Monte Carlo method

Not to be confused withMonte Carlo algorithm.

The approximation of anormal distributionwith a Monte Carlo method

Monte Carlo methods,orMonte Carlo experiments,are a broad class ofcomputationalalgorithmsthat rely onrepeatedrandom samplingto obtain numerical results. The underlying concept is to userandomnessto solve problems that might bedeterministicin principle. The name comes from theMonte Carlo Casinoin Monaco, where the primary developer of the method, mathematicianStanislaw Ulam,was inspired by his uncle's gambling habits.

Monte Carlo methods are mainly used in three distinct problem classes: optimization, numerical integration, and generating draws from a probability distribution. They can also be used to model phenomena with significant uncertainty in inputs, such as calculating the risk of a nuclear power plant failure. Monte Carlo methods are often implemented using computer simulations, and they can provide approximate solutions to problems that are otherwise intractable or too complex to analyze mathematically.

Monte Carlo methods are widely used in various fields of science, engineering, and mathematics, such as physics, chemistry, biology, statistics, artificial intelligence, finance, and cryptography. They have also been applied to social sciences, such as sociology, psychology, and political science. Monte Carlo methods have been recognized as one of the most important and influential ideas of the 20th century, and they have enabled many scientific and technological breakthroughs.

Monte Carlo methods also have some limitations and challenges, such as the trade-off between accuracy and computational cost, the curse of dimensionality, the reliability of random number generators, and the verification and validation of the results.

Overview

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

1. Define a domain of possible inputs
2. Generate inputs randomly from aprobability distributionover the domain
3. Perform adeterministiccomputation of the outputs
4. Aggregate the results

Monte Carlo method applied to approximating the value ofπ

For example, consider aquadrant (circular sector)inscribed in aunit square.Given that the ratio of their areas is⁠π/4⁠,the value ofπcan be approximated using a Monte Carlo method:

1. Draw a square, theninscribea quadrant within it
2. Uniformlyscatter a given number of points over the square
3. Count the number of points inside the quadrant, i.e. having a distance from the origin of less than 1
4. 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 estimateπ.

In this procedure the domain of inputs is the square that circumscribes the quadrant. One can 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 ofπ.

There are two important considerations:

1. If the points are not uniformly distributed, then the approximation will be poor.
2. 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 their use benefitted greatly frompseudorandom number generators,which are far quicker to use than the tables of random numbers that had been previously used for statistical sampling.

Application

Monte Carlo methods are often used inphysicalandmathematicalproblems 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 aprobability distribution.

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

Other examples include modeling phenomena with significantuncertaintyin inputs such as the calculation ofriskin business and, in mathematics, evaluation of multidimensionaldefinite integralswith complicatedboundary conditions.In application to systems engineering problems (space,oil exploration,aircraft design, etc.), Monte Carlo–based predictions of failure,cost overrunsand schedule overruns are routinely better than human intuition or alternative "soft" methods.

In principle, Monte Carlo methods can be used to solve any problem having a probabilistic interpretation. By thelaw of large numbers,integrals described by theexpected valueof some random variable can be approximated by taking theempirical mean(a.k.a.the 'sample mean') of independent samples of the variable. When theprobability distributionof the variable is parameterized, mathematicians often use aMarkov chain Monte Carlo(MCMC) sampler.The central idea is to design a judiciousMarkov chainmodel with a prescribedstationary probability distribution.That is, in the limit, the samples being generated by the MCMC method will be samples from the desired (target) distribution.By theergodic theorem,the stationary distribution is approximated by theempirical measuresof 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 aMarkov processwhose transition probabilities depend on the distributions of the current random states (seeMcKean–Vlasov processes,nonlinear filtering equation).In other instances, a flow of probability distributions with an increasing level of sampling complexity arise (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.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 sampledempirical measures.In contrast with traditional Monte Carlo and MCMC methodologies, thesemean-field particletechniques rely on sequential interacting samples. The terminologymean fieldreflects the fact that each of thesamples(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.

Simple Monte Carlo

Suppose one wants to know theexpected valueμof apopulation(and knows thatμexists), but does not have a formula available to compute it. The simple Monte Carlo method gives an estimate forμby runningnsimulations and averaging the simulations’ results. It has no restrictions on the probability distribution of the inputs to the simulations, requiring only that the inputs are randomly generated and are independent of each other and thatμexists. A sufficiently largenwill produce a value formthat is arbitrarily close toμ;more formally, it will be the case that, for anyε> 0, |μ–m| ≤ε.

Typically, the algorithm to obtainmis

s= 0;
fori= 1 tondo
run the simulation for theithtime, giving resultri;
s=s+ri;
repeat
m=s/n;

An example

Suppose we want to know how many times we should expect to throw three eight-sided dice for the total of the dice throws to be at leastT.We know the expected value exists. The dice throws are randomly distributed and independent of each other. So simple Monte Carlo is applicable:

s= 0;
fori= 1 tondo
throw the three dice untilTis met or first exceeded;ri= the number of throws;
s=s+ri;
repeat
m=s/n;

Ifnis large enough,mwill be withinεofμfor anyε> 0.

Determining a sufficiently largen

General formula

Letε= |μ–m| > 0. Choose the desiredconfidence level– the percent chance that, when the Monte Carlo algorithm completes,mis indeed withinεofμ.Let z be thez-scorecorresponding to that confidence level.

Lets²be the estimated variance, sometimes called the “sample” variance; it is the variance of the results obtained from a relatively small numberkof “sample” simulations. Choose ak;Driels and Shin observe that “even for sample sizes an order of magnitude lower than the number required, the calculation of that number is quite stable."

The following algorithm computess²in one pass while minimizing the possibility that accumulated numerical error produces erroneous results:

s1= 0;
run the simulation for the first time, producing resultr1;
m1=r1;//miis the mean of the firstisimulations
fori = 2 tokdo
run the simulation for theithtime, producing resultri;
δi=ri-mi-1;
mi= mi-1+ (1/i)δi;
si= si-1+ ((i- 1)/i)(δi)2;
repeat
s2=sk/(k- 1);

Note that, when the algorithm completes,m_kis the mean of thekresults.

nis sufficiently large when

$n\geq s^{2}/(z\epsilon )^{2}$

Ifn≤k,thenm_k= m;sufficient sample simulations were done to ensure thatm_kis withinεofμ.Ifn>k,thennsimulations can be run “from scratch,” or, since k simulations have already been done, one can just runn–kmore simulations and add their results into those from the sample simulations:

s=mk*k;
for i =k+ 1 tondo
run the simulation for theithtime, giving resultri;
s=s+ri;
m=s/n;

A formula when simulations' results are bounded

An alternate formula can be used in the special case where all simulation results are bounded above and below.

Choose a value forεthat is twice the maximum allowed difference betweenμandm.Let 0 <δ< 100 be the desired confidence level, expressed as a percentage. Let every simulation resultr₁,r₂,…r_i,…r_nbe such thata≤r_i≤bfor finiteaandb.To have confidence of at leastδthat |μ–m| <ε/2, use a value fornsuch that

$n\geq 2(b-a)^{2}\ln(2/(1-(\delta /100)))/\epsilon ^{2}$

For example, ifδ= 99%, thenn≥ 2(b–a)²ln(2/0.01)/ε²≈ 10.6(b–a)²/ε².

Computational costs

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 incur 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, theembarrassingly parallelnature of the algorithm allows this large cost to be reduced (perhaps to a feasible level) throughparallel computingstrategies in local processors, clusters, cloud computing, GPU, FPGA, etc.

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 usingprobabilisticmetaheuristics(seesimulated annealing).

An early variant of the Monte Carlo method was devised to solve theBuffon's needle problem,in whichπcan be estimated by dropping needles on a floor made of parallel equidistant strips. In the 1930s,Enrico Fermifirst experimented with the Monte Carlo method while studying neutron diffusion, but he did not publish this work.

In the late 1940s,Stanislaw Ulaminvented the modern version of the Markov Chain Monte Carlo method while he was working on nuclear weapons projects at theLos Alamos National Laboratory.In 1946, nuclear weapons physicists at Los Alamos were investigating neutron diffusion in the core of a nuclear weapon.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:

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 toJohn 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 nameMonte Carlo,which refers to theMonte Carlo CasinoinMonacowhere Ulam's uncle would borrow money from relatives to gamble. Monte Carlo methods were central to thesimulationsrequired for further postwar development of nuclear weapons, including the design of the H-bomb, though severely limited by the computational tools at the time. Von Neumann,Nicholas Metropolisand others programmed theENIACcomputer to perform the first fully automated Monte Carlo calculations, of afission weaponcore, in the spring of 1948.In the 1950s Monte Carlo methods were used atLos Alamosfor the development of thehydrogen bomb,and became popularized in the fields ofphysics,physical chemistry,andoperations research.TheRand Corporationand theU.S. Air Forcewere 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 ofHenry P. McKean Jr.on Markov interpretations of a class of nonlinear parabolic partial differential equations arising in fluid mechanics.An earlier pioneering article byTheodore E. Harrisand Herman Kahn, published in 1951, used mean-fieldgenetic-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 ofAlan Turingon genetic type mutation-selection learning machinesand the articles byNils Aall Barricelliat theInstitute for Advanced StudyinPrinceton, New Jersey.

Quantum Monte Carlo,and more specificallydiffusion Monte Carlo methodscan also be interpreted as a mean-field particle Monte Carlo approximation ofFeynman–Kacpath integrals.The origins of Quantum Monte Carlo methods are often attributed to Enrico Fermi andRobert Richtmyerwho developed in 1948 a mean-field particle interpretation of neutron-chain reactions,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.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 ofMarshall N. RosenbluthandArianna W. Rosenbluth.

The use ofSequential Monte Carloin advancedsignal processingandBayesian inferenceis more recent. It was in 1993, that Gordon et al., published in their seminal workthe first application of a Monte Carloresamplingalgorithm 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. Another pioneering article in this field was Genshiro Kitagawa's, on a related "Monte Carlo filter",and the ones by Pierre Del Moraland Himilcon Carvalho, Pierre Del Moral, André Monin and Gérard Saluton 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 theLAAS-CNRS(the Laboratory for Analysis and Architecture of Systems) on radar/sonar and GPS signal processing problems.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.

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,and by Dan Crisan, Pierre Del Moral and Terry Lyons.Further developments in this field were described in 1999 to 2001 by P. Del Moral, A. Guionnet and L. Miclo.

Definitions

There is no consensus on howMonte Carloshould be defined. For example, Ripleydefines most probabilistic modeling asstochastic simulation,withMonte Carlobeing reserved forMonte Carlo integrationand Monte Carlo statistical tests.Sawilowskydistinguishes between asimulation,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).

Here are the examples:

• Simulation: Drawingonepseudo-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.
• 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.
• Monte Carlo simulation: Drawinga large numberof 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 aMonte Carlo simulationof the behavior of repeatedly tossing a coin.

Kalos and Whitlockpoint 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."

Convergence of the Monte Carlo simulation can be checked with theGelman-Rubin statistic.

Monte Carlo and random numbers

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 requiretruly random numbersto be useful (although, for some applications such asprimality testing,unpredictability is vital).Many of the most useful techniques use deterministic,pseudorandomsequences, making it easy to test and re-run simulations. The only quality usually necessary to make goodsimulationsis 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 areuniformly distributedor 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)
• the (pseudo-random) number generator produces values that pass tests for randomness
• there are enough samples to ensure accurate results
• the proper sampling technique is used
• the algorithm used is valid for what is being modeled
• it simulates the phenomenon in question.

Pseudo-random number samplingalgorithms are used to transform uniformly distributed pseudo-random numbers into numbers that are distributed according to a givenprobability distribution.

Low-discrepancy sequencesare 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 calledquasi-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'sRDRANDinstruction set, as compared to those derived from algorithms, like theMersenne Twister,in Monte Carlo simulations of radio flares frombrown dwarfs.No statistically significant difference was found between models generated with typical pseudorandom number generators and RDRAND for trials consisting of the generation of 10⁷random numbers.

Monte Carlo simulation versus "what if" scenarios

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 aprobability distributionfor 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 andtriangular probability distributionsshows 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 (seequantifying 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

Monte Carlo methods are especially useful for simulating phenomena with significantuncertaintyin inputs and systems with manycoupleddegrees of freedom. Areas of application include:

Physical sciences

See also:Monte Carlo method in statistical physics

Monte Carlo methods are very important incomputational physics,physical chemistry,and related applied fields, and have diverse applications from complicatedquantum chromodynamicscalculations to designingheat shieldsandaerodynamicforms as well as in modeling radiation transport for radiation dosimetry calculations.Instatistical physics,Monte Carlo molecular modelingis an alternative to computationalmolecular dynamics,and Monte Carlo methods are used to computestatistical field theoriesof simple particle and polymer systems.Quantum Monte Carlomethods solve themany-body problemfor quantum systems.Inradiation materials science,thebinary collision approximationfor simulatingion implantationis usually based on a Monte Carlo approach to select the next colliding atom.In experimentalparticle physics,Monte Carlo methods are used for designingdetectors,understanding their behavior and comparing experimental data to theory. Inastrophysics,they are used in such diverse manners as to model bothgalaxyevolutionand microwave radiation transmission through a rough planetary surface.Monte Carlo methods are also used in theensemble modelsthat form the basis of modernweather forecasting.

Engineering

Monte Carlo methods are widely used in engineering forsensitivity analysisand quantitativeprobabilisticanalysis inprocess design.The need arises from the interactive, co-linear and non-linear behavior of typical process simulations. For example,

• Inmicroelectronics engineering,Monte Carlo methods are applied to analyze correlated and uncorrelated variations inanaloganddigitalintegrated circuits.
• Ingeostatisticsandgeometallurgy,Monte Carlo methods underpin the design ofmineral processingflowsheetsand contribute toquantitative risk analysis.
• Influid dynamics,in particularrarefied gas dynamics,where the Boltzmann equation is solved for finiteKnudsen numberfluid flows using thedirect simulation Monte Carlomethod in combination with highly efficient computational algorithms.
• Inautonomous robotics,Monte Carlo localizationcan determine the position of a robot. It is often applied to stochastic filters such as theKalman filterorparticle filterthat forms the heart of theSLAM(simultaneous localization and mapping) algorithm.
• Intelecommunications,when planning a wireless network, the design must be proven 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.
• Inreliability engineering,Monte Carlo simulation is used to compute system-level response given the component-level response.
• Insignal processingandBayesian inference,particle filtersandsequential Monte Carlo techniquesare a class ofmean-field particle methodsfor sampling and computing the posterior distribution of a signal process given some noisy and partial observations using interactingempirical measures.

Climate change and radiative forcing

TheIntergovernmental Panel on Climate Changerelies on Monte Carlo methods inprobability density functionanalysis ofradiative forcing.

Computational biology

Monte Carlo methods are used in various fields ofcomputational biology,for example forBayesian inference in phylogeny,or for studying biological systems such as genomes, proteins,or membranes. The systems can be studied in the coarse-grained orab initioframeworks depending on the desired accuracy. Computer simulations allow monitoring of the local environment of a particularmoleculeto see if somechemical reactionis happening for instance. In cases where it is not feasible to conduct a physical experiment,thought experimentscan be conducted (for instance: breaking bonds, introducing impurities at specific sites, changing the local/global structure, or introducing external fields).

Computer graphics

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 therendering equation,making it one of the most physically accurate 3D graphics rendering methods in existence.

Applied statistics

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. Althoughtype I errorand power properties of statistics can be calculated for data drawn from classical theoretical distributions (e.g.,normal curve,Cauchy distribution) forasymptoticconditions (i. e,infinite sample size and infinitesimally small treatment effect), real data often do not have such distributions.
2. To provide implementations ofhypothesis teststhat are more efficient than exact tests such aspermutation tests(which are often impossible to compute) while being more accurate than critical values forasymptotic distributions.
3. To provide a random sample from the posterior distribution inBayesian inference.This sample then approximates and summarizes all the essential features of the posterior.
4. To provide efficient random estimates of the Hessian matrix of the negative log-likelihood function that may be averaged to form an estimate of theFisher informationmatrix.

Monte Carlo methods are also a compromise between approximate randomization and permutation tests. An approximaterandomization testis 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

Main article:Monte Carlo tree search

Monte Carlo methods have been developed into a technique calledMonte-Carlo tree searchthat is useful for searching for the best move in a game. Possible moves are organized in asearch treeand 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:

1. Starting at root node of the tree, select optimal child nodes until a leaf node is reached.
2. Expand the leaf node and choose one of its children.
3. Play a simulated game starting with that node.
4. Use the results of that simulated game to update the node and its ancestors.

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 asGo,Tantrix,Battleship,Havannah,andArimaa.

See also:Computer Go

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 inglobal illuminationcomputations that produce photo-realistic images of virtual 3D models, with applications invideo games,architecture,design,computer generatedfilms,and cinematic special effects.

Search and rescue

TheUS Coast Guardutilizes Monte Carlo methods within its computer modeling softwareSAROPSin order to calculate the probable locations of vessels duringsearch and rescueoperations. 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 ofprobability distributionin order to provide the swiftest and most expedient method of rescue, saving both lives and resources.

Finance and business

See also:Monte Carlo methods in finance,Quasi-Monte Carlo methods in finance,Monte Carlo methods for option pricing,Stochastic modelling (insurance),andStochastic asset model

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 labor 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 financeare often used toevaluate investments in projectsat a business unit or corporate level, or other financial valuations. They can be used to modelproject schedules,where simulations aggregate estimates for worst-case, best-case, and most likely durations for each task to determine outcomes for the overall project.Monte Carlo methods are also used in option pricing, default risk analysis.Additionally, they can be used to estimate the financial impact of medical interventions.

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 forharassmentanddomestic abuse restraining orders.It was proposed to help women succeed in their petitions by providing them with greater advocacy thereby potentially reducing the risk ofrapeandphysical 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.

Library science

Monte Carlo approach had also been used to simulate the number of book publications based on bookgenrein Malaysia. The Monte Carlo simulation utilized previous published National Book publication data and book's price according to book genre in the local market. The Monte Carlo results were used to determine what kind of book genre that Malaysians are fond of and was used to compare book publications betweenMalaysiaandJapan.

Other

Nassim Nicholas Talebwrites about Monte Carlo generators in his 2001 bookFooled by Randomnessas a real instance of thereverse Turing test:a human can be declared unintelligent if their writing cannot be told apart from a generated one.

Use in mathematics

In general, the Monte Carlo methods are used in mathematics to solve various problems by generating suitable random numbers (see alsoRandom 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

Main article:Monte Carlo integration

Monte-Carlo integration works by comparing random points with the value of the function.

Errors reduce by a factor of$\scriptstyle 1/{\sqrt {N}}$.

Deterministicnumerical integrationalgorithms 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, then10¹⁰⁰points are needed for 100 dimensions—far too many to be computed. This is called thecurse of dimensionality.Second, the boundary of a multidimensional region may be very complicated, so it may not be feasible to reduce the problem to aniterated integral.100dimensionsis by no means unusual, since in many physical problems, a "dimension" is equivalent to adegree of freedom.

Monte Carlo methods provide a way out of this exponential increase in computation time. As long as the function in question is reasonablywell-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 thecentral limit theorem,this method displays$\scriptstyle 1/{\sqrt {N}}$convergence—i.e., quadrupling the number of sampled points halves the error, regardless of the number of dimensions.

A refinement of this method, known asimportance samplingin 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 asstratified sampling,recursive stratified sampling,adaptive umbrella samplingor theVEGAS algorithm.

A similar approach, thequasi-Monte Carlo method,useslow-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 theMetropolis–Hastings algorithm,Gibbs sampling,Wang and Landau algorithm,and interacting type MCMC methodologies such as thesequential Monte Carlosamplers.

Simulation and optimization

Main article:Stochastic optimization

Another powerful and very popular application for random numbers in numerical simulation is innumerical 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, acomputer chessprogram could be seen as trying to find the set of, say, 10 moves that produces the best evaluation function at the end. In thetraveling salesman problemthe goal is to minimize distance traveled. There are also applications to engineering design, such asmultidisciplinary design optimization.It has been applied with quasi-one-dimensional models to solve particle dynamics problems by efficiently exploring large configuration space. Referenceis a comprehensive review of many issues related to simulation and optimization.

Thetraveling salesman problemis 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. If instead of the goal being to minimize the total distance traveled to visit each desired destination but rather 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 the optimal path a different simulation is required: 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 the travel decisions to identify the best path to follow taking that uncertainty into account.

Inverse problems

Probabilistic formulation ofinverse problemsleads to the definition of aprobability distributionin the model space. This probability distribution combinespriorinformation 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 normally information on the resolution power of the data is desired. In the general case many parameters are modeled, and an inspection of themarginal probabilitydensities of interest may be impractical, or even useless. But it is possible to pseudorandomly generate a large collection of models according to theposterior probability distributionand 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 thea prioridistribution 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 complexa prioriinformation and data with an arbitrary noise distribution.

Philosophy

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

See also

• Auxiliary-field Monte Carlo
• Biology Monte Carlo method
• Direct simulation Monte Carlo
• Dynamic Monte Carlo method
• Ergodicity
• Genetic algorithms
• Kinetic Monte Carlo
• List of software for Monte Carlo molecular modeling
• Mean-field particle methods
• Monte Carlo method for photon transport
• Monte Carlo methods for electron transport
• Monte Carlo N-Particle Transport Code
• Morris method
• Multilevel Monte Carlo method
• Quasi-Monte Carlo method
• Sobol sequence
• Temporal difference learning

References

Sources

• Anderson, Herbert L.(1986)."Metropolis, Monte Carlo and the MANIAC"(PDF).Los Alamos Science.14:96–108.
• Benov, Dobriyan M. (2016). "The Manhattan Project, the first electronic computer and the Monte Carlo method".Monte Carlo Methods and Applications.22(1): 73–79.doi:10.1515/mcma-2016-0102.S2CID30198383.
• Baeurle, Stephan A. (2009). "Multiscale modeling of polymer materials using field-theoretic methodologies: A survey about recent developments".Journal of Mathematical Chemistry.46(2): 363–426.doi:10.1007/s10910-008-9467-3.S2CID117867762.
• Berg, Bernd A. (2004).Markov Chain Monte Carlo Simulations and Their Statistical Analysis (With Web-Based Fortran Code).Hackensack, NJ: World Scientific.ISBN978-981-238-935-0.
• Binder, Kurt(1995).The Monte Carlo Method in Condensed Matter Physics.New York: Springer.ISBN978-0-387-54369-7.
• Caflisch, R. E.(1998).Monte Carlo and quasi-Monte Carlo methods.Acta Numerica. Vol. 7. Cambridge University Press. pp. 1–49.
• Davenport, J. H. (1992). "Primality testing revisited".Papers from the international symposium on Symbolic and algebraic computation - ISSAC '92.pp. 123–129.CiteSeerX10.1.1.43.9296.doi:10.1145/143242.143290.ISBN978-0-89791-489-5.S2CID17322272.
• Doucet, Arnaud; Freitas, Nando de; Gordon, Neil (2001).Sequential Monte Carlo methods in practice.New York: Springer.ISBN978-0-387-95146-1.
• Eckhardt, Roger (1987)."Stan Ulam, John von Neumann, and the Monte Carlo method"(PDF).Los Alamos Science(15): 131–137.
• Fishman, G. S. (1995).Monte Carlo: Concepts, Algorithms, and Applications.New York: Springer.ISBN978-0-387-94527-9.
• C. Forastero and L. Zamora and D. Guirado and A. Lallena (2010). "A Monte Carlo tool to simulate breast cancer screening programmes".Phys. Med. Biol.55(17): 5213–5229.Bibcode:2010PMB....55.5213F.doi:10.1088/0031-9155/55/17/021.PMID20714045.S2CID30021759.
• Golden, Leslie M. (1979). "The Effect of Surface Roughness on the Transmission of Microwave Radiation Through a Planetary Surface".Icarus.38(3): 451–455.Bibcode:1979Icar...38..451G.doi:10.1016/0019-1035(79)90199-4.
• Gould, Harvey; Tobochnik, Jan (1988).An Introduction to Computer Simulation Methods, Part 2, Applications to Physical Systems.Reading: Addison-Wesley.ISBN978-0-201-16504-3.
• Grinstead, Charles; Snell, J. Laurie (1997).Introduction to Probability.American Mathematical Society.pp.10–11.
• Hammersley, J. M.; Handscomb, D. C. (1975).Monte Carlo Methods.London: Methuen.ISBN978-0-416-52340-9.
• Hartmann, A.K. (2009).Practical Guide to Computer Simulations.World Scientific.ISBN978-981-283-415-7.Archived fromthe originalon February 11, 2009.
• Hubbard, Douglas (2007).How to Measure Anything: Finding the Value of Intangibles in Business.John Wiley & Sons.p.46.ISBN9780470110126.
• Hubbard, Douglas (2009).The Failure of Risk Management: Why It's Broken and How to Fix It.John Wiley & Sons.
• Kahneman, D.; Tversky, A. (1982).Judgement under Uncertainty: Heuristics and Biases.Cambridge University Press.
• Kalos, Malvin H.; Whitlock, Paula A. (2008).Monte Carlo Methods.Wiley-VCH.ISBN978-3-527-40760-6.
• Kroese, D. P.; Taimre, T.; Botev, Z.I. (2011).Handbook of Monte Carlo Methods.New York:John Wiley & Sons.p. 772.ISBN978-0-470-17793-8.
• MacGillivray, H. T.; Dodd, R. J. (1982). "Monte-Carlo simulations of galaxy systems".Astrophysics and Space Science.86(2): 419–435.doi:10.1007/BF00683346.S2CID189849365.
• MacKeown, P. Kevin (1997).Stochastic Simulation in Physics.New York: Springer.ISBN978-981-3083-26-4.
• Metropolis, N.(1987)."The beginning of the Monte Carlo method"(PDF).Los Alamos Science(1987 Special Issue dedicated to Stanislaw Ulam): 125–130.
• Metropolis, N.;Rosenbluth, Arianna W.; Rosenbluth, Marshall N.; Teller, Augusta H.; Teller, Edward (1953)."Equation of State Calculations by Fast Computing Machines".Journal of Chemical Physics.21(6): 1087.Bibcode:1953JChPh..21.1087M.doi:10.1063/1.1699114.OSTI4390578.S2CID1046577.
• Metropolis, N.;Ulam, S.(1949). "The Monte Carlo Method".Journal of the American Statistical Association.44(247): 335–341.doi:10.1080/01621459.1949.10483310.JSTOR2280232.PMID18139350.
• Milik, M.; Skolnick, J. (January 1993). "Insertion of peptide chains into lipid membranes: an off-lattice Monte Carlo dynamics model".Proteins.15(1): 10–25.doi:10.1002/prot.340150104.PMID8451235.S2CID7450512.
• Mosegaard, Klaus; Tarantola, Albert (1995)."Monte Carlo sampling of solutions to inverse problems"(PDF).J. Geophys. Res.100(B7): 12431–12447.Bibcode:1995JGR...10012431M.doi:10.1029/94JB03097.Archived fromthe original(PDF)on March 10, 2021.RetrievedNovember 1,2017.
• Ojeda, P.; Garcia, M.; Londono, A.; Chen, N.Y. (February 2009)."Monte Carlo Simulations of Proteins in Cages: Influence of Confinement on the Stability of Intermediate States".Biophys. J.96(3): 1076–1082.Bibcode:2009BpJ....96.1076O.doi:10.1529/biophysj.107.125369.PMC2716574.PMID18849410.
• 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.
• Int Panis, L.; Rabl, A.; De Nocker, L.; Torfs, R. (2002). Sturm, P. (ed.). "Diesel or Petrol? An environmental comparison hampered by uncertainty".Mitteilungen Institut für Verbrennungskraftmaschinen und Thermodynamik.Heft 81 Vol 1. Technische Universität Graz Austria: 48–54.
• Press, William H.; Teukolsky, Saul A.; Vetterling, William T.; Flannery, Brian P. (1996) [1986].Numerical Recipes in Fortran 77: The Art of Scientific Computing.Fortran Numerical Recipes. Vol. 1 (2nd ed.).Cambridge University Press.ISBN978-0-521-43064-7.
• Ripley, B. D. (1987).Stochastic Simulation.Wiley & Sons.
• Robert, C.; Casella, G. (2004).Monte Carlo Statistical Methods(2nd ed.). New York: Springer.ISBN978-0-387-21239-5.
• Rubinstein, R. Y.; Kroese, D. P. (2007).Simulation and the Monte Carlo Method(2nd ed.). New York: John Wiley & Sons.ISBN978-0-470-17793-8.
• Savvides, Savvakis C. (1994)."Risk Analysis in Investment Appraisal"(PDF).Project Appraisal Journal.9(1).doi:10.2139/ssrn.265905.S2CID2809643.
• Sawilowsky, Shlomo S.; Fahoome, Gail C. (2003).Statistics via Monte Carlo Simulation with Fortran.Rochester Hills, MI: JMASM.ISBN978-0-9740236-0-1.
• Sawilowsky, Shlomo S. (2003)."You think you've got trivials?".Journal of Modern Applied Statistical Methods.2(1): 218–225.doi:10.22237/jmasm/1051748460.
• 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. Archived fromthe original(PDF)on May 25, 2012.RetrievedMarch 15,2011.
• Szirmay-Kalos, László (2008).Monte Carlo Methods in Global Illumination - Photo-realistic Rendering with Randomization.VDM Verlag Dr. Mueller e.K.ISBN978-3-8364-7919-6.
• Tarantola, Albert(2005).Inverse Problem Theory.Philadelphia: Society for Industrial and Applied Mathematics.ISBN978-0-89871-572-9.
• Vose, David (2008).Risk Analysis, A Quantitative Guide(3rd ed.).John Wiley & Sons.ISBN9780470512845.
• Mazhdrakov, Metodi; Benov, Dobriyan; Valkanov, Nikolai (2018).The Monte Carlo Method. Engineering Applications.ACMO Academic Press.ISBN978-619-90684-3-4.

External links

• Media related toMonte Carlo methodat Wikimedia Commons