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'''蒙特卡罗方法''' '''Monte Carlo methods'''，或称'''蒙特卡罗实验''' '''Monte Carlo experiments'''，是一大类计算算法的集合，它们依赖于重复的随机抽样来获得数值结果。其基本概念是利用随机性来解决原则上可能是确定性的问题。它们通常用于物理和数学问题，当问题很棘手或无法使用其他方法时，往往它们最有用。蒙特卡罗方法主要用于3个问题类: 优化，数值积分，依据概率分布生成图像。

'''蒙特卡罗方法''' '''Monte Carlo methods'''，或称'''蒙特卡罗实验''' '''Monte Carlo experiments'''，是一大类计算算法的集合，它们依赖于重复的随机抽样来获得数值结果。其基本概念是利用随机性来解决原则上可能是确定性的问题。它们通常用于物理和数学问题，当问题很棘手或无法使用其他方法时，往往它们最有用。蒙特卡罗方法主要用于3个问题类: 优化，数值积分，依据概率分布生成图像。

In physics-related problems, Monte Carlo methods are useful for simulating systems with many [[coupling (physics)|coupled]] [[degrees of freedom]], such as fluids, disordered materials, strongly coupled solids, and cellular structures (see [[cellular Potts model]], [[interacting particle systems]], [[McKean–Vlasov process]]es, [[kinetic theory of gases|kinetic models of gases]]).

In physics-related problems, Monte Carlo methods are useful for simulating systems with many [[coupling (physics)|coupled]] [[degrees of freedom]], such as fluids, disordered materials, strongly coupled solids, and cellular structures (see [[cellular Potts model]], [[interacting particle systems]], [[McKean–Vlasov process]]es, [[kinetic theory of gases|kinetic models of gases]]).

In physics-related problems, Monte Carlo methods are useful for simulating systems with many coupled degrees of freedom, such as fluids, disordered materials, strongly coupled solids, and cellular structures (see cellular Potts model, interacting particle systems, McKean–Vlasov processes, kinetic models of gases).

In physics-related problems, Monte Carlo methods are useful for simulating systems with many coupled degrees of freedom, such as fluids, disordered materials, strongly coupled solids, and cellular structures (see cellular Potts model, interacting particle systems, McKean–Vlasov processes, kinetic models of gases).

在与物理相关的问题中，蒙特卡罗方法可用于模拟具有多个耦合自由度的系统，如流体、无序材料、强耦合固体和细胞结构(参见细胞波茨模型、相互作用粒子系统、 McKean-Vlasov 过程、气体动力学模型)。

在物理相关的问题中，蒙特卡罗方法可用于模拟具有多个耦合自由度的系统，如流体、无序材料、强耦合固体和细胞结构(参见细胞波茨模型、相互作用粒子系统、 '''麦肯-弗拉索夫 McKean-Vlasov''' 过程、气体动力学模型)。

 

 

Other examples include modeling phenomena with significant [[uncertainty]] in inputs such as the calculation of [[risk]] in business and, in mathematics, evaluation of multidimensional [[Integral|definite integral]]s with complicated [[boundary conditions]].  In application to systems engineering problems (space, [[oil exploration]], aircraft design, etc.), Monte Carlo&ndash;based predictions of failure, [[cost overrun]]s and schedule overruns are routinely better than human intuition or alternative "soft" methods.<ref>{{cite journal|last1 = Hubbard|first1 = Douglas|last2=Samuelson|first2 = Douglas A. |date = October 2009 |title=Modeling Without Measurements |url=http://viewer.zmags.com/publication/357348e6#/357348e6/28|journal = OR/MS Today|pages = 28–33}}</ref>

Other examples include modeling phenomena with significant [[uncertainty]] in inputs such as the calculation of [[risk]] in business and, in mathematics, evaluation of multidimensional [[Integral|definite integral]]s with complicated [[boundary conditions]].  In application to systems engineering problems (space, [[oil exploration]], aircraft design, etc.), Monte Carlo&ndash;based predictions of failure, [[cost overrun]]s and schedule overruns are routinely better than human intuition or alternative "soft" methods.<ref>{{cite journal|last1 = Hubbard|first1 = Douglas|last2=Samuelson|first2 = Douglas A. |date = October 2009 |title=Modeling Without Measurements |url=http://viewer.zmags.com/publication/357348e6#/357348e6/28|journal = OR/MS Today|pages = 28–33}}</ref>

Other examples include modeling phenomena with significant uncertainty in inputs such as the calculation of risk in business and, in mathematics, evaluation of multidimensional definite integrals with complicated boundary conditions.  In application to systems engineering problems (space, oil exploration, aircraft design, etc.), Monte Carlo&ndash;based predictions of failure, cost overruns and schedule overruns are routinely better than human intuition or alternative "soft" methods.

Other examples include modeling phenomena with significant uncertainty in inputs such as the calculation of risk in business and, in mathematics, evaluation of multidimensional definite integrals with complicated boundary conditions.  In application to systems engineering problems (space, oil exploration, aircraft design, etc.), Monte Carlo&ndash;based predictions of failure, cost overruns and schedule overruns are routinely better than human intuition or alternative "soft" methods.

其他例子包括在输入中存在明显不确定性的建模现象，例如商业中的风险计算，以及在数学中对具有复杂边界条件的多维定积分的评价。应用于系统工程问题(空间、石油勘探、飞机设计等)。) ，蒙特卡洛——基于失败、成本超支和进度超支的预测通常比人类直觉或其他“软”方法更好。

其他例子包括对输入中具有重大不确定性的现象进行建模，如商业中的风险计算，以及在数学中对具有复杂边界条件的多维定积分进行评估。 在系统工程问题(空间、石油勘探、飞机设计等)的应用中，基于蒙特卡罗的故障预测、成本超支和进度超支通常比人类的直觉或其他的“软性”方法更好。  

 

 

In principle, Monte Carlo methods can be used to solve any problem having a probabilistic interpretation. By the [[law of large numbers]], integrals described by the [[expected value]] of some random variable can be approximated by taking the [[Sample mean and sample covariance|empirical mean]] (a.k.a. the sample mean) of independent samples of the variable. When the [[probability distribution]] of the variable is parametrized, mathematicians often use a [[Markov chain Monte Carlo]] (MCMC) sampler.<ref>{{Cite journal|title = Equation of State Calculations by Fast Computing Machines|journal = The Journal of Chemical Physics|date = 1953-06-01|issn = 0021-9606|pages = 1087–1092|volume = 21|issue = 6|doi = 10.1063/1.1699114|first1 = Nicholas|last1 = Metropolis|first2 = Arianna W.|last2 = Rosenbluth|first3 = Marshall N.|last3 = Rosenbluth|first4 = Augusta H.|last4 = Teller|first5 = Edward|last5 = Teller|bibcode=1953JChPh..21.1087M|s2cid = 1046577|url = https://semanticscholar.org/paper/f6a13f116e270dde9d67848495f801cdb8efa25d}}</ref><ref>{{Cite journal|title = Monte Carlo sampling methods using Markov chains and their applications|journal = Biometrika|date = 1970-04-01|issn = 0006-3444|pages = 97–109|volume = 57|issue = 1|doi = 10.1093/biomet/57.1.97|first = W. K.|last = Hastings|bibcode = 1970Bimka..57...97H|s2cid = 21204149|url = https://semanticscholar.org/paper/143d2e02ab91ae6259576ac50b664b8647af8988}}</ref><ref>{{Cite journal|title = The Multiple-Try Method and Local Optimization in Metropolis Sampling|journal = Journal of the American Statistical Association|date = 2000-03-01|issn = 0162-1459|pages = 121–134|volume = 95|issue = 449|doi = 10.1080/01621459.2000.10473908|first1 = Jun S.|last1 = Liu|first2 = Faming|last2 = Liang|first3 = Wing Hung|last3 = Wong|s2cid = 123468109|url = https://semanticscholar.org/paper/ff17c129a8d32bb7dc7206230da612e94bd24b9f}}</ref> The central idea is to design a judicious [[Markov chain]] model with a prescribed [[stationary probability distribution]]. That is, in the limit, the samples being generated by the MCMC method will be samples from the desired (target) distribution.<ref>{{cite journal | last1 = Spall | first1 = J. C. | year = 2003 | title = Estimation via Markov Chain Monte Carlo | doi = 10.1109/MCS.2003.1188770 | journal = IEEE Control Systems Magazine | volume = 23 | issue = 2| pages = 34–45 }}</ref><ref>{{Cite journal |doi = 10.1109/MCS.2018.2876959|title = Stationarity and Convergence of the Metropolis-Hastings Algorithm: Insights into Theoretical Aspects|journal = IEEE Control Systems Magazine |volume = 39|pages = 56–67|year = 2019|last1 = Hill|first1 = Stacy D.|last2 = Spall|first2 = James C.|s2cid = 58672766}}</ref> By the [[ergodic theorem]], the stationary distribution is approximated by the [[empirical measure]]s of the random states of the MCMC sampler.

In principle, Monte Carlo methods can be used to solve any problem having a probabilistic interpretation. By the [[law of large numbers]], integrals described by the [[expected value]] of some random variable can be approximated by taking the [[Sample mean and sample covariance|empirical mean]] (a.k.a. the sample mean) of independent samples of the variable. When the [[probability distribution]] of the variable is parametrized, mathematicians often use a [[Markov chain Monte Carlo]] (MCMC) sampler.<ref>{{Cite journal|title = Equation of State Calculations by Fast Computing Machines|journal = The Journal of Chemical Physics|date = 1953-06-01|issn = 0021-9606|pages = 1087–1092|volume = 21|issue = 6|doi = 10.1063/1.1699114|first1 = Nicholas|last1 = Metropolis|first2 = Arianna W.|last2 = Rosenbluth|first3 = Marshall N.|last3 = Rosenbluth|first4 = Augusta H.|last4 = Teller|first5 = Edward|last5 = Teller|bibcode=1953JChPh..21.1087M|s2cid = 1046577|url = https://semanticscholar.org/paper/f6a13f116e270dde9d67848495f801cdb8efa25d}}</ref><ref>{{Cite journal|title = Monte Carlo sampling methods using Markov chains and their applications|journal = Biometrika|date = 1970-04-01|issn = 0006-3444|pages = 97–109|volume = 57|issue = 1|doi = 10.1093/biomet/57.1.97|first = W. K.|last = Hastings|bibcode = 1970Bimka..57...97H|s2cid = 21204149|url = https://semanticscholar.org/paper/143d2e02ab91ae6259576ac50b664b8647af8988}}</ref><ref>{{Cite journal|title = The Multiple-Try Method and Local Optimization in Metropolis Sampling|journal = Journal of the American Statistical Association|date = 2000-03-01|issn = 0162-1459|pages = 121–134|volume = 95|issue = 449|doi = 10.1080/01621459.2000.10473908|first1 = Jun S.|last1 = Liu|first2 = Faming|last2 = Liang|first3 = Wing Hung|last3 = Wong|s2cid = 123468109|url = https://semanticscholar.org/paper/ff17c129a8d32bb7dc7206230da612e94bd24b9f}}</ref> The central idea is to design a judicious [[Markov chain]] model with a prescribed [[stationary probability distribution]]. That is, in the limit, the samples being generated by the MCMC method will be samples from the desired (target) distribution.<ref>{{cite journal | last1 = Spall | first1 = J. C. | year = 2003 | title = Estimation via Markov Chain Monte Carlo | doi = 10.1109/MCS.2003.1188770 | journal = IEEE Control Systems Magazine | volume = 23 | issue = 2| pages = 34–45 }}</ref><ref>{{Cite journal |doi = 10.1109/MCS.2018.2876959|title = Stationarity and Convergence of the Metropolis-Hastings Algorithm: Insights into Theoretical Aspects|journal = IEEE Control Systems Magazine |volume = 39|pages = 56–67|year = 2019|last1 = Hill|first1 = Stacy D.|last2 = Spall|first2 = James C.|s2cid = 58672766}}</ref> By the [[ergodic theorem]], the stationary distribution is approximated by the [[empirical measure]]s of the random states of the MCMC sampler.

In principle, Monte Carlo methods can be used to solve any problem having a probabilistic interpretation. By the law of large numbers, integrals described by the expected value of some random variable can be approximated by taking the empirical mean (a.k.a. the sample mean) of independent samples of the variable. When the probability distribution of the variable is parametrized, mathematicians often use a Markov chain Monte Carlo (MCMC) sampler. The central idea is to design a judicious Markov chain model with a prescribed stationary probability distribution. That is, in the limit, the samples being generated by the MCMC method will be samples from the desired (target) distribution. By the ergodic theorem, the stationary distribution is approximated by the empirical measures of the random states of the MCMC sampler.

In principle, Monte Carlo methods can be used to solve any problem having a probabilistic interpretation. By the law of large numbers, integrals described by the expected value of some random variable can be approximated by taking the empirical mean (a.k.a. the sample mean) of independent samples of the variable. When the probability distribution of the variable is parametrized, mathematicians often use a Markov chain Monte Carlo (MCMC) sampler. The central idea is to design a judicious Markov chain model with a prescribed stationary probability distribution. That is, in the limit, the samples being generated by the MCMC method will be samples from the desired (target) distribution. By the ergodic theorem, the stationary distribution is approximated by the empirical measures of the random states of the MCMC sampler.

原则上，蒙特卡罗方法可以用来解决任何具有概率解释的问题。利用大数定律，用随机变量的期望值所描述的积分可以用经验均值(简称经验均值)来近似。样本平均值)的独立样本的变量。当变量的概率分布被参数化时，数学家们经常使用马尔科夫蒙特卡洛取样器。其核心思想是设计一个具有规定平稳概率分布的明智的马尔可夫链模型。也就是说，在极限情况下，由 MCMC 方法生成的样本将是来自期望(目标)分布的样本。利用遍历定理，用 MCMC 采样器随机状态的经验测度近似平稳分布。

理论上，蒙特卡罗方法可以用来解决任何具有概率解释的问题。 根据大数定律，用某个随机变量的期望值描述的积分可以用该变量独立样本的经验均值(即样本均值)来近似。 当变量的概率分布被参数化时，数学家们经常使用马尔可夫链蒙特卡罗(MCMC)采样器。 中心思想是设计一个具有给定平稳概率分布的有效马尔可夫链模型。 也就是说，在极限情况下，MCMC方法生成的样本将成为来自期望(目标)分布的样本。 通过遍历定理，平稳分布可以用MCMC采样器随机状态的经验测度来近似。  

 

 

In other problems, the objective is generating draws from a sequence of probability distributions satisfying a nonlinear evolution equation. These flows of probability distributions can always be interpreted as the distributions of the random states of a [[Markov process]] whose transition probabilities depend on the distributions of the current random states (see [[McKean–Vlasov process]]es, [[particle filter|nonlinear filtering equation]]).<ref name="kol10">{{cite book|last = Kolokoltsov|first = Vassili|title = Nonlinear Markov processes|year = 2010|publisher = Cambridge Univ. Press|pages = 375}}</ref><ref name="dp13">{{cite book|last = Del Moral|first = Pierre|title = Mean field simulation for Monte Carlo integration|year = 2013|publisher = Chapman & Hall/CRC Press|quote = Monographs on Statistics & Applied Probability|url = http://www.crcpress.com/product/isbn/9781466504059|pages = 626}}</ref> In other instances we are given a flow of probability distributions with an increasing level of sampling complexity (path spaces models with an increasing time horizon, Boltzmann–Gibbs measures associated with decreasing temperature parameters, and many others). These models can also be seen as the evolution of the law of the random states of a nonlinear Markov chain.<ref name="dp13" /><ref>{{Cite journal|title = Sequential Monte Carlo samplers | last1 = Del Moral | first1 = P | last2 = Doucet | first2 = A | last3 = Jasra | first3 = A | year = 2006 |doi=10.1111/j.1467-9868.2006.00553.x|volume=68| issue = 3 |journal=Journal of the Royal Statistical Society, Series B|pages=411–436|arxiv = cond-mat/0212648| s2cid = 12074789 }}</ref> A natural way to simulate these sophisticated nonlinear Markov processes is to sample multiple copies of the process, replacing in the evolution equation the unknown distributions of the random states by the sampled [[empirical measure]]s. In contrast with traditional Monte Carlo and MCMC methodologies these [[Mean field particle methods|mean field particle]] techniques rely on sequential interacting samples. The terminology ''mean field'' reflects the fact that each of the ''samples'' (a.k.a. particles, individuals, walkers, agents, creatures, or phenotypes) interacts with the empirical measures of the process. When the size of the system tends to infinity, these random empirical measures converge to the deterministic distribution of the random states of the nonlinear Markov chain, so that the statistical interaction between particles vanishes.

In other problems, the objective is generating draws from a sequence of probability distributions satisfying a nonlinear evolution equation. These flows of probability distributions can always be interpreted as the distributions of the random states of a [[Markov process]] whose transition probabilities depend on the distributions of the current random states (see [[McKean–Vlasov process]]es, [[particle filter|nonlinear filtering equation]]).<ref name="kol10">{{cite book|last = Kolokoltsov|first = Vassili|title = Nonlinear Markov processes|year = 2010|publisher = Cambridge Univ. Press|pages = 375}}</ref><ref name="dp13">{{cite book|last = Del Moral|first = Pierre|title = Mean field simulation for Monte Carlo integration|year = 2013|publisher = Chapman & Hall/CRC Press|quote = Monographs on Statistics & Applied Probability|url = http://www.crcpress.com/product/isbn/9781466504059|pages = 626}}</ref> In other instances we are given a flow of probability distributions with an increasing level of sampling complexity (path spaces models with an increasing time horizon, Boltzmann–Gibbs measures associated with decreasing temperature parameters, and many others). These models can also be seen as the evolution of the law of the random states of a nonlinear Markov chain.<ref name="dp13" /><ref>{{Cite journal|title = Sequential Monte Carlo samplers | last1 = Del Moral | first1 = P | last2 = Doucet | first2 = A | last3 = Jasra | first3 = A | year = 2006 |doi=10.1111/j.1467-9868.2006.00553.x|volume=68| issue = 3 |journal=Journal of the Royal Statistical Society, Series B|pages=411–436|arxiv = cond-mat/0212648| s2cid = 12074789 }}</ref> A natural way to simulate these sophisticated nonlinear Markov processes is to sample multiple copies of the process, replacing in the evolution equation the unknown distributions of the random states by the sampled [[empirical measure]]s. In contrast with traditional Monte Carlo and MCMC methodologies these [[Mean field particle methods|mean field particle]] techniques rely on sequential interacting samples. The terminology ''mean field'' reflects the fact that each of the ''samples'' (a.k.a. particles, individuals, walkers, agents, creatures, or phenotypes) interacts with the empirical measures of the process. When the size of the system tends to infinity, these random empirical measures converge to the deterministic distribution of the random states of the nonlinear Markov chain, so that the statistical interaction between particles vanishes.