===Search and rescue 搜寻与救援===

The [[US Coast Guard]] utilizes Monte Carlo methods within its computer modeling software [[SAROPS]] in order to calculate the probable locations of vessels during [[search and rescue]] operations. Each simulation can generate as many as ten thousand data points that are randomly distributed based upon provided variables.<ref>{{cite web|url=http://insights.dice.com/2014/01/03/how-the-coast-guard-uses-analytics-to-search-for-those-lost-at-sea|title=How the Coast Guard Uses Analytics to Search for Those Lost at Sea|work=Dice Insights|date=2014-01-03}}</ref> Search patterns are then generated based upon extrapolations of these data in order to optimize the probability of containment (POC) and the probability of detection (POD), which together will equal an overall probability of success (POS). Ultimately this serves as a practical application of [[probability distribution]] in order to provide the swiftest and most expedient method of rescue, saving both lives and resources.<ref>{{cite web|url=http://www.ifremer.fr/web-com/sar2011/Presentations/SARWS2011_STONE_L.pdf|title=Search Modeling and Optimization in USCG's Search and Rescue Optimal Planning System (SAROPS)|author1=Lawrence D. Stone|author2=Thomas M. Kratzke|author3=John R. Frost|website=Ifremer.fr|accessdate=28 October 2017}}</ref>

The [[US Coast Guard]] utilizes Monte Carlo methods within its computer modeling software [[SAROPS]] in order to calculate the probable locations of vessels during [[search and rescue]] operations. Each simulation can generate as many as ten thousand data points that are randomly distributed based upon provided variables.<ref name=":54">"How the Coast Guard Uses Analytics to Search for Those Lost at Sea". ''Dice Insights''. 2014-01-03.</ref> Search patterns are then generated based upon extrapolations of these data in order to optimize the probability of containment (POC) and the probability of detection (POD), which together will equal an overall probability of success (POS). Ultimately this serves as a practical application of [[probability distribution]] in order to provide the swiftest and most expedient method of rescue, saving both lives and resources.<ref name=":55">Lawrence D. Stone; Thomas M. Kratzke; John R. Frost. "Search Modeling and Optimization in USCG's Search and Rescue Optimal Planning System (SAROPS)" (PDF). ''Ifremer.fr''. Retrieved 28 October 2017.</ref>

The US Coast Guard utilizes Monte Carlo methods within its computer modeling software SAROPS in order to calculate the probable locations of vessels during search and rescue operations. Each simulation can generate as many as ten thousand data points that are randomly distributed based upon provided variables. Search patterns are then generated based upon extrapolations of these data in order to optimize the probability of containment (POC) and the probability of detection (POD), which together will equal an overall probability of success (POS). Ultimately this serves as a practical application of probability distribution in order to provide the swiftest and most expedient method of rescue, saving both lives and resources.

The US Coast Guard utilizes Monte Carlo methods within its computer modeling software SAROPS in order to calculate the probable locations of vessels during search and rescue operations. Each simulation can generate as many as ten thousand data points that are randomly distributed based upon provided variables. Search patterns are then generated based upon extrapolations of these data in order to optimize the probability of containment (POC) and the probability of detection (POD), which together will equal an overall probability of success (POS). Ultimately this serves as a practical application of probability distribution in order to provide the swiftest and most expedient method of rescue, saving both lives and resources.

美国海岸警卫队在其计算机建模软件 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。然后根据这些数据的推断生成搜索模式，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，这作为概率分布的一个实际应用，以提供最迅速和最便捷的救援方法，拯救生命和资源。

美国海岸警卫队在其计算机建模软件 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" /> 然后根据这些数据的推断生成搜索模式，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，这作为概率分布的一个实际应用，以提供最迅速和最便捷的救援方法，拯救生命和资源。<ref name=":55" />

===Finance and business 金融与商业===

===Finance and business 金融与商业===

{{See also|Monte Carlo methods in finance| Quasi-Monte Carlo methods in finance| Monte Carlo methods for option pricing| Stochastic modelling (insurance) | Stochastic asset model}}

{{See also|Monte Carlo methods in finance| Quasi-Monte Carlo methods in finance| Monte Carlo methods for option pricing| Stochastic modelling (insurance) | Stochastic asset model}}

[[Monte Carlo methods in finance]] are often used to [[Corporate finance#Quantifying uncertainty|evaluate investments in projects]] at a business unit or corporate level, or other financial valuations. They can be used to model [[project management|project schedules]], where simulations aggregate estimates for worst-case, best-case, and most likely durations for each task to determine outcomes for the overall project.[https://risk.octigo.pl/] Monte Carlo methods are also used in option pricing, default risk analysis.<ref>{{Cite book|title = An Introduction to Particle Methods with Financial Applications|publisher = Springer Berlin Heidelberg|journal = Numerical Methods in Finance|date = 2012|isbn = 978-3-642-25745-2|pages = 3–49|series = Springer Proceedings in Mathematics|volume = 12|first1 = René|last1 = Carmona|first2 = Pierre|last2 = Del Moral|first3 = Peng|last3 = Hu|first4 = Nadia|last4 = Oudjane|editor-first = René A.|editor-last = Carmona|editor2-first = Pierre Del|editor2-last = Moral|editor3-first = Peng|editor3-last = Hu|editor4-first = Nadia|display-editors = 3 |editor4-last = Oudjane|doi=10.1007/978-3-642-25746-9_1|citeseerx = 10.1.1.359.7957}}</ref><ref>{{Cite book |volume = 12|doi=10.1007/978-3-642-25746-9|series = Springer Proceedings in Mathematics|year = 2012|isbn = 978-3-642-25745-2|url = https://basepub.dauphine.fr/handle/123456789/11498|title=Numerical Methods in Finance|last1=Carmona|first1=René|last2=Del Moral|first2=Pierre|last3=Hu|first3=Peng|last4=Oudjane|first4=Nadia}}</ref><ref name="kr11">{{cite book|last1 = Kroese|first1 = D. P.|last2 = Taimre|first2 = T.|last3 = Botev|first3 = Z. I. |title = Handbook of Monte Carlo Methods|year = 2011|publisher = John Wiley & Sons}}</ref> Additionally, they can be used to estimate the financial impact of medical interventions.<ref>{{Cite journal |doi = 10.1371/journal.pone.0189718|pmid = 29284026|pmc = 5746244|title = A Monte Carlo simulation approach for estimating the health and economic impact of interventions provided at a student-run clinic|journal = PLOS ONE|volume = 12|issue = 12|pages = e0189718|year = 2017|last1 = Arenas|first1 = Daniel J.|last2 = Lett|first2 = Lanair A.|last3 = Klusaritz|first3 = Heather|last4 = Teitelman|first4 = Anne M.|bibcode = 2017PLoSO..1289718A}}</ref>

[[Monte Carlo methods in finance]] are often used to [[Corporate finance#Quantifying uncertainty|evaluate investments in projects]] at a business unit or corporate level, or other financial valuations. They can be used to model [[project management|project schedules]], where simulations aggregate estimates for worst-case, best-case, and most likely durations for each task to determine outcomes for the overall project.[https://risk.octigo.pl/] Monte Carlo methods are also used in option pricing, default risk analysis.<ref name=":56">Carmona, René; Del Moral, Pierre; Hu, Peng; Oudjane, Nadia (2012). Carmona, René A.; Moral, Pierre Del; Hu, Peng; et al. (eds.). ''An Introduction to Particle Methods with Financial Applications''. ''Numerical Methods in Finance''. Springer Proceedings in Mathematics. '''12'''. Springer Berlin Heidelberg. pp. 3–49. CiteSeerX 10.1.1.359.7957. doi:10.1007/978-3-642-25746-9_1. ISBN <bdi>978-3-642-25745-2</bdi>.</ref><ref name=":57">Carmona, René; Del Moral, Pierre; Hu, Peng; Oudjane, Nadia (2012). ''Numerical Methods in Finance''. Springer Proceedings in Mathematics. '''12'''. doi:10.1007/978-3-642-25746-9. ISBN <bdi>978-3-642-25745-2</bdi>.</ref><ref name="kr11">Kroese, D. P.; Taimre, T.; Botev, Z. I. (2011). ''Handbook of Monte Carlo Methods''. John Wiley & Sons.</ref> Additionally, they can be used to estimate the financial impact of medical interventions.<ref name=":58">Arenas, Daniel J.; Lett, Lanair A.; Klusaritz, Heather; Teitelman, Anne M. (2017). "A Monte Carlo simulation approach for estimating the health and economic impact of interventions provided at a student-run clinic". ''PLOS ONE''. '''12''' (12): e0189718. Bibcode:2017PLoSO..1289718A. doi:10.1371/journal.pone.0189718. PMC 5746244. <nowiki>PMID 29284026</nowiki>.</ref>

===Law===

===Law===

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">{{cite web|url=http://legalaidresearch.org/wp-content/uploads/Research-Increasing-Access-to-REstraining-Order-for-Low-Income-Victims-of-DV-A-Cost-Benefit-Analysis-of-the-Proposed-Domestic-Abuse-Grant-Program.pdf| title=Increasing Access to Restraining Orders for Low Income Victims of Domestic Violence: A Cost-Benefit Analysis of the Proposed Domestic Abuse Grant Program |publisher=[[State Bar of Wisconsin]] |date=December 2006 |accessdate=2016-12-12|last1=Elwart|first1=Liz|last2=Emerson|first2=Nina|last3=Enders|first3=Christina|last4=Fumia|first4=Dani|last5=Murphy|first5=Kevin|url-status=dead|archive-url=https://web.archive.org/web/20181106220526/https://legalaidresearch.org/wp-content/uploads/Research-Increasing-Access-to-REstraining-Order-for-Low-Income-Victims-of-DV-A-Cost-Benefit-Analysis-of-the-Proposed-Domestic-Abuse-Grant-Program.pdf|archive-date=6 November 2018}}</ref>

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>

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 and 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.

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 and 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.

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。

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" />

==Use in mathematics==

==Use in mathematics==

错误减少一个因素 < math > scriptstyle 1/sqrt { n } <nowiki></math ></nowiki>|链接=Special:FilePath/Monte-Carlo_method_(errors).png]]

错误减少一个因素 < math > scriptstyle 1/sqrt { n } <nowiki></math ></nowiki>|链接=Special:FilePath/Monte-Carlo_method_(errors).png]]

Deterministic [[numerical integration]] algorithms work well in a small number of dimensions, but encounter two problems when the functions have many variables. First, the number of function evaluations needed increases rapidly with the number of dimensions. For example, if 10 evaluations provide adequate accuracy in one dimension, then [[googol|10¹⁰⁰]] points are needed for 100 dimensions—far too many to be computed. This is called the [[curse of dimensionality]]. Second, the boundary of a multidimensional region may be very complicated, so it may not be feasible to reduce the problem to an [[iterated integral]].<ref name="Press">{{harvnb|Press|Teukolsky|Vetterling|Flannery|1996}}</ref> 100 [[dimension]]s is by no means unusual, since in many physical problems, a "dimension" is equivalent to a [[degrees of freedom (physics and chemistry)|degree of freedom]].

Deterministic [[numerical integration]] algorithms work well in a small number of dimensions, but encounter two problems when the functions have many variables. First, the number of function evaluations needed increases rapidly with the number of dimensions. For example, if 10 evaluations provide adequate accuracy in one dimension, then [[googol|10¹⁰⁰]] points are needed for 100 dimensions—far too many to be computed. This is called the [[curse of dimensionality]]. Second, the boundary of a multidimensional region may be very complicated, so it may not be feasible to reduce the problem to an [[iterated integral]].<ref name="Press">Press, William H.; Teukolsky, Saul A.; Vetterling, William T.; Flannery, Brian P. (1996) [1986]. ''Numerical Recipes in Fortran 77: The Art of Scientific Computing''. Fortran Numerical Recipes. '''1''' (2nd ed.). Cambridge University Press. ISBN <bdi>978-0-521-43064-7</bdi>.</ref> 100 [[dimension]]s is by no means unusual, since in many physical problems, a "dimension" is equivalent to a [[degrees of freedom (physics and chemistry)|degree of freedom]].

Deterministic numerical integration algorithms work well in a small number of dimensions, but encounter two problems when the functions have many variables. First, the number of function evaluations needed increases rapidly with the number of dimensions. For example, if 10 evaluations provide adequate accuracy in one dimension, then 10¹⁰⁰ points are needed for 100 dimensions—far too many to be computed. This is called the curse of dimensionality. Second, the boundary of a multidimensional region may be very complicated, so it may not be feasible to reduce the problem to an iterated integral. 100 dimensions is by no means unusual, since in many physical problems, a "dimension" is equivalent to a degree of freedom.

Deterministic numerical integration algorithms work well in a small number of dimensions, but encounter two problems when the functions have many variables. First, the number of function evaluations needed increases rapidly with the number of dimensions. For example, if 10 evaluations provide adequate accuracy in one dimension, then 10¹⁰⁰ points are needed for 100 dimensions—far too many to be computed. This is called the curse of dimensionality. Second, the boundary of a multidimensional region may be very complicated, so it may not be feasible to reduce the problem to an iterated integral. 100 dimensions is by no means unusual, since in many physical problems, a "dimension" is equivalent to a degree of freedom.

确定性数值积分算法在少数维上运行良好，但在函数具有多个变量时会遇到两个问题。首先，随着维数的增加，需要进行的功能评估的数量迅速增加。例如，如果10个评估在一个维度上提供了足够的精确度，那么100个维度需要10个 < sup > 100  点，这太多了以至于无法计算。这就是所谓的维数灾难。其次，多维区域的边界可能非常复杂，因此将问题简化为迭代积分可能是不可行的。100维绝对不是不寻常的，因为在许多物理问题中，一个“维度”等同于一个自由度。

确定性数值积分算法在少数维上运行良好，但在函数具有多个变量时会遇到两个问题。首先，随着维数的增加，需要进行的功能评估的数量迅速增加。例如，如果10个评估在一个维度上提供了足够的精确度，那么100个维度需要10个 < sup > 100  点，这太多了以至于无法计算。这就是所谓的维数灾难。其次，多维区域的边界可能非常复杂，因此将问题简化为迭代积分可能是不可行的。<ref name="Press" /> 100维绝对不是不寻常的，因为在许多物理问题中，一个“维度”等同于一个自由度。

Monte Carlo methods provide a way out of this exponential increase in computation time. As long as the function in question is reasonably [[well-behaved]], it can be estimated by randomly selecting points in 100-dimensional space, and taking some kind of average of the function values at these points. By the [[central limit theorem]], this method displays <math>\scriptstyle 1/\sqrt{N}</math> convergence—i.e., quadrupling the number of sampled points halves the error, regardless of the number of dimensions.<ref name="Press" />

Monte Carlo methods provide a way out of this exponential increase in computation time. As long as the function in question is reasonably [[well-behaved]], it can be estimated by randomly selecting points in 100-dimensional space, and taking some kind of average of the function values at these points. By the [[central limit theorem]], this method displays <math>\scriptstyle 1/\sqrt{N}</math> convergence—i.e., quadrupling the number of sampled points halves the error, regardless of the number of dimensions.<ref name="Press" />

Monte Carlo methods provide a way out of this exponential increase in computation time. As long as the function in question is reasonably well-behaved, it can be estimated by randomly selecting points in 100-dimensional space, and taking some kind of average of the function values at these points. By the central limit theorem, this method displays <math>\scriptstyle 1/\sqrt{N}</math> convergence—i.e., quadrupling the number of sampled points halves the error, regardless of the number of dimensions. or the VEGAS algorithm.

Monte Carlo methods provide a way out of this exponential increase in computation time. As long as the function in question is reasonably well-behaved, it can be estimated by randomly selecting points in 100-dimensional space, and taking some kind of average of the function values at these points. By the central limit theorem, this method displays <math>\scriptstyle 1/\sqrt{N}</math> convergence—i.e., quadrupling the number of sampled points halves the error, regardless of the number of dimensions.  

蒙特卡罗方法提供了一种方法来摆脱这种指数增长的计算时间。只要所涉及的函数具有合理的性质，就可以在100维空间中随机选取一些点，并在这些点上取某种函数值的平均值来估计。通过中心极限定理，这个方法显示 < math > scriptstyle 1/sqrt { n } <nowiki></math ></nowiki> 收敛，即，不管维数多少，将采样点的数目翻两番，误差减半。或者拉斯维加斯算法。

蒙特卡罗方法提供了一种方法来摆脱这种指数增长的计算时间。只要所涉及的函数具有合理的性质，就可以在100维空间中随机选取一些点，并在这些点上取某种函数值的平均值来估计。通过中心极限定理，这个方法显示 < math > scriptstyle 1/sqrt { n } <nowiki></math ></nowiki> 收敛，即，不管维数多少，将采样点的数目翻两番，误差减半。<ref name="Press" />

A refinement of this method, known as [[importance sampling]] in statistics, involves sampling the points randomly, but more frequently where the integrand is large. To do this precisely one would have to already know the integral, but one can approximate the integral by an integral of a similar function or use adaptive routines such as [[stratified sampling]], [[Monte Carlo integration#Recursive stratified sampling|recursive stratified sampling]], adaptive umbrella sampling<ref>{{cite journal|last=MEZEI|first=M|title=Adaptive umbrella sampling: Self-consistent determination of the non-Boltzmann bias|journal=Journal of Computational Physics|date=31 December 1986|volume=68|issue=1|pages=237–248|doi=10.1016/0021-9991(87)90054-4|bibcode = 1987JCoPh..68..237M}}</ref><ref>{{cite journal|last1=Bartels|first1=Christian|last2=Karplus|first2=Martin|title=Probability Distributions for Complex Systems: Adaptive Umbrella Sampling of the Potential Energy|journal=The Journal of Physical Chemistry B|date=31 December 1997|volume=102|issue=5|pages=865–880|doi=10.1021/jp972280j}}</ref> or the [[VEGAS algorithm]].

A refinement of this method, known as [[importance sampling]] in statistics, involves sampling the points randomly, but more frequently where the integrand is large. To do this precisely one would have to already know the integral, but one can approximate the integral by an integral of a similar function or use adaptive routines such as [[stratified sampling]], [[Monte Carlo integration#Recursive stratified sampling|recursive stratified sampling]], adaptive umbrella sampling<ref name=":59">MEZEI, M (31 December 1986). "Adaptive umbrella sampling: Self-consistent determination of the non-Boltzmann bias". ''Journal of Computational Physics''. '''68''' (1): 237–248. Bibcode:1987JCoPh..68..237M. doi:10.1016/0021-9991(87)90054-4.</ref><ref name=":60">Bartels, Christian; Karplus, Martin (31 December 1997). "Probability Distributions for Complex Systems: Adaptive Umbrella Sampling of the Potential Energy". ''The Journal of Physical Chemistry B''. '''102''' (5): 865–880. doi:10.1021/jp972280j.</ref> or the [[VEGAS algorithm]].

A similar approach, the [[quasi-Monte Carlo method]], uses [[low-discrepancy sequence]]s. These sequences "fill" the area better and sample the most important points more frequently, so quasi-Monte Carlo methods can often converge on the integral more quickly.

A similar approach, the [[quasi-Monte Carlo method]], uses [[low-discrepancy sequence]]s. These sequences "fill" the area better and sample the most important points more frequently, so quasi-Monte Carlo methods can often converge on the integral more quickly.

一个类似的方法，拟蒙特卡罗方法，使用低差异序列。这些序列能更好地“填充”区域，更频繁地采样最重要的点，因此拟蒙特卡罗方法往往能更快地收敛于积分。

一个类似的方法，拟蒙特卡罗方法，使用低差异序列。这些序列能更好地“填充”区域，更频繁地采样最重要的点，因此拟蒙特卡罗方法往往能更快地收敛于积分。

Another class of methods for sampling points in a volume is to simulate random walks over it ([[Markov chain Monte Carlo]]). Such methods include the [[Metropolis–Hastings algorithm]], [[Gibbs sampling]], [[Wang and Landau algorithm]], and interacting type MCMC methodologies such as the [[Particle filter|sequential Monte Carlo]] samplers.<ref>{{Cite journal|title = Sequential Monte Carlo samplers|journal = Journal of the Royal Statistical Society, Series B|doi=10.1111/j.1467-9868.2006.00553.x|volume=68|issue = 3|pages=411–436|year = 2006|last1 = Del Moral|first1 = Pierre|last2 = Doucet|first2 = Arnaud|last3 = Jasra|first3 = Ajay|arxiv = cond-mat/0212648|s2cid = 12074789}}</ref>

Another class of methods for sampling points in a volume is to simulate random walks over it ([[Markov chain Monte Carlo]]). Such methods include the [[Metropolis–Hastings algorithm]], [[Gibbs sampling]], [[Wang and Landau algorithm]], and interacting type MCMC methodologies such as the [[Particle filter|sequential Monte Carlo]] samplers.<ref name=":61">Del Moral, Pierre; Doucet, Arnaud; Jasra, Ajay (2006). "Sequential Monte Carlo samplers". ''Journal of the Royal Statistical Society, Series B''. '''68''' (3): 411–436. arXiv:cond-mat/0212648. doi:10.1111/j.1467-9868.2006.00553.x. S2CID 12074789.</ref>

Another class of methods for sampling points in a volume is to simulate random walks over it (Markov chain Monte Carlo). Such methods include the Metropolis–Hastings algorithm, Gibbs sampling, Wang and Landau algorithm, and interacting type MCMC methodologies such as the sequential Monte Carlo samplers.

Another class of methods for sampling points in a volume is to simulate random walks over it (Markov chain Monte Carlo). Such methods include the Metropolis–Hastings algorithm, Gibbs sampling, Wang and Landau algorithm, and interacting type MCMC methodologies such as the sequential Monte Carlo samplers.

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。

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" />

=== Simulation and optimization ===

=== Simulation and optimization ===

{{Main|Stochastic optimization}}

{{Main|Stochastic optimization}}

Another powerful and very popular application for random numbers in numerical simulation is in [[Optimization (mathematics)|numerical optimization]]. The problem is to minimize (or maximize) functions of some vector that often has many dimensions. Many problems can be phrased in this way: for example, a [[computer chess]] program could be seen as trying to find the set of, say, 10 moves that produces the best evaluation function at the end. In the [[traveling salesman problem]] the goal is to minimize distance traveled. There are also applications to engineering design, such as [[multidisciplinary design optimization]]. It has been applied with quasi-one-dimensional models to solve particle dynamics problems by efficiently exploring large configuration space. Reference<ref>Spall, J. C. (2003), ''Introduction to Stochastic Search and Optimization: Estimation, Simulation, and Control'', Wiley, Hoboken, NJ. http://www.jhuapl.edu/ISSO</ref> is a comprehensive review of many issues related to simulation and optimization.

Another powerful and very popular application for random numbers in numerical simulation is in [[Optimization (mathematics)|numerical optimization]]. The problem is to minimize (or maximize) functions of some vector that often has many dimensions. Many problems can be phrased in this way: for example, a [[computer chess]] program could be seen as trying to find the set of, say, 10 moves that produces the best evaluation function at the end. In the [[traveling salesman problem]] the goal is to minimize distance traveled. There are also applications to engineering design, such as [[multidisciplinary design optimization]]. It has been applied with quasi-one-dimensional models to solve particle dynamics problems by efficiently exploring large configuration space. Reference<ref name=":62">Spall, J. C. (2003), ''Introduction to Stochastic Search and Optimization: Estimation, Simulation, and Control'', Wiley, Hoboken, NJ. http://www.jhuapl.edu/ISSO</ref> is a comprehensive review of many issues related to simulation and optimization.

Another powerful and very popular application for random numbers in numerical simulation is in numerical optimization. The problem is to minimize (or maximize) functions of some vector that often has many dimensions. Many problems can be phrased in this way: for example, a computer chess program could be seen as trying to find the set of, say, 10 moves that produces the best evaluation function at the end. In the traveling salesman problem the goal is to minimize distance traveled. There are also applications to engineering design, such as multidisciplinary design optimization. It has been applied with quasi-one-dimensional models to solve particle dynamics problems by efficiently exploring large configuration space. Reference is a comprehensive review of many issues related to simulation and optimization.

Another powerful and very popular application for random numbers in numerical simulation is in numerical optimization. The problem is to minimize (or maximize) functions of some vector that often has many dimensions. Many problems can be phrased in this way: for example, a computer chess program could be seen as trying to find the set of, say, 10 moves that produces the best evaluation function at the end. In the traveling salesman problem the goal is to minimize distance traveled. There are also applications to engineering design, such as multidisciplinary design optimization. It has been applied with quasi-one-dimensional models to solve particle dynamics problems by efficiently exploring large configuration space. Reference is a comprehensive review of many issues related to simulation and optimization.

另一个强大的和非常流行的应用随机数在数值模拟是在数值优化。问题在于如何最小化(或最大化)某些向量的函数，这些向量通常具有多个维度。许多问题可以这样表述: 例如，一个计算机国际象棋程序可以被视为试图找到一组，比如说，10步棋，最终产生最好的评价函数。在旅行商问题中，目标是使旅行距离最小。在工程设计中也有一些应用，如多学科设计优化。它已被应用于准一维模型，以解决粒子动力学问题，有效地探索大型位形空间。参考文献是对许多与模拟和优化有关的问题的全面回顾。

另一个强大的和非常流行的应用随机数在数值模拟是在数值优化。问题在于如何最小化(或最大化)某些向量的函数，这些向量通常具有多个维度。许多问题可以这样表述: 例如，一个计算机国际象棋程序可以被视为试图找到一组，比如说，10步棋，最终产生最好的评价函数。在旅行商问题中，目标是使旅行距离最小。在工程设计中也有一些应用，如多学科设计优化。它已被应用于准一维模型，以解决粒子动力学问题，有效地探索大型位形空间。参考文献<ref name=":62" />是对许多与模拟和优化有关的问题的全面回顾。

The [[traveling salesman problem]] is what is called a conventional optimization problem. That is, all the facts (distances between each destination point) needed to determine the optimal path to follow are known with certainty and the goal is to run through the possible travel choices to come up with the one with the lowest total distance. However, let's assume that instead of wanting to minimize the total distance traveled to visit each desired destination, we wanted to minimize the total time needed to reach each destination. This goes beyond conventional optimization since travel time is inherently uncertain (traffic jams, time of day, etc.). As a result, to determine our optimal path we would want to use simulation - optimization to first understand the range of potential times it could take to go from one point to another (represented by a probability distribution in this case rather than a specific distance) and then optimize our travel decisions to identify the best path to follow taking that uncertainty into account.

The [[traveling salesman problem]] is what is called a conventional optimization problem. That is, all the facts (distances between each destination point) needed to determine the optimal path to follow are known with certainty and the goal is to run through the possible travel choices to come up with the one with the lowest total distance. However, let's assume that instead of wanting to minimize the total distance traveled to visit each desired destination, we wanted to minimize the total time needed to reach each destination. This goes beyond conventional optimization since travel time is inherently uncertain (traffic jams, time of day, etc.). As a result, to determine our optimal path we would want to use simulation - optimization to first understand the range of potential times it could take to go from one point to another (represented by a probability distribution in this case rather than a specific distance) and then optimize our travel decisions to identify the best path to follow taking that uncertainty into account.

当分析一个反问题时，获得一个最大似然模型通常是不够的，因为我们通常也希望有关于数据的分辨率的信息。在一般情况下，我们可能有许多模型参数，检查的边际概率密度的兴趣可能是不切实际的，甚至无用的。但是，根据《后验概率可以伪随机生成大量的模型集合，并以这样一种方式分析和显示模型，模型属性的相对可能性信息被传达给观众，这是可能的。这可以通过一个有效的蒙特卡罗方法安全管理系统来实现，即使在没有黎曼显式公式安全管理先验概率的情况下。

当分析一个反问题时，获得一个最大似然模型通常是不够的，因为我们通常也希望有关于数据的分辨率的信息。在一般情况下，我们可能有许多模型参数，检查的边际概率密度的兴趣可能是不切实际的，甚至无用的。但是，根据《后验概率可以伪随机生成大量的模型集合，并以这样一种方式分析和显示模型，模型属性的相对可能性信息被传达给观众，这是可能的。这可以通过一个有效的蒙特卡罗方法安全管理系统来实现，即使在没有黎曼显式公式安全管理先验概率的情况下。

The best-known importance sampling method, the Metropolis algorithm, can be generalized, and this gives a method that allows analysis of (possibly highly nonlinear) inverse problems with complex ''a priori'' information and data with an arbitrary noise distribution.<ref>{{harvnb|Mosegaard|Tarantola|1995}}</ref><ref>{{harvnb|Tarantola|2005}}</ref>

The best-known importance sampling method, the Metropolis algorithm, can be generalized, and this gives a method that allows analysis of (possibly highly nonlinear) inverse problems with complex ''a priori'' information and data with an arbitrary noise distribution.<ref name=":63">Mosegaard, Klaus; Tarantola, Albert (1995). "Monte Carlo sampling of solutions to inverse problems" (PDF). ''J. Geophys. Res''. '''100''' (B7): 12431–12447. Bibcode:1995JGR...10012431M. doi:10.1029/94JB03097.</ref><ref name=":64">Tarantola, Albert (2005). ''Inverse Problem Theory''. Philadelphia: Society for Industrial and Applied Mathematics. ISBN <bdi>978-0-89871-572-9</bdi>.</ref>

The best-known importance sampling method, the Metropolis algorithm, can be generalized, and this gives a method that allows analysis of (possibly highly nonlinear) inverse problems with complex a priori information and data with an arbitrary noise distribution.

The best-known importance sampling method, the Metropolis algorithm, can be generalized, and this gives a method that allows analysis of (possibly highly nonlinear) inverse problems with complex a priori information and data with an arbitrary noise distribution.

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。

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" />

===Philosophy===

===Philosophy===

Popular exposition of the Monte Carlo Method was conducted by McCracken<ref>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>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>Grüne-Yanoff, T., & Weirich, P. (2010). The philosophy and epistemology of simulation: A review, Simulation & Gaming, 41(1), pp. 20-50</ref>.

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>.

Popular exposition of the Monte Carlo Method was conducted by McCracken. Method's general philosophy was discussed by Elishakoff and Grüne-Yanoff and Weirich.

Popular exposition of the Monte Carlo Method was conducted by McCracken. Method's general philosophy was discussed by Elishakoff and Grüne-Yanoff and Weirich.

由 McCracken 主持的蒙特卡罗方法博览会的普及展览。方法的一般哲学由 Elishakoff、 Grüne-Yanoff 和 weurich 讨论。

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" />

== See also ==

== See also ==