蒙特卡罗模拟

（重定向自蒙特卡洛模拟

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:[1] optimization, numerical integration, and generating draws from a probability distribution.

In physics-related problems, Monte Carlo methods are useful for simulating systems with many coupled degrees of freedom, such as fluids, disordered materials, strongly coupled solids, and cellular structures (see cellular Potts model, interacting particle systems, McKean–Vlasov processes, kinetic models of gases).

Other examples include modeling phenomena with significant uncertainty in inputs such as the calculation of risk in business and, in mathematics, evaluation of multidimensional definite integrals with complicated boundary conditions. In application to systems engineering problems (space, oil exploration, aircraft design, etc.), Monte Carlo–based predictions of failure, cost overruns and schedule overruns are routinely better than human intuition or alternative "soft" methods.[2]

In principle, Monte Carlo methods can be used to solve any problem having a probabilistic interpretation. By the law of large numbers, integrals described by the expected value of some random variable can be approximated by taking the empirical mean (a.k.a. the sample mean) of independent samples of the variable. When the probability distribution of the variable is parametrized, mathematicians often use a Markov chain Monte Carlo (MCMC) sampler.[3][4][5] The central idea is to design a judicious Markov chain model with a prescribed stationary probability distribution. That is, in the limit, the samples being generated by the MCMC method will be samples from the desired (target) distribution.[6][7] By the ergodic theorem, the stationary distribution is approximated by the empirical measures of the random states of the MCMC sampler.

In other problems, the objective is generating draws from a sequence of probability distributions satisfying a nonlinear evolution equation. These flows of probability distributions can always be interpreted as the distributions of the random states of a Markov process whose transition probabilities depend on the distributions of the current random states (see McKean–Vlasov processes, nonlinear filtering equation).[8][9] In other instances we are given a flow of probability distributions with an increasing level of sampling complexity (path spaces models with an increasing time horizon, Boltzmann–Gibbs measures associated with decreasing temperature parameters, and many others). These models can also be seen as the evolution of the law of the random states of a nonlinear Markov chain.[9][10] A natural way to simulate these sophisticated nonlinear Markov processes is to sample multiple copies of the process, replacing in the evolution equation the unknown distributions of the random states by the sampled empirical measures. In contrast with traditional Monte Carlo and MCMC methodologies these mean field particle techniques rely on sequential interacting samples. The terminology mean field reflects the fact that each of the samples (a.k.a. particles, individuals, walkers, agents, creatures, or phenotypes) interacts with the empirical measures of the process. When the size of the system tends to infinity, these random empirical measures converge to the deterministic distribution of the random states of the nonlinear Markov chain, so that the statistical interaction between particles vanishes.

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

Overview 概述

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

1. Define a domain of possible inputs
Define a domain of possible inputs


1、定义可能输入的域

1. Generate inputs randomly from a probability distribution over the domain
Generate inputs randomly from a probability distribution over the domain


2、从域上的概率分布随机生成输入

1. Perform a deterministic computation on the inputs
Perform a deterministic computation on the inputs


3、对输入进行确定性计算

1. Aggregate the results
Aggregate the results


4、汇总结果

Monte Carlo method applied to approximating the value of 模板:Pi. 蒙特卡罗方法应用于估计π值。
Monte Carlo method applied to approximating the value of .


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

1. Draw a square, then inscribe a quadrant within it
Draw a square, then inscribe a quadrant within it


1、画一个正方形，然后在其中划出一个四分之一圆

1. Uniformly scatter a given number of points over the square
Uniformly scatter a given number of points over the square


2、在正方形上均匀散布给定数量的点

1. Count the number of points inside the quadrant, i.e. having a distance from the origin of less than 1
Count the number of points inside the quadrant, i.e. having a distance from the origin of less than 1


3、计算四分之一圆内的点数，即满足距离原点小于1的

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


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

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

There are two important considerations:

1. 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.


1、如果这些点不是均匀分布的，那么近似效果就会很差。

1. 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.


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

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

History 历史

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

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. Despite having most of the necessary data, such as the average distance a neutron would travel in a substance before it collided with an atomic nucleus and how much energy the neutron was likely to give off following a collision, the Los Alamos physicists were unable to solve the problem using conventional, deterministic mathematical methods. Ulam proposed using random experiments. He recounts his inspiration as follows:

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

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

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.

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 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.

Quantum Monte Carlo, and more specifically diffusion Monte Carlo methods can also be interpreted as a mean field particle Monte Carlo approximation of FeynmanKac path integrals.[27][28][29][30][31][32][33] The origins of Quantum Monte Carlo methods are often attributed to Enrico Fermi and Robert Richtmyer who developed in 1948 a mean field particle interpretation of neutron-chain reactions,[34] but the first heuristic-like and genetic type particle algorithm (a.k.a. Resampled or Reconfiguration Monte Carlo methods) for estimating ground state energies of quantum systems (in reduced matrix models) is due to Jack H. Hetherington in 1984[33] In molecular chemistry, the use of genetic heuristic-like particle methodologies (a.k.a. pruning and enrichment strategies) can be traced back to 1955 with the seminal work of Marshall N. Rosenbluth and Arianna W. Rosenbluth.[35]

The use of 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[36] the first application of a Monte Carlo 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",[37] and the ones by Pierre Del Moral[38] and Himilcon Carvalho, Pierre Del Moral, André Monin and Gérard Salut[39] on particle filters published in the mid-1990s. Particle filters were also developed in signal processing in 1989–1992 by P. Del Moral, J. C. Noyer, G. Rigal, and G. Salut in the LAAS-CNRS in a series of restricted and classified research reports with STCAN (Service Technique des Constructions et Armes Navales), the IT company DIGILOG, and the LAAS-CNRS (the Laboratory for Analysis and Architecture of Systems) on radar/sonar and GPS signal processing problems.[40][41][42][43][44][45] These Sequential Monte Carlo methodologies can be interpreted as an acceptance-rejection sampler equipped with an interacting recycling mechanism.

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.[38][46]

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,[47][48][49] and by Dan Crisan, Pierre Del Moral and Terry Lyons.[50] Further developments in this field were developed in 2000 by P. Del Moral, A. Guionnet and L. Miclo.[28][51][52]

20世纪90年代末，丹·克里桑 Dan Crisan杰西卡·盖恩斯 Jessica Gaines特里·利昂斯 Terry Lyons[47][48][49] 以及丹·克里桑、皮埃尔·德尔·莫勒尔和特里·利昂斯也发展了具有不同种群大小的分支型粒子方法。[50] 2000年，皮埃尔·德尔·莫勒尔、爱丽丝·吉奥内 A. Guionnet洛朗·米克洛 L. Miclo进一步发展了这一领域。[28][51][52]

Definitions 定义

There is no consensus on how Monte Carlo should be defined. For example, Ripley[53] defines most probabilistic modeling as stochastic simulation, with Monte Carlo being reserved for Monte Carlo integration and Monte Carlo statistical tests. Sawilowsky[54] distinguishes between a simulation, a Monte Carlo method, and a Monte Carlo simulation: a simulation is a fictitious representation of reality, a Monte Carlo method is a technique that can be used to solve a mathematical or statistical problem, and a Monte Carlo simulation uses repeated sampling to obtain the statistical properties of some phenomenon (or behavior). Examples:

• Simulation: Drawing one pseudo-random uniform variable from the interval [0,1] can be used to simulate the tossing of a coin: If the value is less than or equal to 0.50 designate the outcome as heads, but if the value is greater than 0.50 designate the outcome as tails. This is a simulation, but not a Monte Carlo simulation.
• 模拟：从区间[0,1]中绘制一个伪随机均匀变量可以用来模拟抛硬币：如果值小于或等于0.50，则结果为正面，但如果值大于0.50，则结果为反面。这是一个模拟，但不是蒙特卡洛模拟。
• Monte Carlo method: Pouring out a box of coins on a table, and then computing the ratio of coins that land heads versus tails is a Monte Carlo method of determining the behavior of repeated coin tosses, but it is not a simulation.
• 蒙特卡洛方法：将一盒硬币倒在桌子上，然后计算正面与反面落地的硬币比例。这是一种确定重复掷硬币行为的蒙特卡洛方法，但它不是模拟。
• Monte Carlo simulation: Drawing ''a large number'' of pseudo-random uniform variables from the interval [0,1] at one time, or once at many different times, and assigning values less than or equal to 0.50 as heads and greater than 0.50 as tails, is a ''Monte Carlo simulation'' of the behavior of repeatedly tossing a coin.
• 蒙特卡罗模拟法：一次或多次从区间[0,1]中绘制“大量”伪随机均匀变量，赋值小于或等于0.50作为正面，大于0.50为反面。这是一个多次掷硬币的“蒙特卡罗模拟”行为。

Kalos and Whitlock point out that such distinctions are not always easy to maintain. For example, the emission of radiation from atoms is a natural stochastic process. It can be simulated directly, or its average behavior can be described by stochastic equations that can themselves be solved using Monte Carlo methods. "Indeed, the same computer code can be viewed simultaneously as a 'natural simulation' or as a solution of the equations by natural sampling."

Monte Carlo and random numbers 蒙特卡罗和随机数

The main idea behind this method is that the results are computed based on repeated random sampling and statistical analysis. The Monte Carlo simulation is, in fact, random experimentations, in the case that, the results of these experiments are not well known. Monte Carlo simulations are typically characterized by many unknown parameters, many of which are difficult to obtain experimentally. Monte Carlo simulation methods do not always require truly random numbers to be useful (although, for some applications such as primality testing, unpredictability is vital). Many of the most useful techniques use deterministic, pseudorandom sequences, making it easy to test and re-run simulations. The only quality usually necessary to make good simulations is for the pseudo-random sequence to appear "random enough" in a certain sense.

What this means depends on the application, but typically they should pass a series of statistical tests. Testing that the numbers are uniformly distributed or follow another desired distribution when a large enough number of elements of the sequence are considered is one of the simplest and most common ones. Weak correlations between successive samples are also often desirable/necessary.

Sawilowsky lists the characteristics of a high-quality Monte Carlo simulation:

• the (pseudo-random) number generator has certain characteristics (e.g. a long "period" before the sequence repeats)
• (伪随机数)生成器具有某些特征(例如，序列重复之前有一个很长的“周期”)
• the (pseudo-random) number generator produces values that pass tests for randomness
• (伪随机数)生成器可以产生能通过随机性测试的值
• there are enough samples to ensure accurate results
• 有足够的样本来确保准确的结果
• the proper sampling technique is used
• 使用适当的取样方法
• the algorithm used is valid for what is being modeled
• 使用的算法对建模内容是有效的
• it simulates the phenomenon in question.
• 它可以对问题中的现象进行模拟。

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 methods.

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.[58]

Monte Carlo simulation versus "what if" scenarios 蒙特卡罗模拟与“假设”情景

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.

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. 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 probability distributions shows that the Monte Carlo analysis has a narrower range than the "what if" analysis. This is because the "what if" analysis gives equal weight to all scenarios (see 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".

Applications 应用

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:

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:

Physical sciences 物理科学

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.[61][62][63] 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.[35][64] Quantum Monte Carlo methods solve the many-body problem for quantum systems.[8][9][27] In radiation materials science, the binary collision approximation for simulating ion implantation is usually based on a Monte Carlo approach to select the next colliding atom.[65] 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[66] and microwave radiation transmission through a rough planetary surface.[67] 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 radiation materials science, the binary collision approximation for simulating ion implantation is usually based on a Monte Carlo approach to select the next colliding atom. 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.

Engineering 工程学

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,

• In microelectronics engineering, Monte Carlo methods are applied to analyze correlated and uncorrelated variations in analog and digital integrated circuits.
• 微电子工程 Microelectronics Engineering中，蒙特卡罗方法被用于分析模拟和数字集成电路中相关和不相关的变化。
• In wind energy yield analysis, the predicted energy output of a wind farm during its lifetime is calculated giving different levels of uncertainty (P90, P50, etc.)
• 在风能产量分析中，考虑不同的不确定性(P90、P50等)，计算风电场在其生命周期内的预测发电量。
• Impacts of pollution are simulated[69] and diesel compared with petrol.[70]
• 模拟了污染产生的影响，[69] 并将柴油和汽油进行了比较。[70]
• In fluid dynamics, in particular rarefied gas dynamics, where the Boltzmann equation is solved for finite Knudsen number fluid flows using the direct simulation Monte Carlo[71] method in combination with highly efficient computational algorithms.[72]
• 在流体动力学，特别是稀薄气体动力学 Rarefied Gas Dynamics中，采用直接模拟蒙特卡罗方法 Direct Simulation Monte Carlo[71] 结合高效计算算法求解有限努森数 Knudsen Number流体的玻尔兹曼方程。[72]
• In autonomous robotics, Monte Carlo localization can determine the position of a robot. It is often applied to stochastic filters such as the Kalman filter or particle filter that forms the heart of the SLAM (simultaneous localization and mapping) algorithm.
• 在自主机器人中，蒙特卡洛定位 Monte Carlo Localization可以确定机器人的位置。它通常应用于随机滤波器，如卡尔曼滤波器 Kalman Filter粒子滤波器 Particle Filter，构成同步定位和映射算法 SLAM (Simultaneous Localization and Mapping)的核心。
• In telecommunications, when planning a wireless network, design must be proved to work for a wide variety of scenarios that depend mainly on the number of users, their locations and the services they want to use. Monte Carlo methods are typically used to generate these users and their states. The network performance is then evaluated and, if results are not satisfactory, the network design goes through an optimization process.
• 在电信行业，在规划无线网络时，必须证明设计适用于各种不同用户数量、用户位置和他们想使用的服务的场景。蒙特卡罗方法通常用于生成这些用户及其状态。然后对网络性能进行评估，如果结果不能令人满意，则进行网络设计优化。
• In reliability engineering, Monte Carlo simulation is used to compute system-level response given the component-level response. For example, for a transportation network subject to an earthquake event, Monte Carlo simulation can be used to assess the k-terminal reliability of the network given the failure probability of its components, e.g. bridges, roadways, etc.[73][74][75] Another profound example is the application of the Monte Carlo method to solve the G-Renewal equation of the generalized renewal process.[76]
• 在可靠性工程中，给定部件级响应，蒙特卡罗仿真用来计算系统级响应。例如，对于一个受地震事件影响的交通网络，给定其组件，如桥梁、道路等的失效概率，蒙特卡洛模拟可以用来评估网络的k-终端可靠性。[73][74][75] 另一个意义深远的例子是应用蒙特卡罗方法求解广义更新过程 Generalized Renewal Process的G-更新方程。[76]
• In signal processing and Bayesian inference, particle filters and sequential Monte Carlo techniques are a class of mean field particle methods for sampling and computing the posterior distribution of a signal process given some noisy and partial observations using interacting empirical measures.
• 在信号处理和贝叶斯推断中，粒子滤波器和序列蒙特卡罗技术是一类平均场粒子方法，用于对给定噪声和局部观测的信号过程进行采样和计算后验分布，使用相互作用的经验测度。
• In groundwater modeling, Monte Carlo methods are utilized to generate a large number of realizations of heterogeneous parameter field for model uncertainty quantification or parameter inversion.
• 在地下水模拟中，利用蒙特卡罗方法产生了大量的非均质参数场实现，用于模型不确定性量化或参数反演。[77]

Climate change and radiative forcing 气候变化与辐射强迫

The Intergovernmental Panel on Climate Change relies on Monte Carlo methods in probability density function analysis of radiative forcing.

Probability density function (PDF) of ERF due to total GHG, aerosol forcing and total anthropogenic forcing. The GHG consists of WMGHG, ozone and stratospheric water vapour. The PDFs are generated based on uncertainties provided in Table 8.6. The combination of the individual RF agents to derive total forcing over the Industrial Era are done by Monte Carlo simulations and based on the method in Boucher and Haywood (2001). PDF of the ERF from surface albedo changes and combined contrails and contrail-induced cirrus are included in the total anthropogenic forcing, but not shown as a separate PDF. We currently do not have ERF estimates for some forcing mechanisms: ozone, land use, solar, etc.

Computational biology 计算生物学

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,[79] or membranes.[80]The systems can be studied in the coarse-grained or ab initio frameworks depending on the desired accuracy. Computer simulations allow us to monitor the local environment of a particular molecule to see if some chemical reaction is happening for instance. In cases where it is not feasible to conduct a physical experiment, thought experiments can be conducted (for instance: breaking bonds, introducing impurities at specific sites, changing the local/global structure, or introducing external fields).

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, or membranes. The systems can be studied in the coarse-grained or ab initio frameworks depending on the desired accuracy. Computer simulations allow us to monitor the local environment of a particular molecule to see if some chemical reaction is happening for instance. In cases where it is not feasible to conduct a physical experiment, thought experiments can be conducted (for instance: breaking bonds, introducing impurities at specific sites, changing the local/global structure, or introducing external fields).

Computer graphics 计算机图形学

Path tracing, occasionally referred to as Monte Carlo ray tracing, renders a 3D scene by randomly tracing samples of possible light paths. Repeated sampling of any given pixel will eventually cause the average of the samples to converge on the correct solution of the rendering equation, making it one of the most physically accurate 3D graphics rendering methods in existence.

Path tracing, occasionally referred to as Monte Carlo ray tracing, renders a 3D scene by randomly tracing samples of possible light paths. Repeated sampling of any given pixel will eventually cause the average of the samples to converge on the correct solution of the rendering equation, making it one of the most physically accurate 3D graphics rendering methods in existence.

Applied statistics 应用统计学

The standards for Monte Carlo experiments in statistics were set by Sawilowsky.[81] In applied statistics, Monte Carlo methods may be used for at least four purposes:

The standards for Monte Carlo experiments in statistics were set by Sawilowsky. In applied statistics, Monte Carlo methods may be used for at least four purposes:

1. 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.[82]
2. 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. 比较在现实数据条件下小样本的竞争统计。虽然根据经典理论分布(例如，正态曲线，柯西分布 Cauchy distribution)数据的渐近条件(即，无限大的样本量和无限小的处理效果) ， i 型误差和统计的幂次特性可以进行计算，但是实际数据往往没有这样的分布。[82]
1. To provide implementations of hypothesis tests that are more efficient than exact tests such as permutation tests (which are often impossible to compute) while being more accurate than critical values for asymptotic distributions.
2. To provide implementations of hypothesis tests that are more efficient than exact tests such as permutation tests (which are often impossible to compute) while being more accurate than critical values for asymptotic distributions. 提供比精确检验更有效的假设检验的实现，例如排列检验(通常无法计算) ，同时比渐近分布的临界值更精确。
1. To provide a random sample from the posterior distribution in Bayesian inference. This sample then approximates and summarizes all the essential features of the posterior.
2. To provide a random sample from the posterior distribution in Bayesian inference. This sample then approximates and summarizes all the essential features of the posterior. 提供一份来自后验概率贝叶斯推断的随机样本。然后基于这个样本进行近似和总结后验的所有基本特征。
1. To provide efficient random estimates of the Hessian matrix of the negative log-likelihood function that may be averaged to form an estimate of the Fisher information matrix.[83][84]
2. To provide efficient random estimates of the Hessian matrix of the negative log-likelihood function that may be averaged to form an estimate of the Fisher information matrix. 提供负对数似然函数的海赛矩阵的有效随机估计，这些估计的平均值可以形成费雪信息量 Fisher Information矩阵的估计。[83][84]

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).

Artificial intelligence for games 人工智能在游戏中的应用

Monte Carlo methods have been developed into a technique called Monte-Carlo tree search that is useful for searching for the best move in a game. Possible moves are organized in a search tree and many random simulations are used to estimate the long-term potential of each move. A black box simulator represents the opponent's moves.[85]

Monte Carlo methods have been developed into a technique called Monte-Carlo tree search that is useful for searching for the best move in a game. Possible moves are organized in a search tree and many random simulations are used to estimate the long-term potential of each move. A black box simulator represents the opponent's moves.

The Monte Carlo tree search (MCTS) method has four steps:[86]

The Monte Carlo tree search (MCTS) method has four steps:

1. Starting at root node of the tree, select optimal child nodes until a leaf node is reached.
2. Starting at root node of the tree, select optimal child nodes until a leaf node is reached. 从树的根节点开始，选择最佳的子节点，直到达到叶节点。
1. Expand the leaf node and choose one of its children.
2. Expand the leaf node and choose one of its children. 展开叶节点并选择其中一个子节点。
1. Play a simulated game starting with that node.
2. Play a simulated game starting with that node. 以该节点开始玩一个模拟游戏。
1. Use the results of that simulated game to update the node and its ancestors.
2. 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.

Monte Carlo Tree Search has been used successfully to play games such as Go,[87] Tantrix,[88] Battleship,[89] Havannah,[90] and Arimaa.[91]

Monte Carlo Tree Search has been used successfully to play games such as Go, Tantrix, Battleship, Havannah, and Arimaa.

Design and visuals 设计与视觉效果

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 games, architecture, design, computer generated films, and cinematic special effects.[92]

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 games, architecture, design, computer generated films, and cinematic special effects.

Search and rescue 搜寻与救援

The US Coast Guard utilizes Monte Carlo methods within its computer modeling software SAROPS in order to calculate the probable locations of vessels during search and rescue operations. Each simulation can generate as many as ten thousand data points that are randomly distributed based upon provided variables.[93] Search patterns are then generated based upon extrapolations of these data in order to optimize the probability of containment (POC) and the probability of detection (POD), which together will equal an overall probability of success (POS). Ultimately this serves as a practical application of probability distribution in order to provide the swiftest and most expedient method of rescue, saving both lives and resources.[94]

The US Coast Guard utilizes Monte Carlo methods within its computer modeling software SAROPS in order to calculate the probable locations of vessels during search and rescue operations. Each simulation can generate as many as ten thousand data points that are randomly distributed based upon provided variables. Search patterns are then generated based upon extrapolations of these data in order to optimize the probability of containment (POC) and the probability of detection (POD), which together will equal an overall probability of success (POS). Ultimately this serves as a practical application of probability distribution in order to provide the swiftest and most expedient method of rescue, saving both lives and resources.

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 methods in finance are often used to evaluate investments in projects at a business unit or corporate level, or other financial valuations. They can be used to model project schedules, where simulations aggregate estimates for worst-case, best-case, and most likely durations for each task to determine outcomes for the overall project.[1] Monte Carlo methods are also used in option pricing, default risk analysis.[95][96][97] Additionally, they can be used to estimate the financial impact of medical interventions.[98]

Law 法律

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.[99]

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.

Use in mathematics 数学应用

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.

Integration 积分

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
Errors reduce by a factor of $\displaystyle{ \scriptstyle 1/\sqrt{N} }$Errors reduce by a factor of $\displaystyle{ \scriptstyle 1/\sqrt{N} }$错误由于$\displaystyle{ \scriptstyle 1/\sqrt{N} }$而减少

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] 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.

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 $\displaystyle{ \scriptstyle 1/\sqrt{N} }$ convergence—i.e., quadrupling the number of sampled points halves the error, regardless of the number of dimensions.[100]

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 $\displaystyle{ \scriptstyle 1/\sqrt{N} }$ convergence—i.e., quadrupling the number of sampled points halves the error, regardless of the number of dimensions.

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, recursive stratified sampling, adaptive umbrella sampling[101][102] or the VEGAS algorithm.

A similar approach, the quasi-Monte Carlo method, uses low-discrepancy sequences. 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 sequences. 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 sequential Monte Carlo samplers.[103]

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 sequential Monte Carlo samplers.

Simulation and optimization 模拟与优化

Another powerful and very popular application for random numbers in numerical simulation is in 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[104] 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.

Inverse problems 逆问题

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). 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.[105][106]

Philosophy 哲学

Popular exposition of the Monte Carlo Method was conducted by McCracken[107]. Method's general philosophy was discussed by Elishakoff[108] and Grüne-Yanoff and Weirich[109].

Popular exposition of the Monte Carlo Method was conducted by McCracken. Method's general philosophy was discussed by Elishakoff and Grüne-Yanoff and Weirich.

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• Int Panis, L.; de Nocker, L.; De Vlieger, I.; Torfs, R. (2001). "Trends and uncertainty in air pollution impacts and external costs of Belgian passenger car traffic". International Journal of Vehicle Design. 27 (1–4): 183–194. doi:10.1504/IJVD.2001.001963.
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Category:Numerical analysis

Category:Statistical mechanics

• Sawilowsky, Shlomo S.; Fahoome, Gail C. (2003). Statistics via Monte Carlo Simulation with Fortran. Rochester Hills, MI: JMASM. ISBN 978-0-9740236-0-1.

Category:Computational physics

Category:Sampling techniques

• Silver, David; Veness, Joel (2010). "Monte-Carlo Planning in Large POMDPs" (PDF). In Lafferty, J.; Williams, C. K. I.; Shawe-Taylor, J.; Zemel, R. S.; Culotta, A. (eds.). Advances in Neural Information Processing Systems 23. Neural Information Processing Systems 2010. Neural Information Processing Systems Foundation.

Category:Statistical approximations

• Szirmay-Kalos, László (2008). Monte Carlo Methods in Global Illumination - Photo-realistic Rendering with Randomization. VDM Verlag Dr. Mueller e.K.. ISBN 978-3-8364-7919-6.

Category:Stochastic simulation

Category:Randomized algorithms

• Vose, David (2008). Risk Analysis, A Quantitative Guide (3rd ed.). John Wiley & Sons.

Category:Risk analysis methodologies

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