Editing Monte Carlo method (section)

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

===Physical sciences===
{{Computational physics}}
{{See also|Monte Carlo method in statistical physics}}
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 shield]]s and [[aerodynamics|aerodynamic]] forms as well as in modeling radiation transport for radiation dosimetry calculations.<ref>{{cite journal |doi=10.1088/0031-9155/59/4/R151 |pmid=24486639 |volume=59 |issue=4 |title=GPU-based high-performance computing for radiation therapy |journal=Physics in Medicine and Biology |pages=R151–R182 |bibcode=2014PMB....59R.151J |year=2014 |author-last1=Jia |author-first1=Xun |author-last2=Ziegenhein |author-first2=Peter |author-last3=Jiang |author-first3=Steve B |pmc=4003902 }}</ref><ref>{{cite journal |doi=10.1088/0031-9155/59/6/R183 |volume=59 |issue=6 |title=Advances in kilovoltage x-ray beam dosimetry | journal=Physics in Medicine and Biology |pages=R183–R231 |bibcode=2014PMB....59R.183H |pmid=24584183 |date=Mar 2014 |author-last1=Hill |author-first1=R. | last2=Healy |author-first2=B. |author-last3=Holloway |author-first3=L. |author-last4=Kuncic |author-first4=Z. |author-last5=Thwaites |author-first5=D. |author-last6=Baldock |author-first6=C. |s2cid=18082594 }}</ref><ref>{{cite journal |doi=10.1088/0031-9155/51/13/R17 |pmid=16790908 |volume=51 |issue=13 |title=Fifty years of Monte Carlo simulations for medical physics |journal=Physics in Medicine and Biology |pages=R287–R301 |bibcode=2006PMB....51R.287R |year=2006 |author-last1=Rogers |author-first1=D.W.O. |s2cid=12066026 }}</ref> In [[statistical physics]], [[Monte Carlo molecular modeling]] is an alternative to computational [[molecular dynamics]], and Monte Carlo methods are used to compute [[statistical field theory|statistical field theories]] of simple particle and polymer systems.<ref name=":0" /><ref>{{harvnb|Baeurle|2009}}</ref> [[Quantum Monte Carlo]] methods solve the [[many-body problem]] for quantum systems.<ref name="kol10" /><ref name="dp13" /><ref name="dp04" /> In [[Radiation material science|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.<ref>{{cite journal|author-last1=Möller |author-first1=W. |author-last2=Eckstein |author-first2=W. |date=March 1, 1984 |title=Tridyn — A TRIM simulation code including dynamic composition changes |journal=Nuclear Instruments and Methods in Physics Research Section B: Beam Interactions with Materials and Atoms |volume=2 |issue=1 |pages=814–818 |doi=10.1016/0168-583X(84)90321-5 |bibcode=1984NIMPB...2..814M}}</ref> In experimental [[particle physics]], Monte Carlo methods are used for designing [[particle detector|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<ref>{{harvnb|MacGillivray|Dodd|1982}}</ref> and microwave radiation transmission through a rough planetary surface.<ref>{{harvnb|Golden|1979}}</ref> Monte Carlo methods are also used in the [[Ensemble forecasting|ensemble models]] that form the basis of modern [[Numerical weather prediction|weather forecasting]].

===Engineering===
Monte Carlo methods are widely used in engineering for [[sensitivity analysis]] and quantitative [[probabilistic]] analysis in [[Process design (chemical engineering)|process design]]. The need arises from the interactive, co-linear and non-linear behavior of typical process simulations. For example,
* In [[microelectronics|microelectronics engineering]], Monte Carlo methods are applied to analyze correlated and uncorrelated variations in [[Analog signal|analog]] and [[Digital data|digital]] [[integrated circuits]].
* In [[geostatistics]] and [[geometallurgy]], Monte Carlo methods underpin the design of [[mineral processing]] [[process flow diagram|flowsheets]] and contribute to [[quantitative risk analysis]].{{sfn|Mazhdrakov|Benov|Valkanov|2018|p=250}}
* In [[fluid dynamics]], in particular [[gas dynamics|rarefied gas dynamics]], where the Boltzmann equation is solved for finite [[Knudsen number]] fluid flows using the [[direct simulation Monte Carlo]]<ref>G. A. Bird, Molecular Gas Dynamics, Clarendon, Oxford (1976)</ref> method in combination with highly efficient computational algorithms.<ref>{{cite journal |author-last1=Dietrich |author-first1=S. |author-last2=Boyd |author-first2=I. |year=1996 |title=A Scalar optimized parallel implementation of the DSMC technique |journal=Journal of Computational Physics |volume=126 |issue=2 |pages=328–42 |doi=10.1006/jcph.1996.0141 |bibcode=1996JCoPh.126..328D |doi-access=free }}</ref>
* 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 [[Simultaneous localization and mapping|SLAM]] (simultaneous localization and mapping) algorithm.
* In [[telecommunications]], 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.
* In [[reliability engineering]], Monte Carlo simulation is used to compute system-level response given the component-level response.
* In [[signal processing]] and [[Bayesian inference]], [[particle filter]]s and [[sequential Monte Carlo method|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 measure]]s.<ref>{{cite journal |author-last1=Chen |author-first1=Shang-Ying |author-last2=Hsu |author-first2=Kuo-Chin |author-last3=Fan |author-first3=Chia-Ming |title=Improvement of generalized finite difference method for stochastic subsurface flow modeling |journal=Journal of Computational Physics |date=March 15, 2021 |volume=429 |pages=110002 |doi=10.1016/J.JCP.2020.110002 |bibcode=2021JCoPh.42910002C |s2cid=228828681 }}</ref>

===Climate change and radiative forcing===
The [[IPCC|Intergovernmental Panel on Climate Change]] relies on Monte Carlo methods in [[probability density function]] analysis of [[radiative forcing]].<ref>{{cite book|title=Climate Change 2013 The Physical Science Basis |date=2013 |publisher=[[Cambridge University Press]] |isbn=978-1-107-66182-0 |page=697 |url=http://www.climatechange2013.org/images/report/WG1AR5_ALL_FINAL.pdf |access-date=July 6, 2023}}</ref>

===Computational biology===
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,{{sfn|Ojeda|Garcia|Londono|Chen|2009}} or membranes.{{sfn|Milik|Skolnick|1993}}
The systems can be studied in the coarse-grained or ''ab initio'' frameworks depending on the desired accuracy.
Computer simulations allow monitoring of the local environment of a particular [[biomolecule|molecule]] to see if some [[chemical reaction]] is happening for instance. In cases where it is not feasible to conduct a physical experiment, [[thought experiment]]s can be conducted (for instance: breaking bonds, introducing impurities at specific sites, changing the local/global structure, or introducing external fields).

===Computer graphics===
[[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===
The standards for Monte Carlo experiments in statistics were set by Sawilowsky.<ref>{{cite journal |author-last1=Cassey |author-last2=Smith |year=2014 |title=Simulating confidence for the Ellison-Glaeser Index |journal=Journal of Urban Economics |volume=81 |page=93 |doi=10.1016/j.jue.2014.02.005}}</ref> In applied statistics, Monte Carlo methods may be used for at least four purposes:
# 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.<ref>{{harvnb|Sawilowsky|Fahoome|2003}}</ref>
# To provide implementations of [[Statistical hypothesis testing|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 distribution]]s.
# 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.
# 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.<ref>{{cite journal |doi=10.1198/106186005X78800 |title=Monte Carlo Computation of the Fisher Information Matrix in Nonstandard Settings |journal=Journal of Computational and Graphical Statistics |volume=14 |issue=4 |pages=889–909 |year=2005 |author-last1=Spall |author-first1=James C. |citeseerx=10.1.1.142.738 |s2cid=16090098}}</ref><ref>{{cite journal |doi=10.1016/j.csda.2009.09.018 |title=Efficient Monte Carlo computation of Fisher information matrix using prior information |journal=Computational Statistics & Data Analysis |volume=54 |issue=2 |pages=272–289 |year=2010 |author-last1=Das |author-first1=Sonjoy |author-last2=Spall |author-first2=James C. |author-last3=Ghanem |author-first3=Roger}}</ref>

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).

{{anchor|Monte Carlo tree search}}

===Artificial intelligence for games===
{{Main|Monte Carlo tree search}}
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.<ref>{{cite web|url=http://sander.landofsand.com/publications/Monte-Carlo_Tree_Search_-_A_New_Framework_for_Game_AI.pdf |title=Monte-Carlo Tree Search: A New Framework for Game AI |author-first1=Guillaume |author-last1=Chaslot |author-first2=Sander |author-last2=Bakkes |author-first3=Istvan |author-last3=Szita |author-first4=Pieter |author-last4=Spronck |website=Sander.landofsand.com |access-date=October 28, 2017}}</ref>

The Monte Carlo tree search (MCTS) method has four steps:<ref>{{cite web|url=http://mcts.ai/about/index.html |title=Monte Carlo Tree Search - About|access-date=May 15, 2013 |archive-url=https://web.archive.org/web/20151129023043/http://mcts.ai/about/index.html |archive-date=November 29, 2015 |url-status=dead}}</ref>
# Starting at root node of the tree, select optimal child nodes until a leaf node is reached.
# Expand the leaf node and choose one of its children.
# Play a simulated game starting with that node.
# 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 as [[Go (game)|Go]],<ref>{{cite book|chapter=Parallel Monte-Carlo Tree Search |doi=10.1007/978-3-540-87608-3_6 |volume=5131 |pages=60–71 |series=Lecture Notes in Computer Science |year=2008 |author-last1=Chaslot |author-first1=Guillaume M. J. -B |author-last2=Winands |author-first2=Mark H. M. |author-last3=Van Den Herik |author-first3=H. Jaap |title=Computers and Games |isbn=978-3-540-87607-6 |citeseerx=10.1.1.159.4373}}</ref> [[Tantrix]],<ref>{{cite report|url=https://www.tantrix.com/Tantrix/TRobot/MCTS%20Final%20Report.pdf |title=Monte-Carlo Tree Search in the game of Tantrix: Cosc490 Final Report |author-last=Bruns |author-first=Pete}}</ref> [[Battleship (game)|Battleship]],<ref>{{cite web |url=http://www0.cs.ucl.ac.uk/staff/D.Silver/web/Publications_files/pomcp.pdf |title=Monte-Carlo Planning in Large POMDPs |author-first1=David |author-last1=Silver |author-first2=Joel |author-last2=Veness |website=0.cs.ucl.ac.uk |access-date=October 28, 2017 |archive-date=July 18, 2016 |archive-url=https://web.archive.org/web/20160718050040/http://www0.cs.ucl.ac.uk/staff/d.silver/web/Publications_files/pomcp.pdf |url-status=dead }}</ref> [[Havannah (board game)|Havannah]],<ref>{{cite book|chapter=Improving Monte–Carlo Tree Search in Havannah |doi=10.1007/978-3-642-17928-0_10 |volume=6515 |pages=105–115|bibcode=2011LNCS.6515..105L |series=Lecture Notes in Computer Science |year=2011 |author-last1=Lorentz |author-first1=Richard J. |title=Computers and Games |isbn=978-3-642-17927-3}}</ref> and [[Arimaa]].<ref>{{cite web|url=http://www.arimaa.com/arimaa/papers/ThomasJakl/bc-thesis.pdf |author-first=Tomas |author-last=Jakl |title=Arimaa challenge – comparison study of MCTS versus alpha-beta methods |website=Arimaa.com |access-date=October 28, 2017}}</ref>

{{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 in [[global illumination]] computations that produce photo-realistic images of virtual 3D models, with applications in [[video game]]s, [[architecture]], [[design]], computer generated [[film]]s, and cinematic special effects.{{sfn|Szirmay-Kalos|2008}}

===Search and rescue===
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.<ref>{{cite web|url=http://insights.dice.com/2014/01/03/how-the-coast-guard-uses-analytics-to-search-for-those-lost-at-sea |title=How the Coast Guard Uses Analytics to Search for Those Lost at Sea |work=Dice Insights |date=January 3, 2014}}</ref> 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.<ref>{{cite web|url=http://www.ifremer.fr/web-com/sar2011/Presentations/SARWS2011_STONE_L.pdf |title=Search Modeling and Optimization in USCG's Search and Rescue Optimal Planning System (SAROPS) |author-first1=Lawrence D. |author-last1=Stone |author-first2=Thomas M. |author-last2=Kratzke |author-first3=John R. |author-last3=Frost |website=Ifremer.fr |access-date=October 28, 2017}}</ref>

===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) | Stochastic 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 finance]] are often used to [[Corporate finance#Quantifying uncertainty|evaluate investments in projects]] at a business unit or corporate level, or other financial valuations. They can be used to model [[project management|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.<ref>{{Cite web |title=Project Risk Simulation (BETA) |url=https://risk.octigo.pl/ |access-date=2024-05-21 |website=risk.octigo.pl}}</ref> Monte Carlo methods are also used in option pricing, default risk analysis.<ref>{{cite book|chapter=An Introduction to Particle Methods with Financial Applications |publisher=Springer Berlin Heidelberg |title=Numerical Methods in Finance |date=2012 |isbn=978-3-642-25745-2 |pages=3–49 |series=Springer Proceedings in Mathematics |volume=12 |author-first1=René |author-last1=Carmona |author-first2=Pierre |author-last2=Del Moral |author-first3=Peng |author-last3=Hu |author-first4=Nadia |author-last4=Oudjane |editor-first1=René A. |editor-last1=Carmona |editor-first2= Pierre Del |editor-last2=Moral |editor-first3=Peng |editor-last3=Hu |editor-first4=Nadia |display-editors=3 |editor-last4=Oudjane |doi=10.1007/978-3-642-25746-9_1 |citeseerx=10.1.1.359.7957}}</ref><ref name="kr11">{{cite book|author-last1=Kroese |author-first1=D. P. |author-last2=Taimre |author-first2=T. |author-last3=Botev |author-first3=Z. I. |title=Handbook of Monte Carlo Methods |year=2011 |publisher=John Wiley & Sons}}</ref> Additionally, they can be used to estimate the financial impact of medical interventions.<ref>{{cite journal |doi=10.1371/journal.pone.0189718 |pmid=29284026 |pmc=5746244 |title=A Monte Carlo simulation approach for estimating the health and economic impact of interventions provided at a student-run clinic |journal=[[PLOS ONE]] |volume=12 |issue=12 |pages=e0189718 |year=2017 |author-last1=Arenas |author-first1=Daniel J. |author-last2=Lett |author-first2=Lanair A. |author-last3=Klusaritz |author-first3=Heather |author-last4=Teitelman |author-first4=Anne M. |bibcode=2017PLoSO..1289718A |doi-access=free}}</ref>

===Law===
A Monte Carlo approach was used for evaluating the potential value of a proposed program to help female petitioners in Wisconsin be successful in their applications for [[Harassment Restraining Order|harassment]] and [[Domestic Abuse Restraining Order|domestic abuse restraining orders]]. It was proposed to help women succeed in their petitions by providing them with greater advocacy thereby potentially reducing the risk of [[rape]] and [[physical assault]]. However, there were many variables in play that could not be estimated perfectly, including the effectiveness of restraining orders, the success rate of petitioners both with and without advocacy, and many others. The study ran trials that varied these variables to come up with an overall estimate of the success level of the proposed program as a whole.<ref name="montecarloanalysis">{{cite web|url=http://legalaidresearch.org/wp-content/uploads/Research-Increasing-Access-to-REstraining-Order-for-Low-Income-Victims-of-DV-A-Cost-Benefit-Analysis-of-the-Proposed-Domestic-Abuse-Grant-Program.pdf |title=Increasing Access to Restraining Orders for Low Income Victims of Domestic Violence: A Cost-Benefit Analysis of the Proposed Domestic Abuse Grant Program |publisher=[[State Bar of Wisconsin]] |date=December 2006 |access-date=December 12, 2016 |author-last1=Elwart |author-first1=Liz |author-last2=Emerson |author-first2=Nina |author-last3=Enders |author-first3=Christina |author-last4=Fumia |author-first4=Dani |author-last5=Murphy |author-first5=Kevin |url-status=dead |archive-url=https://web.archive.org/web/20181106220526/https://legalaidresearch.org/wp-content/uploads/Research-Increasing-Access-to-REstraining-Order-for-Low-Income-Victims-of-DV-A-Cost-Benefit-Analysis-of-the-Proposed-Domestic-Abuse-Grant-Program.pdf |archive-date=November 6, 2018}}</ref>

===Library science===
Monte Carlo approach had also been used to simulate the number of book publications based on book [[Literary genre|genre]] in 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 between [[Malaysia]] and [[Japan]].<ref>{{Cite journal|author-last=Dahlan |author-first=Hadi Akbar |date=October 29, 2021 |title=Perbandingan Penerbitan dan Harga Buku Mengikut Genre di Malaysia dan Jepun Menggunakan Data Akses Terbuka dan Simulasi Monte Carlo |url=http://web.usm.my/km/39(2)2021/KM39022021_8.pdf |journal=Kajian Malaysia |volume=39 |issue=2 |pages=179–202 |doi=10.21315/km2021.39.2.8|s2cid=240435973 }}</ref>

===Other===
<!--From old version of [[Postmodernism generator]] page-->
[[Nassim Nicholas Taleb]] writes about Monte Carlo generators in his 2001 book ''[[Fooled by Randomness]]'' as a real instance of the [[reverse Turing test]]: a human can be declared unintelligent if their writing cannot be told apart from a generated one.