更改

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

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

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

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

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

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

 

 

Uses of Monte Carlo methods require large amounts of random numbers, and it was their use that spurred the development of [[pseudorandom number generator]]s{{Citation needed|reason=I can't find any reference in the linked page saying that PRNG development was spurred by Monte Carlo methods and there's no reference here.  This assertion, between the commas, needs a quotable source backing up that PRNGs development happened faster than it would have otherwise done specifically because of the use of MC methods.  This statement seems to me to be conjecture, in the absence of a citation.|date=November 2019}}, 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 generator]]s{{Citation needed|reason=I can't find any reference in the linked page saying that PRNG development was spurred by Monte Carlo methods and there's no reference here.  This assertion, between the commas, needs a quotable source backing up that PRNGs development happened faster than it would have otherwise done specifically because of the use of MC methods.  This statement seems to me to be conjecture, in the absence of a citation.|date=November 2019}}, 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.

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 ==

== History ==

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).

在蒙特卡罗方法模型开发之前，模拟测试了一个先前已知的确定性问题，并且使用统计抽样来估计模拟中的不确定性。蒙特卡罗模拟颠覆了这种方法，使用概率启发式方法解决确定性问题(见模拟退火)。

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)。

 

 

An early variant of the Monte Carlo method was devised to solve the [[Buffon's needle problem]], in which {{pi}} 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.{{sfn|Metropolis|1987}}  

An early variant of the Monte Carlo method was devised to solve the [[Buffon's needle problem]], in which {{pi}} 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.{{sfn|Metropolis|1987}}  

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，Stanislaw Ulam 发明了现代版的马尔科夫蒙特卡洛方法，当时他正在洛斯阿拉莫斯国家实验室研究核武器项目。乌兰取得突破性进展之后，约翰·冯·诺伊曼马上意识到了它的重要性。冯 · 诺依曼编写了 ENIAC 计算机执行蒙特卡罗计算的程序。1946年，洛斯阿拉莫斯的核武器物理学家正在研究裂变材料中的中子扩散。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。乌兰姆建议使用随机实验。他的灵感如下:

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:

 

{{quote|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)|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.{{sfn|Eckhardt|1987}}}}

{{quote|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)|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.{{sfn|Eckhardt|1987}}}}

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，在那里乌拉姆的叔叔会从亲戚那里借钱去赌博。使用“真正随机”的随机数列表是非常慢的，但冯 · 诺依曼开发了一种计算伪随机数生成器的方法，使用平方取中法。尽管这种方法被批评为粗糙，冯 · 诺依曼意识到这一点: 他证明这种方法比任何其他方法都快，并指出，当它出错时，这种方法是如此明显，不像方法可能会微妙地不正确。

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。

 

 

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.

蒙特卡罗方法是曼哈顿计划所需要的模拟的中心，尽管在当时受到计算工具的严重限制。20世纪50年代，它们在洛斯阿拉莫斯用于研制氢弹的早期工作，并在物理学、物理化学和运筹学领域得到普及。兰德公司和美国空军是当时负责资助和传播蒙特卡洛方法信息的两个主要组织，他们开始在许多不同的领域找到广泛的应用。

蒙特卡罗方法是曼哈顿计划所需要的模拟的中心，尽管在当时受到计算工具的严重限制。20世纪50年代，它们在洛斯阿拉莫斯用于研制氢弹的早期工作，并在物理学、物理化学和运筹学领域得到普及。兰德公司和美国空军是当时负责资助和传播蒙特卡洛方法信息的两个主要组织，他们开始在许多不同的领域找到广泛的应用。

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>

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。

[[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

卡洛斯和惠特洛克指出，这种区别并不总是容易维持。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。”

卡洛斯和惠特洛克指出，这种区别并不总是容易维持。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。”

The use of [[Sequential Monte Carlo method|Sequential Monte Carlo]] in advanced [[signal processing]] and [[Bayesian inference]] is more recent. It was in 1993, that Gordon et al., published in their seminal work<ref>{{Cite journal|title = Novel approach to nonlinear/non-Gaussian Bayesian state estimation |journal =  IEE Proceedings F - Radar and Signal Processing|date = April 1993|issn = 0956-375X|pages = 107–113|volume = 140|issue = 2|first1 = N.J.|last1 = Gordon|first2 = D.J.|last2 = Salmond|first3 = A.F.M.|last3 = Smith|doi=10.1049/ip-f-2.1993.0015|s2cid = 12644877|url = https://semanticscholar.org/paper/65484334a5cd4cabf6e5f7a17f606f07e2acf625}}</ref> the first application of a Monte Carlo [[Resampling (statistics)|resampling]] algorithm in Bayesian statistical inference. The authors named their algorithm 'the bootstrap filter', and demonstrated that compared to other filtering methods, their bootstrap algorithm does not require any assumption about that state-space or the noise of the system. We also quote another pioneering article in this field of Genshiro Kitagawa on a related "Monte Carlo filter",<ref>{{cite journal

The use of [[Sequential Monte Carlo method|Sequential Monte Carlo]] in advanced [[signal processing]] and [[Bayesian inference]] is more recent. It was in 1993, that Gordon et al., published in their seminal work<ref>{{Cite journal|title = Novel approach to nonlinear/non-Gaussian Bayesian state estimation |journal =  IEE Proceedings F - Radar and Signal Processing|date = April 1993|issn = 0956-375X|pages = 107–113|volume = 140|issue = 2|first1 = N.J.|last1 = Gordon|first2 = D.J.|last2 = Salmond|first3 = A.F.M.|last3 = Smith|doi=10.1049/ip-f-2.1993.0015|s2cid = 12644877|url = https://semanticscholar.org/paper/65484334a5cd4cabf6e5f7a17f606f07e2acf625}}</ref> the first application of a Monte Carlo [[Resampling (statistics)|resampling]] algorithm in Bayesian statistical inference. The authors named their algorithm 'the bootstrap filter', and demonstrated that compared to other filtering methods, their bootstrap algorithm does not require any assumption about that state-space or the noise of the system. We also quote another pioneering article in this field of Genshiro Kitagawa on a related "Monte Carlo filter",<ref>{{cite journal

蒙特卡罗模拟通常需要拥有属性许多未知参数，其中许多参数很难通过实验获得。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，对于素数测试等一些应用，不可预测性是至关重要的)。许多最有用的技术使用确定性，伪随机序列，使它很容易测试和重新运行模拟。伪随机序列在某种意义上表现出足够的“足够随机” ，这是进行良好模拟所必需的唯一品质。

蒙特卡罗模拟通常需要拥有属性许多未知参数，其中许多参数很难通过实验获得。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，对于素数测试等一些应用，不可预测性是至关重要的)。许多最有用的技术使用确定性，伪随机序列，使它很容易测试和重新运行模拟。伪随机序列在某种意义上表现出足够的“足够随机” ，这是进行良好模拟所必需的唯一品质。

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

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

这意味着什么取决于应用程序，但通常应该通过一系列统计测试。当考虑序列中足够多的元素时，检验这些数是均匀分布的还是遵循另一个期望的分布是最简单和最常见的方法之一。连续样本之间的弱相关性通常也是可取的/必要的。

这意味着什么取决于应用程序，但通常应该通过一系列统计测试。当考虑序列中足够多的元素时，检验这些数是均匀分布的还是遵循另一个期望的分布是最简单和最常见的方法之一。连续样本之间的弱相关性通常也是可取的/必要的。

Branching type particle methodologies with varying population sizes were also developed in the end of the 1990s by Dan Crisan, Jessica Gaines and Terry Lyons,<ref name=":42">{{cite journal|last1 = Crisan|first1 = Dan|last2 = Gaines|first2 = Jessica|last3 = Lyons|first3 = Terry|title = Convergence of a branching particle method to the solution of the Zakai|journal = SIAM Journal on Applied Mathematics|date = 1998|volume = 58|issue = 5|pages = 1568–1590|doi = 10.1137/s0036139996307371|s2cid = 39982562|url = https://semanticscholar.org/paper/99e8759a243cd0568b0f32cbace2ad0525b16bb6}}</ref><ref>{{cite journal|last1 = Crisan|first1 = Dan|last2 = Lyons|first2 = Terry|title = Nonlinear filtering and measure-valued processes|journal = Probability Theory and Related Fields|date = 1997|volume = 109|issue = 2|pages = 217–244|doi = 10.1007/s004400050131|s2cid = 119809371}}</ref><ref>{{cite journal|last1 = Crisan|first1 = Dan|last2 = Lyons|first2 = Terry|title = A particle approximation of the solution of the Kushner–Stratonovitch equation|journal = Probability Theory and Related Fields|date = 1999|volume = 115|issue = 4|pages = 549–578|doi = 10.1007/s004400050249|s2cid = 117725141}}</ref> and by Dan Crisan, Pierre Del Moral and Terry Lyons.<ref name=":52">{{cite journal|last1 = Crisan|first1 = Dan|last2 = Del Moral|first2 = Pierre|last3 = Lyons|first3 = Terry|title = Discrete filtering using branching and interacting particle systems|journal = Markov Processes and Related Fields|date = 1999|volume = 5|issue = 3|pages = 293–318|url = http://web.maths.unsw.edu.au/~peterdel-moral/crisan98discrete.pdf}}</ref> Further developments in this field were developed in 2000 by P. Del Moral, A. Guionnet and L. Miclo.<ref name="dmm002"/><ref name="dg99">{{cite journal|last1 = Del Moral|first1 = Pierre|last2 = Guionnet|first2 = Alice|title = On the stability of Measure Valued Processes with Applications to filtering|journal = C. R. Acad. Sci. Paris|date = 1999|volume = 39|issue = 1|pages = 429–434}}</ref><ref name="dg01">{{cite journal|last1 = Del Moral|first1 = Pierre|last2 = Guionnet|first2 = Alice|title = On the stability of interacting processes with applications to filtering and genetic algorithms|journal = Annales de l'Institut Henri Poincaré|date = 2001|volume = 37|issue = 2|pages = 155–194|url = http://web.maths.unsw.edu.au/~peterdel-moral/ihp.ps|doi = 10.1016/s0246-0203(00)01064-5|bibcode=2001AnIHP..37..155D}}</ref>

Branching type particle methodologies with varying population sizes were also developed in the end of the 1990s by Dan Crisan, Jessica Gaines and Terry Lyons,<ref name=":42">{{cite journal|last1 = Crisan|first1 = Dan|last2 = Gaines|first2 = Jessica|last3 = Lyons|first3 = Terry|title = Convergence of a branching particle method to the solution of the Zakai|journal = SIAM Journal on Applied Mathematics|date = 1998|volume = 58|issue = 5|pages = 1568–1590|doi = 10.1137/s0036139996307371|s2cid = 39982562|url = https://semanticscholar.org/paper/99e8759a243cd0568b0f32cbace2ad0525b16bb6}}</ref><ref>{{cite journal|last1 = Crisan|first1 = Dan|last2 = Lyons|first2 = Terry|title = Nonlinear filtering and measure-valued processes|journal = Probability Theory and Related Fields|date = 1997|volume = 109|issue = 2|pages = 217–244|doi = 10.1007/s004400050131|s2cid = 119809371}}</ref><ref>{{cite journal|last1 = Crisan|first1 = Dan|last2 = Lyons|first2 = Terry|title = A particle approximation of the solution of the Kushner–Stratonovitch equation|journal = Probability Theory and Related Fields|date = 1999|volume = 115|issue = 4|pages = 549–578|doi = 10.1007/s004400050249|s2cid = 117725141}}</ref> and by Dan Crisan, Pierre Del Moral and Terry Lyons.<ref name=":52">{{cite journal|last1 = Crisan|first1 = Dan|last2 = Del Moral|first2 = Pierre|last3 = Lyons|first3 = Terry|title = Discrete filtering using branching and interacting particle systems|journal = Markov Processes and Related Fields|date = 1999|volume = 5|issue = 3|pages = 293–318|url = http://web.maths.unsw.edu.au/~peterdel-moral/crisan98discrete.pdf}}</ref> Further developments in this field were developed in 2000 by P. Del Moral, A. Guionnet and L. Miclo.<ref name="dmm002"/><ref name="dg99">{{cite journal|last1 = Del Moral|first1 = Pierre|last2 = Guionnet|first2 = Alice|title = On the stability of Measure Valued Processes with Applications to filtering|journal = C. R. Acad. Sci. Paris|date = 1999|volume = 39|issue = 1|pages = 429–434}}</ref><ref name="dg01">{{cite journal|last1 = Del Moral|first1 = Pierre|last2 = Guionnet|first2 = Alice|title = On the stability of interacting processes with applications to filtering and genetic algorithms|journal = Annales de l'Institut Henri Poincaré|date = 2001|volume = 37|issue = 2|pages = 155–194|url = http://web.maths.unsw.edu.au/~peterdel-moral/ihp.ps|doi = 10.1016/s0246-0203(00)01064-5|bibcode=2001AnIHP..37..155D}}</ref>

萨维罗斯基列出了高质量蒙特卡罗模拟的特点:

萨维罗斯基列出了高质量蒙特卡罗模拟的特点:

==Definitions==

==Definitions==

A Monte Carlo method simulation is defined as any method that utilizes sequences of random numbers to perform the simulation.  Monte Carlo simulations are applied to many topics including quantum chromodynamics, cancer radiation therapy, traffic flow, stellar evolution and VLSI design.  All these simulations require the use of random numbers and therefore pseudorandom number generators, which makes creating random-like numbers very important.

A Monte Carlo method simulation is defined as any method that utilizes sequences of random numbers to perform the simulation.  Monte Carlo simulations are applied to many topics including quantum chromodynamics, cancer radiation therapy, traffic flow, stellar evolution and VLSI design.  All these simulations require the use of random numbers and therefore pseudorandom number generators, which makes creating random-like numbers very important.

*it simulates the phenomenon in question.

*it simulates the phenomenon in question.

[[Pseudo-random number sampling]] algorithms are used to transform uniformly distributed pseudo-random numbers into numbers that are distributed according to a given [[probability distribution]].

[[Pseudo-random number sampling]] algorithms are used to transform uniformly distributed pseudo-random numbers into numbers that are distributed according to a given [[probability distribution]].

蒙特卡罗方法尤其适用于模拟输入和多自由度耦合系统中具有明显不确定性的现象。申请范围包括:

蒙特卡罗方法尤其适用于模拟输入和多自由度耦合系统中具有明显不确定性的现象。申请范围包括:

[[Low-discrepancy sequences]] are often used instead of random sampling from a space as they ensure even coverage and normally have a faster order of convergence than Monte Carlo simulations using random or pseudorandom sequences. Methods based on their use are called [[quasi-Monte Carlo method]]s.

[[Low-discrepancy sequences]] are often used instead of random sampling from a space as they ensure even coverage and normally have a faster order of convergence than Monte Carlo simulations using random or pseudorandom sequences. Methods based on their use are called [[quasi-Monte Carlo method]]s.

In an effort to assess the impact of random number quality on Monte Carlo simulation outcomes, astrophysical researchers tested cryptographically-secure pseudorandom numbers generated via Intel's [[RDRAND]] instruction set, as compared to those derived from algorithms, like the [[Mersenne Twister]], in Monte Carlo simulations of radio flares from [[brown dwarfs]].  RDRAND is the closest pseudorandom number generator to a true random number generator. No statistically significant difference was found between models generated with typical pseudorandom number generators and RDRAND for trials consisting of the generation of 10⁷ random numbers.<ref>{{cite journal|last1=Route|first1=Matthew|title=Radio-flaring Ultracool Dwarf Population Synthesis|journal=The Astrophysical Journal|date=August 10, 2017|volume=845|issue=1|page=66|doi=10.3847/1538-4357/aa7ede|arxiv=1707.02212|bibcode=2017ApJ...845...66R|s2cid=118895524}}</ref>

In an effort to assess the impact of random number quality on Monte Carlo simulation outcomes, astrophysical researchers tested cryptographically-secure pseudorandom numbers generated via Intel's [[RDRAND]] instruction set, as compared to those derived from algorithms, like the [[Mersenne Twister]], in Monte Carlo simulations of radio flares from [[brown dwarfs]].  RDRAND is the closest pseudorandom number generator to a true random number generator. No statistically significant difference was found between models generated with typical pseudorandom number generators and RDRAND for trials consisting of the generation of 10⁷ random numbers.<ref>{{cite journal|last1=Route|first1=Matthew|title=Radio-flaring Ultracool Dwarf Population Synthesis|journal=The Astrophysical Journal|date=August 10, 2017|volume=845|issue=1|page=66|doi=10.3847/1538-4357/aa7ede|arxiv=1707.02212|bibcode=2017ApJ...845...66R|s2cid=118895524}}</ref>

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 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.

A ''Monte Carlo method'' simulation is defined as any method that utilizes sequences of random numbers to perform the simulation.  Monte Carlo simulations are applied to many topics including [[quantum chromodynamics]], cancer radiation therapy, traffic flow, [[stellar evolution]] and VLSI design.  All these simulations require the use of random numbers and therefore [[pseudorandom number generator]]s, which makes creating random-like numbers very important.

A ''Monte Carlo method'' simulation is defined as any method that utilizes sequences of random numbers to perform the simulation.  Monte Carlo simulations are applied to many topics including [[quantum chromodynamics]], cancer radiation therapy, traffic flow, [[stellar evolution]] and VLSI design.  All these simulations require the use of random numbers and therefore [[pseudorandom number generator]]s, which makes creating random-like numbers very important.

Monte Carlo methods are widely used in engineering for sensitivity analysis and quantitative probabilistic analysis in process design. The need arises from the interactive, co-linear and non-linear behavior of typical process simulations. For example,

Monte Carlo methods are widely used in engineering for sensitivity analysis and quantitative probabilistic analysis in process design. The need arises from the interactive, co-linear and non-linear behavior of typical process simulations. For example,

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>{{harvnb|Vose|2000|page=13}}</ref>

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>{{harvnb|Vose|2000|page=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".

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".

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。

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。

The systems can be studied in the coarse-grained or ab initio frameworks depending on the desired accuracy.  

The systems can be studied in the coarse-grained or ab initio frameworks depending on the desired accuracy.  

Monte Carlo methods are especially useful for simulating phenomena with significant [[uncertainty]] in inputs and systems with many [[coupling (physics)|coupled]] degrees of freedom. Areas of application include:

Monte Carlo methods are especially useful for simulating phenomena with significant [[uncertainty]] in inputs and systems with many [[coupling (physics)|coupled]] degrees of freedom. Areas of application include:

===Physical sciences===

===Physical sciences===

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 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:

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.

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.

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。

===Climate change and radiative forcing===

===Climate change and radiative forcing===

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:

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.

展开叶节点并选择其中一个子节点。

展开叶节点并选择其中一个子节点。

Play a simulated game starting with that node.

Play a simulated game starting with that node.

使用模拟游戏的结果来更新节点及其祖先。

使用模拟游戏的结果来更新节点及其祖先。

===Computational biology===

===Computational biology===

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 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>

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>

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).

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).

===Computer graphics===

===Computer graphics===

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。

===Applied statistics===

===Applied statistics===

蒙特卡罗模拟通常用于评估影响不同决策方案结果的风险和不确定性。蒙特卡洛模拟允许商业风险分析师在销售量、商品和劳动力价格、利率和汇率等变量中考虑不确定性的总体影响，以及不同风险事件的影响，如合同的取消或税法的改变。

蒙特卡罗模拟通常用于评估影响不同决策方案结果的风险和不确定性。蒙特卡洛模拟允许商业风险分析师在销售量、商品和劳动力价格、利率和汇率等变量中考虑不确定性的总体影响，以及不同风险事件的影响，如合同的取消或税法的改变。

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).

金融领域的蒙特卡罗方法通常用于评估一个业务单位或公司层面的项目投资，或其他金融估值。它们可以用于对项目进度表进行建模，模拟对每个任务的最坏情况、最好情况和最可能的持续时间进行聚合估计，以确定整个项目的结果。蒙特卡罗方法也用于期权定价，违约风险分析 https://risk.octigo.pl/。此外，它们还可以用来评估医疗干预措施的财务影响。

金融领域的蒙特卡罗方法通常用于评估一个业务单位或公司层面的项目投资，或其他金融估值。它们可以用于对项目进度表进行建模，模拟对每个任务的最坏情况、最好情况和最可能的持续时间进行聚合估计，以确定整个项目的结果。蒙特卡罗方法也用于期权定价，违约风险分析 https://risk.octigo.pl/。此外，它们还可以用来评估医疗干预措施的财务影响。

{{anchor|Monte Carlo tree search}}

{{anchor|Monte Carlo tree search}}

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.

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

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

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>{{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>

错误减少一个因素 < math > scriptstyle 1/sqrt { n } </math >  

错误减少一个因素 < math > scriptstyle 1/sqrt { n } </math >  

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

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

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。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。

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 } </math > 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 } </math > 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。

{{See also|Computer Go}}

{{See also|Computer Go}}

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。

===Design and visuals===

===Design and visuals===

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>{{harvnb|Szirmay–Kalos|2008}}</ref>

===Search and rescue===

===Search and rescue===

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。参考文献是对许多与模拟和优化有关的问题的全面回顾。

===Finance and business===

===Finance and business===

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

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

Probabilistic formulation of inverse problems leads to the definition of a probability distribution in the model space. This probability distribution combines prior information with new information obtained by measuring some observable parameters (data).

Probabilistic formulation of inverse problems leads to the definition of a probability distribution in the model space. This probability distribution combines prior information with new information obtained by measuring some observable parameters (data).

因为，在一般情况下，连接数据和模型参数的理论是非线性的，模型空间中的后验概率可能不容易描述(它可能是多模态的，一些矩可能没有定义，等等。).

因为，在一般情况下，连接数据和模型参数的理论是非线性的，模型空间中的后验概率可能不容易描述(它可能是多模态的，一些矩可能没有定义，等等。).

===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">{{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>

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.

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

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

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.

=== Integration ===

=== Integration ===

{{Main|Monte Carlo integration}}

{{Main|Monte Carlo integration}}[[File:Monte-carlo2.gif|thumb|Monte-Carlo integration works by comparing random points with the value of the function|链接=Special:FilePath/Monte-carlo2.gif]]

 

 

 

[[File:Monte-carlo2.gif|thumb|Monte-Carlo integration works by comparing random points with the value of the function|链接=Special:FilePath/Monte-carlo2.gif]]

[[File:Monte-Carlo method (errors).png|thumb|Errors reduce by a factor of <math>\scriptstyle 1/\sqrt{N}</math>|链接=Special:FilePath/Monte-Carlo_method_(errors).png]]

[[File:Monte-Carlo method (errors).png|thumb|Errors reduce by a factor of <math>\scriptstyle 1/\sqrt{N}</math>|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>{{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]].

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/>

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>{{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 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.

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>{{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>

=== Simulation and optimization ===

=== 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>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>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.

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.

===Inverse problems===

===Inverse problems===

As, in the general case, the theory linking data with model parameters is nonlinear, the posterior probability in the model space may not be easy to describe (it may be multimodal, some moments may not be defined, etc.).

As, in the general case, the theory linking data with model parameters is nonlinear, the posterior probability in the model space may not be easy to describe (it may be multimodal, some moments may not be defined, etc.).

When analyzing an inverse problem, obtaining a maximum likelihood model is usually not sufficient, as we normally also wish to have information on the resolution power of the data. In the general case we may have many model parameters, and an inspection of the [[marginal probability]] densities of interest may be impractical, or even useless. But it is possible to pseudorandomly generate a large collection of models according to the [[posterior probability distribution]] and to analyze and display the models in such a way that information on the relative likelihoods of model properties is conveyed to the spectator. This can be accomplished by means of an efficient Monte Carlo method, even in cases where no explicit formula for the ''a priori'' distribution is available.

When analyzing an inverse problem, obtaining a maximum likelihood model is usually not sufficient, as we normally also wish to have information on the resolution power of the data. In the general case we may have many model parameters, and an inspection of the [[marginal probability]] densities of interest may be impractical, or even useless. But it is possible to pseudorandomly generate a large collection of models according to the [[posterior probability distribution]] and to analyze and display the models in such a way that information on the relative likelihoods of model properties is conveyed to the spectator. This can be accomplished by means of an efficient Monte Carlo method, even in cases where no explicit formula for the ''a priori'' distribution is available.

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>{{harvnb|Mosegaard|Tarantola|1995}}</ref><ref>{{harvnb|Tarantola|2005}}</ref>

===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>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>.

== See also ==

== See also ==

{{Portal|Mathematics}}

{{Portal|Mathematics}}

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== References ==

== References ==

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{{Reflist}}

=== Sources ===

=== Sources ===