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Monte Carlo methods, or Monte Carlo experiments, are a broad class of computational algorithms that rely on repeated random sampling to obtain numerical results. The underlying concept is to use randomness to solve problems that might be deterministic in principle. They are often used in physical and mathematical problems and are most useful when it is difficult or impossible to use other approaches. Monte Carlo methods are mainly used in three problem classes: optimization, numerical integration, and generating draws from a probability distribution.

Monte Carlo methods, or Monte Carlo experiments, are a broad class of computational algorithms that rely on repeated random sampling to obtain numerical results. The underlying concept is to use randomness to solve problems that might be deterministic in principle. They are often used in physical and mathematical problems and are most useful when it is difficult or impossible to use other approaches. Monte Carlo methods are mainly used in three problem classes: optimization, numerical integration, and generating draws from a probability distribution.

蒙特卡罗方法，或称蒙特卡罗方法方法，是一类广泛的计算算法，它依赖于重复的随机抽样来获得数值结果。其基本概念是利用随机性来解决原则上可能是确定性的问题。它们通常用于物理和数学问题，当很难或不可能使用其他方法时，它们最有用。蒙特卡罗方法主要用于3个问题类: 优化，数值积分，从概率分布生成绘图。

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

[[File:Pi 30K.gif|thumb|right| Monte Carlo method applied to approximating the value of {{pi}}.]]

[[File:Pi 30K.gif|thumb|right| Monte Carlo method applied to approximating the value of {{pi}}.|链接=Special:FilePath/Pi_30K.gif]]

  Monte Carlo method applied to approximating the value of .

  Monte Carlo method applied to approximating the value of .

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 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. 100 dimensions is by no means unusual, since in many physical problems, a "dimension" is equivalent to a 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 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. 100 dimensions is by no means unusual, since in many physical problems, a "dimension" is equivalent to a degree of freedom.

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 </sup > 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。

 

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

[[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>]]

[[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]]