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第355行： − − Monte Carlo methods are especially useful for simulating phenomena with significant uncertainty in inputs and systems with many coupled degrees of freedom. Areas of application include: − − 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: 第426行：
第422行： + + + + 第440行：
第440行： + + − − Monte Carlo methods are widely used in engineering for [[sensitivity analysis]] and quantitative [[probabilistic]] analysis in [[Process design (chemical engineering)|process design]]. The need arises from the interactive, co-linear and non-linear behavior of typical process simulations. For example, 第538行：
第538行： − − Monte-Carlo integration works by comparing random points with the value of the function − − 蒙特卡罗积分是通过比较随机点和函数值来工作的 − − Errors reduce by a factor of <math>\scriptstyle 1/\sqrt{N}</math> − − 错误减少一个因素 < math > scriptstyle 1/sqrt { n } <nowiki></math ></nowiki> − 第609行：
第600行： − + + + + + − [[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]]+
→Definitions
[[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]].
[[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.
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:
Monte Carlo methods are especially useful for simulating phenomena with significant uncertainty in inputs and systems with many coupled degrees of freedom. Areas of application include:
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:
===Physical sciences===
===Physical sciences===
===Engineering===
===Engineering===
Monte Carlo methods are widely used in engineering for [[sensitivity analysis]] and quantitative [[probabilistic]] analysis in [[Process design (chemical engineering)|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,
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,
蒙特卡罗方法被广泛应用于工程设计中的敏感度分析和工艺设计中的定量概率分析。这种需求来源于典型过程模拟的交互性、共线性和非线性行为。比如说,
蒙特卡罗方法被广泛应用于工程设计中的敏感度分析和工艺设计中的定量概率分析。这种需求来源于典型过程模拟的交互性、共线性和非线性行为。比如说,
* In [[microelectronics|microelectronics engineering]], Monte Carlo methods are applied to analyze correlated and uncorrelated variations in [[Analog signal|analog]] and [[Digital data|digital]] [[integrated circuits]].
* In [[microelectronics|microelectronics engineering]], Monte Carlo methods are applied to analyze correlated and uncorrelated variations in [[Analog signal|analog]] and [[Digital data|digital]] [[integrated circuits]].
#Play a simulated game starting with that node.
#Play a simulated game starting with that node.
#Play a simulated game starting with that node. 以该节点开始玩一个模拟游戏。
#Play a simulated game starting with that node. 以该节点开始玩一个模拟游戏。
#Use the results of that simulated game to update the node and its ancestors.
#Use the results of that simulated game to update the node and its ancestors.
#Use the results of that simulated game to update the node and its ancestors. 使用模拟游戏的结果来更新节点及其祖先。
#Use the results of that simulated game to update the node and its ancestors. 使用模拟游戏的结果来更新节点及其祖先。
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.
=== Integration ===
=== 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]]
{{Main|Monte Carlo integration}}[[File:Monte-carlo2.gif|thumb|Monte-Carlo integration works by comparing random points with the value of the functionMonte-Carlo integration works by comparing random points with the value of the function
蒙特卡罗积分是通过比较随机点和函数值来工作的|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>Errors reduce by a factor of <math>\scriptstyle 1/\sqrt{N}</math>
错误减少一个因素 < math > scriptstyle 1/sqrt { n } <nowiki></math ></nowiki>|链接=Special:FilePath/Monte-Carlo_method_(errors).png]]
Deterministic [[numerical integration]] algorithms work well in a small number of dimensions, but encounter two problems when the functions have many variables. First, the number of function evaluations needed increases rapidly with the number of dimensions. For example, if 10 evaluations provide adequate accuracy in one dimension, then [[googol|10<sup>100</sup>]] 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|10<sup>100</sup>]] 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]].