更改

在其他问题中，目标是从满足非线性发展方程的概率分布序列生成图。这些概率分布流总是可以解释为马尔可夫过程的随机状态的分布，其转移概率依赖于当前随机状态的分布(见麦肯-弗拉索夫 McKean-Vlasov过程，非线性滤波方程)。在其他情况下，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 Boltzmann-Gibbs'''测度，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。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 MCMC 方法相比，这些平均场粒子技术依赖于连续的相互作用样本。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。当系统的大小趋近于无穷时，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，目标是从满足非线性发展方程的概率分布序列生成图。这些概率分布流总是可以解释为马尔可夫过程的随机状态的分布，其转移概率依赖于当前随机状态的分布(见麦肯-弗拉索夫 McKean-Vlasov过程，非线性滤波方程)。在其他情况下，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 Boltzmann-Gibbs'''测度，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。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 MCMC 方法相比，这些平均场粒子技术依赖于连续的相互作用样本。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。当系统的大小趋近于无穷时，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。

== Overview（下面的步骤格式和维基原文差别很大，序号都是1，不知道为啥） ==

== Overview 概述 ==

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

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

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

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

== History ==

== 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 [[Probability|probabilistic]] [[metaheuristic]]s (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 [[Probability|probabilistic]] [[metaheuristic]]s (see [[simulated annealing]]).

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>

The theory of more sophisticated mean field type particle Monte Carlo methods had certainly started by the mid-1960s, with the work of Henry P. McKean Jr. on Markov interpretations of a class of nonlinear parabolic partial differential equations arising in fluid mechanics. We also quote an earlier pioneering article by Theodore E. Harris and Herman Kahn, published in 1951, using mean field genetic-type Monte Carlo methods for estimating particle transmission energies. 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 and the articles by Nils Aall Barricelli at the Institute for Advanced Study in Princeton, New Jersey.

The theory of more sophisticated mean field type particle Monte Carlo methods had certainly started by the mid-1960s, with the work of Henry P. McKean Jr. on Markov interpretations of a class of nonlinear parabolic partial differential equations arising in fluid mechanics. We also quote an earlier pioneering article by Theodore E. Harris and Herman Kahn, published in 1951, using mean field genetic-type Monte Carlo methods for estimating particle transmission energies. 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 and the articles by Nils Aall Barricelli at the Institute for Advanced Study in Princeton, New Jersey.

更为复杂的平均场型粒子蒙特卡罗方法的理论自20世纪60年代中期开始，由 Henry p. McKean jr. 关于流体力学中一类非线性抛物型偏微分方程的 Markov 解释的工作开始。我们还引用了西奥多 · e · 哈里斯(Theodore e. Harris)和赫尔曼 · 卡恩(Herman Kahn)在1951年发表的一篇开创性文章，该文使用平均场遗传型蒙特卡罗方法估算粒子传输能量。平均场遗传型蒙特卡罗方法也被用作启发式自然搜索算法。进化计算中的元启发式算法。这些平均场计算技术的起源可以追溯到1950年和1954年，其中包括阿兰 · 图灵关于基因类型突变选择学习机的工作，以及新泽西州普林斯顿高级研究所的尼尔斯 · 阿尔 · 巴里切利的文章。

更复杂的平均场型粒子蒙特卡罗方法的理论产生于20世纪60年代中期，最初来自于'''小亨利·麦基恩 Henry P. McKean Jr.''' 研究流体力学中出现的一类非线性抛物型偏微分方程的马尔可夫解释。我们还引用Theodore E. Harris和Herman Kahn在1951年发表的一篇较早的开创性文章，使用平均场遗传型蒙特卡罗方法来估计粒子传输能量平均场遗传型蒙特卡罗方法在进化计算中也被用作启发式自然搜索算法(又称元启发式)。这些平均场计算技术的起源可以追溯到1950年和1954年，当时艾伦·图灵(Alan Turing)在基因类型突变-选择学习机器[24]上的工作，以及新泽西州普林斯顿高等研究院(Institute for Advanced Study)的尼尔斯·阿尔·巴里里利(Nils Aall Barricelli)的文章。

[[Quantum Monte Carlo]], and more specifically [[Diffusion Monte Carlo|diffusion Monte Carlo methods]] can also be interpreted as a mean field particle Monte Carlo approximation of [[Richard Feynman|Feynman]]–[[Mark Kac|Kac]] path integrals.<ref name="dp04">{{cite book |last = Del Moral |first = Pierre|title = Feynman–Kac formulae. Genealogical and interacting particle approximations |year = 2004 |publisher = Springer |quote = Series: Probability and Applications |url = https://www.springer.com/mathematics/probability/book/978-0-387-20268-6 |page = 575 |isbn = 9780387202686|series = Probability and Its Applications}}</ref><ref name="dmm002">{{cite book

[[Quantum Monte Carlo]], and more specifically [[Diffusion Monte Carlo|diffusion Monte Carlo methods]] can also be interpreted as a mean field particle Monte Carlo approximation of [[Richard Feynman|Feynman]]–[[Mark Kac|Kac]] path integrals.<ref name="dp04">{{cite book |last = Del Moral |first = Pierre|title = Feynman–Kac formulae. Genealogical and interacting particle approximations |year = 2004 |publisher = Springer |quote = Series: Probability and Applications |url = https://www.springer.com/mathematics/probability/book/978-0-387-20268-6 |page = 575 |isbn = 9780387202686|series = Probability and Its Applications}}</ref><ref name="dmm002">{{cite book