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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 name=":8">{{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 name=":8">{{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.

在其他问题中，要实现的目标则是从满足非线性演化方程的概率分布序列中生成图像。这些概率分布流总是可以解释为'''马尔可夫过程 Markov process'''的随机状态的分布，其转移概率依赖于当前随机状态的分布(见麦肯-弗拉索夫过程，'''非线性滤波方程 Nonlinear Filtering Equation''')。<ref name="kol10" /><ref name="dp13" /> 其他情况下，我们给出了采样复杂度不断增加的概率分布流(如时间范围不断增加的路径空间模型，与温度参数降低有联系的'''玻尔兹曼—吉布斯测度 Boltzmann-Gibbs Measures'''，以及许多其他例子)。这些模型也可以看作是一个非线性马尔可夫链的随机状态规律的演化。<ref name="dp13" /><ref name=":8" /> 模拟这些复杂非线性马尔可夫过程的一个自然的方法是对过程的多个副本进行抽样，用抽样的经验测度替代演化方程中未知的随机状态分布。与传统的蒙特卡罗和马尔科夫链蒙特卡洛方法相比，这些'''平均场粒子 Mean Field Particle'''技术依赖于连续相互作用的样本。平均场一词反映了每个样本(也就是粒子、个体、步行者、媒介、生物或表现型)与过程的经验测量相互作用的事实。当系统大小趋近于无穷时，这些随机经验测度收敛于非线性马尔可夫链随机状态的确定性分布，从而使粒子之间的统计相互作用消失。

在其他问题中，要实现的目标则是从满足非线性演化方程的概率分布序列中生成图像。这些概率分布流总是可以解释为'''马尔可夫过程 Markov process'''的随机状态的分布，其转移概率依赖于当前随机状态的分布(见麦肯-弗拉索夫过程，'''非线性滤波方程 Nonlinear Filtering Equation''')。<ref name="kol10" /><ref name="dp13" /> 其他情况下，我们给出了采样复杂度不断增加的概率分布流(如时间范围不断增加的路径空间模型，与温度参数降低有联系的'''玻尔兹曼—吉布斯测度 Boltzmann-Gibbs Measures'''，以及许多其他例子)。这些模型也可以看作是一个非线性马尔可夫链的随机状态规律的演化。<ref name="dp13" /><ref name=":8" /> 模拟这些复杂非线性马尔可夫过程的一个自然的方法是对过程的多个副本进行抽样，用抽样的经验测度替代演化方程中未知的随机状态分布。与传统的蒙特卡罗和马尔科夫链蒙特卡洛方法相比，这些'''平均场粒子 Mean Field Particle'''技术依赖于连续相互作用的样本。平均场一词反映了每个样本(也就是粒子、个体、步行者、媒介、生物或表现型)与过程的经验测量相互作用的事实。当系统大小趋近于无穷时，这些随机经验测度收敛于非线性马尔可夫链随机状态的确定性分布，从而使粒子之间的统计相互作用消失。

Despite its conceptual and algorithmic simplicity, the computational cost associated with a Monte Carlo simulation can be staggeringly high. In general the method requires many samples to get a good approximation, which may incurs an arbitrarily large total runtime if the processing time of a single sample is high. Although this is a severe limitation in very complex problems, the embarrassingly parallel nature of the algorithm allows this large cost to be reduced (perhaps to a feasible level) through parallel computing strategies in local processors, clusters, cloud computing, GPU, FPGA etc.

Despite its conceptual and algorithmic simplicity, the computational cost associated with a Monte Carlo simulation can be staggeringly high. In general the method requires many samples to get a good approximation, which may incurs an arbitrarily large total runtime if the processing time of a single sample is high. Although this is a severe limitation in very complex problems, the embarrassingly parallel nature of the algorithm allows this large cost to be reduced (perhaps to a feasible level) through parallel computing strategies in local processors, clusters, cloud computing, GPU, FPGA etc.

尽管其概念和算法简单，但与蒙特卡罗模拟相关的计算成本却是高的惊人。一般情况下，该方法需要大量的样本来获得良好的近似，如果单个样本的处理时间较长，可能会导致总运行时间长度难以控制。尽管在非常复杂的问题中，这是一个严重的限制，但该算法令人尴尬的并行性质允许通过本地处理器、集群、云计算、GPU、FPGA等的并行计算策略来降低高昂的成本(或许降低到可以接受的水平上)。

尽管其概念和算法简单，但与蒙特卡罗模拟相关的计算成本却是高的惊人。一般情况下，该方法需要大量的样本来获得良好的近似，如果单个样本的处理时间较长，可能会导致总运行时间长度难以控制。【11】尽管在非常复杂的问题中，这是一个严重的限制，但该算法令人尴尬的并行性质允许通过本地处理器、集群、云计算、GPU、FPGA等的并行计算策略来降低高昂的成本(或许降低到可以接受的水平上)。【12-15】

== Overview 概述 ==

== Overview 概述 ==

For example, consider a [[circular sector#Quadrant|quadrant (circular sector)]] inscribed in a [[unit square]]. Given that the ratio of their areas is {{sfrac|{{pi}}|4}}, the value of [[pi|{{pi}}]] can be approximated using a Monte Carlo method:{{sfn|Kalos|Whitlock|2008}}

For example, consider a [[circular sector#Quadrant|quadrant (circular sector)]] inscribed in a [[unit square]]. Given that the ratio of their areas is {{sfrac|{{pi}}|4}}, the value of [[pi|{{pi}}]] can be approximated using a Monte Carlo method:{{sfn|Kalos|Whitlock|2008}}

For example, consider a quadrant (circular sector) inscribed in a unit square. Given that the ratio of their areas is |4}}, the value of pi| can be approximated using a Monte Carlo method:

For example, consider a quadrant (circular sector) inscribed in a unit square. Given that the ratio of their areas is π/4, the value of π can be approximated using a Monte Carlo method:【16】

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

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

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

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

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

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

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

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

In this procedure the domain of inputs is the square that circumscribes the quadrant.  We generate random inputs by scattering grains over the square then perform a computation on each input (test whether it falls within the quadrant). Aggregating the results yields our final result, the approximation of {{pi}}.

In this procedure the domain of inputs is the square that circumscribes the quadrant.  We generate random inputs by scattering grains over the square then perform a computation on each input (test whether it falls within the quadrant). Aggregating the results yields our final result, the approximation of {{pi}}.

2、这一过程需要很多点。如果整个正方形中只有几个点是随机放置的，那么这个近似值通常是很差的。平均而言，随着放置更多的点，近似值精度会提升。

2、这一过程需要很多点。如果整个正方形中只有几个点是随机放置的，那么这个近似值通常是很差的。平均而言，随着放置更多的点，近似值精度会提升。

Uses of Monte Carlo methods require large amounts of random numbers, and it was their use that spurred the development of pseudorandom number generators, which were far quicker to use than the tables of random numbers that had been previously used for statistical sampling.

Uses of Monte Carlo methods require large amounts of random numbers, and it was their use that spurred the development of pseudorandom number generators, which were far quicker to use than the tables of random numbers that had been previously used for statistical sampling.

== History 历史 ==

== History 历史 ==

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

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

在开发蒙特卡罗方法之前，人们模拟测试了一个已经解决的确定性问题，统计抽样用于估计模拟中的不确定性。蒙特卡罗模拟将这种方法反转，使用概率元启发式方法解决确定性问题(参见模拟退火)。

在开发蒙特卡罗方法之前，人们模拟测试一个已经解决的确定性问题，通过统计抽样估计模拟中的不确定性。蒙特卡罗模拟将这种方法反转，使用'''概率元启发式方法Probabilistic Metaheuristics'''解决确定性问题(参见'''模拟退火法 Simulated Annealing''')。

 

An early variant of the Monte Carlo method was devised to solve the Buffon's needle problem, in which  can be estimated by dropping needles on a floor made of parallel equidistant strips. In the 1930s, Enrico Fermi first experimented with the Monte Carlo method while studying neutron diffusion, but he did not publish this work.  

An early variant of the Monte Carlo method was devised to solve the Buffon's needle problem, in which  can be estimated by dropping needles on a floor made of parallel equidistant strips. In the 1930s, Enrico Fermi first experimented with the Monte Carlo method while studying neutron diffusion, but he did not publish this work.  

In the late 1940s, [[Stanislaw Ulam]] invented the modern version of the Markov Chain Monte Carlo method while he was working on nuclear weapons projects at the [[Los Alamos National Laboratory]]. Immediately after Ulam's breakthrough, [[John von Neumann]] understood its importance. Von Neumann programmed the [[ENIAC]] computer to perform Monte Carlo calculations. In 1946, nuclear weapons physicists at Los Alamos were investigating neutron diffusion in fissionable material.{{sfn|Metropolis|1987}} Despite having most of the necessary data, such as the average distance a neutron would travel in a substance before it collided with an atomic nucleus and how much energy the neutron was likely to give off following a collision, the Los Alamos physicists were unable to solve the problem using conventional, deterministic mathematical methods. Ulam proposed using random experiments. He recounts his inspiration as follows:

In the late 1940s, [[Stanislaw Ulam]] invented the modern version of the Markov Chain Monte Carlo method while he was working on nuclear weapons projects at the [[Los Alamos National Laboratory]]. Immediately after Ulam's breakthrough, [[John von Neumann]] understood its importance. Von Neumann programmed the [[ENIAC]] computer to perform Monte Carlo calculations. In 1946, nuclear weapons physicists at Los Alamos were investigating neutron diffusion in fissionable material.{{sfn|Metropolis|1987}} Despite having most of the necessary data, such as the average distance a neutron would travel in a substance before it collided with an atomic nucleus and how much energy the neutron was likely to give off following a collision, the Los Alamos physicists were unable to solve the problem using conventional, deterministic mathematical methods. Ulam proposed using random experiments. He recounts his inspiration as follows:

In the late 1940s, Stanislaw Ulam invented the modern version of the Markov Chain Monte Carlo method while he was working on nuclear weapons projects at the Los Alamos National Laboratory. Immediately after Ulam's breakthrough, John von Neumann understood its importance. Von Neumann programmed the ENIAC computer to perform Monte Carlo calculations. In 1946, nuclear weapons physicists at Los Alamos were investigating neutron diffusion in fissionable material. Despite having most of the necessary data, such as the average distance a neutron would travel in a substance before it collided with an atomic nucleus and how much energy the neutron was likely to give off following a collision, the Los Alamos physicists were unable to solve the problem using conventional, deterministic mathematical methods. Ulam proposed using random experiments. He recounts his inspiration as follows:

In the late 1940s, Stanislaw Ulam invented the modern version of the Markov Chain Monte Carlo method while he was working on nuclear weapons projects at the Los Alamos National Laboratory. Immediately after Ulam's breakthrough, John von Neumann understood its importance. Von Neumann programmed the ENIAC computer to perform Monte Carlo calculations. In 1946, nuclear weapons physicists at Los Alamos were investigating neutron diffusion in fissionable material. Despite having most of the necessary data, such as the average distance a neutron would travel in a substance before it collided with an atomic nucleus and how much energy the neutron was likely to give off following a collision, the Los Alamos physicists were unable to solve the problem using conventional, deterministic mathematical methods. Ulam proposed using random experiments. He recounts his inspiration as follows:

20世纪40年代末，'''斯坦尼斯拉夫·乌拉姆 Stanislaw Ulam'''在洛斯阿拉莫斯国家实验室研究核武器项目时，发明了现代版的马尔可夫链蒙特卡罗方法。在乌拉姆的突破之后，约翰·冯·诺伊曼立即意识到了它的重要性。冯·诺伊曼为ENIAC（人类第一台电子数字积分计算机）编写了程序来进行蒙特卡罗计算。1946年，洛斯阿拉莫斯的核武器物理学家正在研究中子在可裂变材料中的扩散。尽管拥有大部分必要的数据，例如中子在与原子核碰撞之前在物质中的平均运行距离，以及碰撞后中子可能释放出多少能量，但洛斯阿拉莫斯的物理学家们无法用传统的、确定性的数学方法解决这个问题。此时乌拉姆建议使用随机实验。他后来回忆当初灵感产生过程:

20世纪40年代末，'''斯坦尼斯拉夫·乌拉姆 Stanislaw Ulam'''在洛斯阿拉莫斯国家实验室研究核武器项目时，发明了现代版的马尔可夫链蒙特卡罗方法。在乌拉姆的突破之后，'''约翰·冯·诺伊曼 John von Neumann'''立即意识到了它的重要性。冯·诺伊曼为ENIAC（人类第一台电子数字积分计算机）编写了程序来进行蒙特卡罗计算。1946年，洛斯阿拉莫斯的核武器物理学家正在研究中子在可裂变材料中的扩散。【17】尽管拥有大部分必要的数据，例如中子在与原子核碰撞之前在物质中的平均运行距离，以及碰撞后中子可能释放出多少能量，但洛斯阿拉莫斯的物理学家们无法用传统的、确定性的数学方法解决这个问题。此时乌拉姆建议使用随机实验。他后来回忆当初灵感产生过程:

The first thoughts and attempts I made to practice [the Monte Carlo Method] were suggested by a question which occurred to me in 1946 as I was convalescing from an illness and playing solitaires. The question was what are the chances that a Canfield solitaire laid out with 52 cards will come out successfully? After spending a lot of time trying to estimate them by pure combinatorial calculations, I wondered whether a more practical method than "abstract thinking" might not be to lay it out say one hundred times and simply observe and count the number of successful plays. This was already possible to envisage with the beginning of the new era of fast computers, and I immediately thought of problems of neutron diffusion and other questions of mathematical physics, and more generally how to change processes described by certain differential equations into an equivalent form interpretable as a succession of random operations. Later [in 1946], I described the idea to John von Neumann, and we began to plan actual calculations.

The first thoughts and attempts I made to practice [the Monte Carlo Method] were suggested by a question which occurred to me in 1946 as I was convalescing from an illness and playing solitaires. The question was what are the chances that a Canfield solitaire laid out with 52 cards will come out successfully? After spending a lot of time trying to estimate them by pure combinatorial calculations, I wondered whether a more practical method than "abstract thinking" might not be to lay it out say one hundred times and simply observe and count the number of successful plays. This was already possible to envisage with the beginning of the new era of fast computers, and I immediately thought of problems of neutron diffusion and other questions of mathematical physics, and more generally how to change processes described by certain differential equations into an equivalent form interpretable as a succession of random operations. Later [in 1946], I described the idea to John von Neumann, and we began to plan actual calculations.

我最初的想法和尝试蒙特卡洛法是在1946年，当时我正从疾病中康复，时常玩单人纸牌游戏。那时我会思考这样一个问题：一盘52张的加菲尔德纸牌成功出牌的几率有多大?在花了大量时间尝试通过纯粹的组合计算来估计它们之后，我想知道是否有一种比“抽象思维”更实际的方法，可能不是将它展开100次，然后简单地观察和计算成功的游戏数量。在快速计算机新时代开始时，这已经是可以想象的了，我立刻想到了中子扩散和其他数学物理的问题，以及更一般的情形—如何将由某些微分方程描述的过程转换成可解释为一系列随机操作的等价形式。后来(1946年)，我向约翰·冯·诺伊曼描述了这个想法，然后我们开始计划实际的计算。

我最初构想和尝试蒙特卡洛法是在1946年，当时我正从疾病中康复，时常玩单人纸牌游戏。那时我会思考这样一个问题：一盘52张的加菲尔德纸牌成功出牌的几率有多大？在花了大量时间尝试通过纯粹的组合计算来估计它们之后，我想知道是否有一种比“抽象思维”更实际的方法，可能不是将它展开100次，然后简单地观察和计算成功的游戏数量。在快速计算机新时代开始时，这已经是可以想象的了，我立刻想到了中子扩散和其他数学物理的问题，以及更一般的情形—如何将由某些微分方程描述的过程转换成可解释为一系列随机操作的等价形式。后来(1946年)，我向约翰·冯·诺伊曼描述了这个想法，然后我们开始计划实际的计算。【18】

 

Being secret, the work of von Neumann and Ulam required a code name.{{sfn|Mazhdrakov|Benov|Valkanov|2018|p=250}} A colleague of von Neumann and Ulam, [[Nicholas Metropolis]], suggested using the name ''Monte Carlo'', which refers to the [[Monte Carlo Casino]] in [[Monaco]] where Ulam's uncle would borrow money from relatives to gamble.{{sfn|Metropolis|1987}} Using [[A Million Random Digits with 100,000 Normal Deviates|lists of "truly random" random numbers]] was extremely slow, but von Neumann developed a way to calculate [[pseudorandom number]]s, using the [[middle-square method]]. Though this method has been criticized as crude, von Neumann was aware of this: he justified it as being faster than any other method at his disposal, and also noted that when it went awry it did so obviously, unlike methods that could be subtly incorrect.<ref>{{cite book |last = Peragine |first = Michael |title = The Universal Mind: The Evolution of Machine Intelligence and Human Psychology |year = 2013 |publisher = Xiphias Press |url = https://books.google.com/books?id=Dvb0DAAAQBAJ&q=he+justified+it+as+being+faster+than+any+other+method+at+his+disposal,+and+also+noted+that+when+it+went+awry+it+did+so+obviously,+unlike+methods+that+could+be+subtly+incorrect.&pg=PT201 |access-date = 2018-12-17 }}</ref>

Being secret, the work of von Neumann and Ulam required a code name.{{sfn|Mazhdrakov|Benov|Valkanov|2018|p=250}} A colleague of von Neumann and Ulam, [[Nicholas Metropolis]], suggested using the name ''Monte Carlo'', which refers to the [[Monte Carlo Casino]] in [[Monaco]] where Ulam's uncle would borrow money from relatives to gamble.{{sfn|Metropolis|1987}} Using [[A Million Random Digits with 100,000 Normal Deviates|lists of "truly random" random numbers]] was extremely slow, but von Neumann developed a way to calculate [[pseudorandom number]]s, using the [[middle-square method]]. Though this method has been criticized as crude, von Neumann was aware of this: he justified it as being faster than any other method at his disposal, and also noted that when it went awry it did so obviously, unlike methods that could be subtly incorrect.<ref>{{cite book |last = Peragine |first = Michael |title = The Universal Mind: The Evolution of Machine Intelligence and Human Psychology |year = 2013 |publisher = Xiphias Press |url = https://books.google.com/books?id=Dvb0DAAAQBAJ&q=he+justified+it+as+being+faster+than+any+other+method+at+his+disposal,+and+also+noted+that+when+it+went+awry+it+did+so+obviously,+unlike+methods+that+could+be+subtly+incorrect.&pg=PT201 |access-date = 2018-12-17 }}</ref>

Being secret, the work of von Neumann and Ulam required a code name. A colleague of von Neumann and Ulam, Nicholas Metropolis, suggested using the name Monte Carlo, which refers to the Monte Carlo Casino in Monaco where Ulam's uncle would borrow money from relatives to gamble. Using lists of "truly random" random numbers was extremely slow, but von Neumann developed a way to calculate pseudorandom numbers, using the middle-square method. Though this method has been criticized as crude, von Neumann was aware of this: he justified it as being faster than any other method at his disposal, and also noted that when it went awry it did so obviously, unlike methods that could be subtly incorrect.

Being secret, the work of von Neumann and Ulam required a code name. A colleague of von Neumann and Ulam, Nicholas Metropolis, suggested using the name Monte Carlo, which refers to the Monte Carlo Casino in Monaco where Ulam's uncle would borrow money from relatives to gamble. Using lists of "truly random" random numbers was extremely slow, but von Neumann developed a way to calculate pseudorandom numbers, using the middle-square method. Though this method has been criticized as crude, von Neumann was aware of this: he justified it as being faster than any other method at his disposal, and also noted that when it went awry it did so obviously, unlike methods that could be subtly incorrect.

冯 · 诺依曼和乌拉姆的工作是秘密进行的，需要一个代号。冯 · 诺依曼和乌拉姆的一位同事，尼古拉斯·梅特罗波利斯建议使用蒙特卡洛这个名字，这个名字指的是摩纳哥的蒙特卡洛赌场，乌拉姆的叔叔会和亲戚借钱然后去那里赌博。使用“真正随机”的随机数列表是非常慢的，然而冯 · 诺依曼使用平方取中法开发了一种计算伪随机数生成器的方法。虽然许多人一直批评这种方法较为粗糙原始，但是冯 · 诺依曼也意识到这一点：他证明这种方法比任何其他方法都快，并指出当它出错时，人们可以轻易发现，不像其他方法产生的错误可能会不易察觉。

冯 · 诺依曼和乌拉姆的工作是秘密进行的，需要一个代号。【19】冯 · 诺依曼和乌拉姆的一位同事，'''尼古拉斯·梅特罗波利斯 Nicholas Metropolis'''建议使用蒙特卡洛这个名字，这个名字指的是摩纳哥的蒙特卡洛赌场，乌拉姆的叔叔会和亲戚借钱然后去那里赌博。【17】使用“真正随机”的随机数列表是非常慢的，然而冯 · 诺依曼使用'''平方取中法 Middle-Square Method'''开发了一种计算伪随机数生成器的方法。虽然许多人一直批评这种方法较为粗糙原始，但是冯 · 诺依曼也意识到这一点：他证明这种方法比任何其他方法都快，并指出当它出错时，人们可以轻易发现，不像其他方法产生的错误可能会不易察觉。<ref name=":9" />【20】

Monte Carlo methods were central to the [[simulation]]s required for the [[Manhattan Project]], though severely limited by the computational tools at the time. In the 1950s they were used at [[Los Alamos National Laboratory|Los Alamos]] for early work relating to the development of the [[hydrogen bomb]], and became popularized in the fields of [[physics]], [[physical chemistry]], and [[operations research]]. The [[Rand Corporation]] and the [[U.S. Air Force]] were two of the major organizations responsible for funding and disseminating information on Monte Carlo methods during this time, and they began to find a wide application in many different fields.

Monte Carlo methods were central to the [[simulation]]s required for the [[Manhattan Project]], though severely limited by the computational tools at the time. In the 1950s they were used at [[Los Alamos National Laboratory|Los Alamos]] for early work relating to the development of the [[hydrogen bomb]], and became popularized in the fields of [[physics]], [[physical chemistry]], and [[operations research]]. The [[Rand Corporation]] and the [[U.S. Air Force]] were two of the major organizations responsible for funding and disseminating information on Monte Carlo methods during this time, and they began to find a wide application in many different fields.

Monte Carlo methods were central to the simulations required for the Manhattan Project, though severely limited by the computational tools at the time. In the 1950s they were used at Los Alamos for early work relating to the development of the hydrogen bomb, and became popularized in the fields of physics, physical chemistry, and operations research. The Rand Corporation and the U.S. Air Force were two of the major organizations responsible for funding and disseminating information on Monte Carlo methods during this time, and they began to find a wide application in many different fields.

Monte Carlo methods were central to the simulations required for the Manhattan Project, though severely limited by the computational tools at the time. In the 1950s they were used at Los Alamos for early work relating to the development of the hydrogen bomb, and became popularized in the fields of physics, physical chemistry, and operations research. The Rand Corporation and the U.S. Air Force were two of the major organizations responsible for funding and disseminating information on Monte Carlo methods during this time, and they began to find a wide application in many different fields.

The theory of more sophisticated mean field type particle Monte Carlo methods had certainly started by the mid-1960s, with the work of [[Henry McKean|Henry P. McKean Jr.]] on Markov interpretations of a class of nonlinear parabolic partial differential equations arising in fluid mechanics.<ref name="mck67">{{cite journal |last = McKean |first = Henry, P. |title = Propagation of chaos for a class of non-linear parabolic equations |journal = Lecture Series in Differential Equations, Catholic Univ. |year = 1967 |volume = 7 |pages = 41–57 }}</ref><ref>{{cite journal |last1 = McKean |first1 = Henry, P. |title = A class of Markov processes associated with nonlinear parabolic equations |journal = Proc. Natl. Acad. Sci. USA |year = 1966 |volume = 56 |issue = 6 |pages = 1907–1911 |doi = 10.1073/pnas.56.6.1907 |pmid = 16591437 |pmc = 220210 |bibcode = 1966PNAS...56.1907M }}</ref> We also quote an earlier pioneering article by [[Ted Harris (mathematician)|Theodore E. Harris]] and Herman Kahn, published in 1951, using mean field [[genetic algorithm|genetic]]-type Monte Carlo methods for estimating particle transmission energies.<ref>{{cite journal |last1 = Herman |first1 = Kahn |last2 = Theodore |first2 = Harris E. |title = Estimation of particle transmission by random sampling |journal = Natl. Bur. Stand. Appl. Math. Ser. |year = 1951 |volume = 12 |pages = 27–30 |url = https://dornsifecms.usc.edu/assets/sites/520/docs/kahnharris.pdf }}</ref> Mean field genetic type Monte Carlo methodologies are also used as heuristic natural search algorithms (a.k.a. [[metaheuristic]]) in evolutionary computing. The origins of these mean field computational techniques can be traced to 1950 and 1954 with the work of [[Alan Turing]] on genetic type mutation-selection learning machines<ref>{{cite journal |last = Turing |first = Alan M. |title = Computing machinery and intelligence|journal = Mind|volume = LIX |issue = 238 |pages = 433–460 |doi = 10.1093/mind/LIX.236.433 |year = 1950 }}</ref> and the articles by [[Nils Aall Barricelli]] at the [[Institute for Advanced Study]] in [[Princeton, New Jersey]].<ref>{{cite journal |last = Barricelli |first = Nils Aall |year = 1954 |author-link = Nils Aall Barricelli |title = Esempi numerici di processi di evoluzione |journal = Methodos |pages = 45–68 }}</ref><ref>{{cite journal |last = Barricelli |first = Nils Aall |year = 1957 |author-link = Nils Aall Barricelli |title = Symbiogenetic evolution processes realized by artificial methods |journal = Methodos |pages = 143–182 }}</ref>

The theory of more sophisticated mean field type particle Monte Carlo methods had certainly started by the mid-1960s, with the work of [[Henry McKean|Henry P. McKean Jr.]] on Markov interpretations of a class of nonlinear parabolic partial differential equations arising in fluid mechanics.<ref name="mck67">{{cite journal |last = McKean |first = Henry, P. |title = Propagation of chaos for a class of non-linear parabolic equations |journal = Lecture Series in Differential Equations, Catholic Univ. |year = 1967 |volume = 7 |pages = 41–57 }}</ref><ref>{{cite journal |last1 = McKean |first1 = Henry, P. |title = A class of Markov processes associated with nonlinear parabolic equations |journal = Proc. Natl. Acad. Sci. USA |year = 1966 |volume = 56 |issue = 6 |pages = 1907–1911 |doi = 10.1073/pnas.56.6.1907 |pmid = 16591437 |pmc = 220210 |bibcode = 1966PNAS...56.1907M }}</ref> We also quote an earlier pioneering article by [[Ted Harris (mathematician)|Theodore E. Harris]] and Herman Kahn, published in 1951, using mean field [[genetic algorithm|genetic]]-type Monte Carlo methods for estimating particle transmission energies.<ref>{{cite journal |last1 = Herman |first1 = Kahn |last2 = Theodore |first2 = Harris E. |title = Estimation of particle transmission by random sampling |journal = Natl. Bur. Stand. Appl. Math. Ser. |year = 1951 |volume = 12 |pages = 27–30 |url = https://dornsifecms.usc.edu/assets/sites/520/docs/kahnharris.pdf }}</ref> Mean field genetic type Monte Carlo methodologies are also used as heuristic natural search algorithms (a.k.a. [[metaheuristic]]) in evolutionary computing. The origins of these mean field computational techniques can be traced to 1950 and 1954 with the work of [[Alan Turing]] on genetic type mutation-selection learning machines<ref>{{cite journal |last = Turing |first = Alan M. |title = Computing machinery and intelligence|journal = Mind|volume = LIX |issue = 238 |pages = 433–460 |doi = 10.1093/mind/LIX.236.433 |year = 1950 }}</ref> and the articles by [[Nils Aall Barricelli]] at the [[Institute for Advanced Study]] in [[Princeton, New Jersey]].<ref>{{cite journal |last = Barricelli |first = Nils Aall |year = 1954 |author-link = Nils Aall Barricelli |title = Esempi numerici di processi di evoluzione |journal = Methodos |pages = 45–68 }}</ref><ref>{{cite journal |last = Barricelli |first = Nils Aall |year = 1957 |author-link = Nils Aall Barricelli |title = Symbiogenetic evolution processes realized by artificial methods |journal = Methodos |pages = 143–182 }}</ref>