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| {{distinguish|Monte Carlo algorithm}}
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| {{Computational physics}}
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| '''Monte Carlo methods''' (or '''Monte Carlo experiments''') are a broad class of [[computation]]al [[algorithm]]s that rely on repeated [[random]] sampling to obtain numerical results; typically one runs simulations many times over in order to obtain the distribution of an unknown probabilistic entity. The name comes from the resemblance of the technique to the act of playing and recording your results in a real gambling casino. They are often used in [[physics|physical]] and [[mathematics|mathematical]] problems and are most useful when it is difficult or impossible to obtain a [[closed-form expression]], or infeasible to apply a [[deterministic algorithm]]. Monte Carlo methods are mainly used in three distinct problems classes: [[optimization]], [[numerical integration]] and generation of draws from a [[probability distribution]].
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| In physics-related problems, Monte Carlo methods are quite useful for simulating systems with many [[coupling (physics)|coupled]] degrees of freedom, such as fluids, disordered materials, strongly coupled solids, and cellular structures (see [[cellular Potts model]]). Other examples include modeling phenomena with significant [[uncertainty]] in inputs, such as the calculation of [[risk]] in business; and, in math, evaluation of multidimensional [[Integral|definite integral]]s with complicated boundary conditions. In application to space and oil exploration problems, Monte Carlo-based predictions of failures, [[cost overrun]]s and schedule overruns are routinely better than human intuition or alternative "soft" methods.<ref>{{harvnb|Hubbard|2009}}</ref>
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| The modern version of the Monte Carlo method was invented in the late 1940s by [[Stanislaw Ulam]], while he was working on nuclear weapons projects at the [[Los Alamos National Laboratory]]. It was named by [[Nicholas Metropolis]], after the [[Monte Carlo Casino]], where Ulam's uncle often gambled.<ref name=Metropolis87>{{harvnb|Metropolis|1987}}</ref> Immediately after Ulam's breakthrough, [[John von Neumann]] understood its importance and programmed the [[ENIAC]] computer to carry out Monte Carlo calculations.
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| == Introduction ==
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| [[File:Pi 30K.gif|thumb|right|Monte Carlo method applied to approximating the value of {{pi}}. After placing 30000 random points, the estimate for {{pi}} is within 0.07% of the actual value. This happens with an approximate probability of 20%.]]
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| Monte Carlo methods vary, but tend to follow a particular pattern:
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| # Define a domain of possible inputs.
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| # Generate inputs randomly from a [[probability distribution]] over the domain.
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| # Perform a [[Deterministic algorithm|deterministic]] computation on the inputs.
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| # Aggregate the results.
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| For example, consider a circle inscribed in a unit square. Given that the circle and the square have a ratio of areas that is {{pi}}/4, the value of [[pi|{{pi}}]] can be approximated using a Monte Carlo method:<ref name=Kalos>{{harvnb|Kalos|Whitlock|2008}}</ref>
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| # Draw a square on the ground, then [[inscribed figure|inscribe]] a circle within it.
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| # [[uniform distribution (continuous)|Uniformly]] scatter some objects of uniform size (grains of rice or sand) over the square.
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| # Count the number of objects inside the circle and the total number of objects.
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| # The ratio of the two counts is an estimate of the ratio of the two areas, which is {{pi}}/4. Multiply the result by 4 to estimate {{pi}}.
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| In this procedure the domain of inputs is the square that circumscribes our circle. We generate random inputs by scattering grains over the square then perform a computation on each input (test whether it falls within the circle). Finally, we aggregate the results to obtain our final result, the approximation of {{pi}}.
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| If the grains are not uniformly distributed, then our approximation will be poor. Secondly, there should be a large number of inputs. The approximation is generally poor if only a few grains are randomly dropped into the whole square. On average, the approximation improves as more grains are dropped.
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| == History ==
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| 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 a [[probabilistic]] [[meta-algorithm|analog]] (see [[Simulated annealing]]).
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| <!-- work on small samples by [[William Sealy Gosset]] was mentioned in an earlier draft, but it is not clear how it was related. -->
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| An early variant of the Monte Carlo method can be seen in the [[Buffon's needle]] experiment, in which {{pi}} can be estimated by dropping needles on a floor made of parallel strips of wood. In the 1930s, [[Enrico Fermi]] first experimented with the Monte Carlo method while studying neutron diffusion, but did not publish anything on it.<ref name=Metropolis87/>
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| In 1946, physicists at [[Los Alamos Scientific Laboratory]] were investigating [[radiation shielding]] and the distance that [[neutrons]] would likely travel through various materials. 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. [[Stanislaw Ulam]] had the idea of using random experiments. He recounts his inspiration as follows:
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| ::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)|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.
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| :::–[[Stanislaw Ulam]]<ref>{{harvnb|Eckhardt|1987}}</ref>
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| Being secret, the work of von Neumann and Ulam required a code name. [[Von Neumann]] chose the name ''Monte Carlo.'' The name refers to the [[Monte Carlo Casino]] in [[Monaco]] where Ulam's uncle would borrow money to gamble.<ref name="Measure Anything pg. 46">{{harvnb|Hubbart|2007}}</ref><ref>{{harvnb|Grinstead|Snell|1997}}</ref><ref>{{harvnb|Anderson |1986}}</ref> Using [[A Million Random Digits with 100,000 Normal Deviates|lists of "truly random" random numbers]] was extremely slow, but von Neumann developed a way to calculate [[pseudorandom number]]s, 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.
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| Monte Carlo methods were central to the [[simulation]]s required for the [[Manhattan Project]], though severely limited by the computational tools at the time. In the 1950s they were used at [[Los Alamos National Laboratory|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.
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| Uses of Monte Carlo methods require large amounts of random numbers, and it was their use that spurred the development of [[pseudorandom number generator]]s, which were far quicker to use than the tables of random numbers that had been previously used for statistical sampling.
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| ==Definitions==
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| There is no consensus on how ''Monte Carlo'' should be defined. For example, Ripley<ref name=Ripley>{{harvnb|Ripley|1987}}</ref> defines most probabilistic modeling as ''[[stochastic simulation]]'', with ''Monte Carlo'' being reserved for [[Monte Carlo integration]] and Monte Carlo statistical tests. [[Shlomo Sawilowsky|Sawilowsky]]<ref name=Sawilowsky>{{harvnb|Sawilowsky|2003}}</ref> 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 determine the properties of some phenomenon (or behavior). Examples:
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| *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.
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| *Monte Carlo method: The area of an irregular figure inscribed in a unit square can be determined by throwing darts at the square and computing the ratio of hits within the irregular figure to the total number of darts thrown. This is a Monte Carlo method of determining area, but not a simulation.
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| *Monte Carlo simulation: Drawing '''a large number''' of pseudo-random uniform variables from the interval (0,1], 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.
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| Kalos and Whitlock<ref name=Kalos/> 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."
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| ===Monte Carlo and random numbers===
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| Monte Carlo simulation methods do not always require [[Random number generation#"True" random numbers vs. pseudorandom numbers|truly random number]]s to be useful — while for some applications, such as [[primality testing]], unpredictability is vital.<ref>{{harvnb|Davenport|1992}}</ref> Many of the most useful techniques use deterministic, [[pseudorandom number generator|pseudorandom]] sequences, making it easy to test and re-run simulations. The only quality usually necessary to make good [[simulation]]s is for the pseudo-random sequence to appear "random enough" in a certain sense.
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| What this means depends on the application, but typically they should pass a series of statistical tests. Testing that the numbers are [[Uniform distribution (continuous)|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 is also often desirable/necessary.
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| Sawilowsky lists the characteristics of a high quality Monte Carlo simulation:<ref name=Sawilowsky/>
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| *the (pseudo-random) number generator has certain characteristics (''e.g.'', a long “period” before the sequence repeats)
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| *the (pseudo-random) number generator produces values that pass tests for randomness
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| *there are enough samples to ensure accurate results
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| *the proper sampling technique is used
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| *the algorithm used is valid for what is being modeled
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| *it simulates the phenomenon in question.
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| [[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]].
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| [[Low-discrepancy sequences]] are often used instead of random sampling from a space as they ensure even coverage and normally have a faster order of convergence than Monte Carlo simulations using random or pseudorandom sequences. Methods based on their use are called [[quasi-Monte Carlo method]]s.
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| === Monte Carlo simulation versus "what if" scenarios ===
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| 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.<ref>{{harvnb|Vose|2000|page=13}}</ref>
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| By contrast, Monte Carlo simulations sample [[probability distribution]] for each variable to produce hundreds or thousands of possible outcomes. The results are analyzed to get probabilities of different outcomes occurring.<ref>{{harvnb|Vose|2000|page=16}}</ref> For example, a comparison of a spreadsheet cost construction model run using traditional “what if” scenarios, and then run again with Monte Carlo simulation and Triangular probability distributions shows that the Monte Carlo analysis has a narrower range than the “what if” analysis.{{Examples|date=May 2012}} This is because the “what if” analysis gives equal weight to all scenarios (see [[Corporate finance#Quantifying uncertainty|quantifying uncertainty in corporate finance]]), while Monte Carlo method hardly samples in the very low probability regions. The samples in such regions are called "rare events".
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| ==Applications==
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| Monte Carlo methods are especially useful for simulating phenomena with significant [[uncertainty]] in inputs and systems with a large number of [[coupling (physics)|coupled]] degrees of freedom. Areas of application include:
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| ===Physical sciences===
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| {{See also|Monte Carlo method in statistical physics}}
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| 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. 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>{{harvnb|Baeurle|2009}}</ref> [[Quantum Monte Carlo]] methods solve the [[many-body problem]] for quantum systems. 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 the evolution of [[Galaxy|galaxies]]<ref>{{harvnb|MacGillivray|Dodd|1982}}</ref> and the transmission of microwave radiation 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]].
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| ===Engineering===
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| 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,
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| * 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]].
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| * In [[geostatistics]] and [[geometallurgy]], Monte Carlo methods underpin the design of [[mineral processing]] [[process flow diagram|flowsheets]] and contribute to quantitative [[Quantitative risk analysis|risk analysis]].
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| * In [[wind energy]] yield analysis, the predicted energy output of a wind farm during its lifetime is calculated giving different levels of uncertainty ([[Percentile|P90]], P50, etc.)
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| * impacts of pollution are simulated<ref name="IntPanis1">{{harvnb|Int Panis|De Nocker|De Vlieger|Torfs|2001}}</ref> and diesel compared with petrol.<ref name="IntPanis2">{{harvnb|Int Panis|Rabl|De Nocker|Torfs|2002}}</ref>
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| * 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.
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| * 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.
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| * In [[reliability engineering]], one can use Monte Carlo simulation to generate [[mean time between failures]] and [[mean time to repair]] for components.
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| ===Computational biology===
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| Monte Carlo methods are used in computational biology, such for as [[Bayesian inference in phylogeny]].
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| Biological systems such as proteins<ref>{{harvnb|Ojeda|et al.|2009}},</ref> membranes,<ref>{{harvnb|Milik|Skolnick|1993}}</ref> images of cancer,<ref>{{harvnb|Forastero|Zamora|Guirado|Lallena|2010}}</ref> are being studied by means of computer simulations.
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| The systems can be studied in the coarse-grained or ''ab initio'' frameworks depending on the desired accuracy.
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| Computer simulations allow us to monitor the local environment of a particular molecule to see if some chemical
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| reaction is happening for instance. We can also conduct [[thought experiment]]s when the physical experiments are not feasible,
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| for instance breaking bonds, introducing impurities at specific sites, changing the local/global structure, or introducing external fields.
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| ===Computer graphics===
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| [[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.
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| ===Applied statistics===
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| In applied statistics, Monte Carlo methods are generally used for two purposes:
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| #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>
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| #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.
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| 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).
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| {{anchor|Monte Carlo tree search}}
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| ===Artificial intelligence for games===
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| {{Main|Monte Carlo tree search}}
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| 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 a large number of random simulations are used to estimate the long-term potential of each move. A black box simulator represents the opponent's moves.<ref>http://sander.landofsand.com/publications/Monte-Carlo_Tree_Search_-_A_New_Framework_for_Game_AI.pdf</ref>
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| The Monte Carlo Tree Search (MCTS) method has four steps:<ref>[http://mcts.ai/about/index.html Monte Carlo Tree Search - About<!-- Bot generated title -->]</ref>
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| #Starting at root node of the tree, select optimal child nodes until a leaf node is reached.
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| #Expand the leaf node and choose one of its children.
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| #Play a simulated game starting with that node.
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| #Use the results of that simulated game to update the node and its ancestors.
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| 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.
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| Monte Carlo Tree Search has been used successfully to play games such as [[Go (game)|Go]],<ref>[http://link.springer.com/chapter/10.1007/978-3-540-87608-3_6 Parallel Monte-Carlo Tree Search - Springer<!-- Bot generated title -->]</ref> [[Tantrix]],<ref>http://www.tantrix.com:4321/Tantrix/TRobot/MCTS%20Final%20Report.pdf</ref> [[Battleship (game)|Battleship]],<ref>http://www0.cs.ucl.ac.uk/staff/D.Silver/web/Publications_files/pomcp.pdf</ref> [[Havannah]],<ref>[http://link.springer.com/chapter/10.1007/978-3-642-17928-0_10 Improving Monte–Carlo Tree Search in Havannah - Springer<!-- Bot generated title -->]</ref> and [[Arimaa]].<ref>http://www.arimaa.com/arimaa/papers/ThomasJakl/bc-thesis.pdf</ref>
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| {{See also|Computer Go}}
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| ===Design and visuals===
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| 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.<ref>{{harvnb|Szirmay-Kalos|2008}}</ref>
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| ===Finance and business===
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| {{See also|Monte Carlo methods for option pricing| Stochastic modelling (insurance) | Stochastic asset model}}
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| [[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 to evaluate [[derivative (finance)|financial derivatives]]. 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.
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| ==Use in mathematics==
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| In general, Monte Carlo methods are used in mathematics to solve various problems by generating suitable random numbers 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.
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| === Integration ===
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| {{Main|Monte Carlo integration}}
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| [[File:Monte-carlo2.gif|thumb|Monte-Carlo integration works by comparing random points with the value of the function]]
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| [[File:Monte-Carlo method (errors).png|thumb|Errors reduce by a factor of <math>\scriptstyle 1/\sqrt{N}</math>]]
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| Deterministic [[numerical integration]] algorithms work well in a small number of dimensions, but encounter two problems when the functions have many variables. First, the number of function evaluations needed increases rapidly with the number of dimensions. For example, if 10 evaluations provide adequate accuracy in one dimension, then [[googol|10<sup>100</sup>]] points are needed for 100 dimensions—far too many to be computed. This is called the [[curse of dimensionality]]. Second, the boundary of a multidimensional region may be very complicated, so it may not be feasible to reduce the problem to a series of nested one-dimensional integrals.<ref name=Press>{{harvnb|Press|Teukolsky|Vetterling|Flannery|1996}}</ref> 100 [[dimension]]s is by no means unusual, since in many physical problems, a "dimension" is equivalent to a [[degrees of freedom (physics and chemistry)|degree of freedom]].
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| 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 <math>\scriptstyle 1/\sqrt{N}</math> convergence—i.e., quadrupling the number of sampled points halves the error, regardless of the number of dimensions.<ref name=Press/>
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| 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]], [[Monte Carlo integration#Recursive stratified sampling|recursive stratified sampling]], adaptive umbrella sampling<ref>{{cite journal|last=MEZEI|first=M|title=Adaptive umbrella sampling: Self-consistent determination of the non-Boltzmann bias|journal=Journal of Computational Physics|date=31 December 1986|volume=68|issue=1|pages=237–248|doi=10.1016/0021-9991(87)90054-4|bibcode = 1987JCoPh..68..237M|ref=harv }}</ref><ref>{{cite journal|last=Bartels|first=Christian|coauthors=Karplus, Martin|title=Probability Distributions for Complex Systems: Adaptive Umbrella Sampling of the Potential Energy|journal=The Journal of Physical Chemistry B|date=31 December 1997|volume=102|issue=5|pages=865–880|doi=10.1021/jp972280j|ref=harv}}</ref> or the [[VEGAS algorithm]].
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| A similar approach, the [[quasi-Monte Carlo method]], uses [[low-discrepancy sequence]]s. 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.
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| 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]] and the [[Wang and Landau algorithm]].
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| === Simulation and optimization ===
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| {{Main|Stochastic optimization}}
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| Another powerful and very popular application for random numbers in numerical simulation is in [[Optimization (mathematics)|numerical optimization]]. The problem is to minimize (or maximize) functions of some vector that often has a large number of 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]].
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| 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.
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| Simulating industry problems using spreadsheet software is a powerful application of Monte Carlo simulation. With a basic spreadsheet tool and some formulas built into the sheet, industry can often attain good solutions without having to procure expensive simulation software.
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| ===Inverse problems===
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| Probabilistic formulation of [[inverse problem]]s leads to the definition of a [[probability distribution]] in the model space. This probability distribution combines [[prior probability|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.).
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| 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 a large number of 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.
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| 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.<ref>{{harvnb|Mosegaard|Tarantola|1995}}</ref><ref>{{harvnb|Tarantola|2005}}</ref>
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| == See also ==
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| {{Portal|Statistics}}
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| * [[Auxiliary field Monte Carlo]]
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| * [[Biology Monte Carlo method]]
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| * [[Comparison of risk analysis Microsoft Excel add-ins]]
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| * [[Direct simulation Monte Carlo]]
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| * [[Dynamic Monte Carlo method]]
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| * [[Kinetic Monte Carlo]]
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| * [[List of software for Monte Carlo molecular modeling]]
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| * [[Monte Carlo method for photon transport]]
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| * [[Monte Carlo methods for electron transport]]
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| * [[Morris method]]
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| * [[Quasi-Monte Carlo method]]
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| * [[Sobol sequence]]
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| ==Notes==
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| {{Reflist | 4}}
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| ==References==
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| *{{Cite journal|first=Stephan A. |last=Baeurle|title=Multiscale modeling of polymer materials using field-theoretic methodologies: A survey about recent developments|journal=Journal of Mathematical Chemistry|volume=46|issue=2|pages=363–426|year=2009|doi=10.1007/s10910-008-9467-3 | ref=harvnb}}
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| *{{Cite book |title=Markov Chain Monte Carlo Simulations and Their Statistical Analysis (With Web-Based Fortran Code) |last=Berg |first=Bernd A. |authorlink= |coauthors= |year=2004 |publisher=World Scientific |location=Hackensack, NJ |isbn=981-238-935-0 |page= |pages= |url= | ref=harvnb}}
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| *{{Cite book |title=The Monte Carlo Method in Condensed Matter Physics |last=Binder |first=Kurt |authorlink=Kurt Binder |coauthors= |year=1995 |publisher=Springer |location=New York |isbn=0-387-54369-4 |page= |pages= |url= | ref=harvnb}}
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| *{{Cite book |title=Monte Carlo and quasi-Monte Carlo methods |last=Caflisch |first=R. E. |authorlink= |coauthors= |year=1998 |series=Acta Numerica |volume=7 |publisher=Cambridge University Press |location= |isbn= |page= |pages=1–49 |url= | ref=harvnb}}
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| *{{Cite book |title=Sequential Monte Carlo methods in practice |last=Doucet |first=Arnaud |authorlink= |coauthors=Freitas, Nando de; Gordon, Neil |year=2001 |publisher=Springer |location=New York |isbn=0-387-95146-6 |page= |pages= |url= | ref=harvnb}}
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| *{{cite journal | last=Eckhardt |first= Roger |year=1987 |title=Stan Ulam, John von Neumann, and the Monte Carlo method|journal= Los Alamos Science, Special Issue |issue=15|pages= 131–137 | ref=harvnb| url = http://www.lanl.gov/history/admin/files/Stan_Ulam_John_von_Neumann_and_the_Monte_Carlo_Method.pdf}}
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| *{{Cite book |title=Monte Carlo: Concepts, Algorithms, and Applications |last=Fishman |first=G. S. |authorlink= |coauthors= |year=1995 |publisher=Springer |location=New York |isbn=0-387-94527-X |page= |pages= |url= | ref=harvnb}}
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| *{{cite journal | title = A Monte Carlo tool to simulate breast cancer screening programmes | author = C. Forastero and L. Zamora and D. Guirado and A. Lallena | journal = Phys. In Med. And Biol. | volume = 55 | issue = 17 | page = 5213 | year = 2010 | doi = 10.1088/0031-9155/55/17/021 | ref=harvnb |bibcode = 2010PMB....55.5213F }}
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| *{{Cite book |title=An Introduction to Computer Simulation Methods, Part 2, Applications to Physical Systems |last=Gould |first=Harvey |authorlink= |coauthors=Tobochnik, Jan |year=1988 |publisher=Addison-Wesley |location=Reading |isbn=0-201-16504-X |page= |pages= |url= | ref=harvnb}}
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| *{{Cite book |first=Charles|last=Grinstead|first2= J. Laurie|last2=Snell |title=Introduction to Probability| pages= 10–11|publisher=[[American Mathematical Society]]|year= 1997 | ref=harvnb}}
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| *{{cite book| last=Hartmann| first= A.K.|year=2009|title=Practical Guide to Computer Simulations|publisher=World Scientific|isbn=978-981-283-415-7|url=http://www.worldscibooks.com/physics/6988.html}}
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| *{{cite book|first=Douglas |last=Hubbard |title=How to Measure Anything: Finding the Value of Intangibles in Business | page= 46 |publisher=[[John Wiley & Sons]] | year= 2007 | ref=harvnb}}
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| *{{cite book|first=Douglas |last=Hubbard |title=The Failure of Risk Management: Why It's Broken and How to Fix It |publisher= [[John Wiley & Sons]] |year=2009 | ref=harvnb}}
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| *{{Cite book |title=Judgement under Uncertainty: Heuristics and Biases |last=Kahneman |first=D. |authorlink= |coauthors=Tversky, A. |year=1982 |publisher=Cambridge University Press |location= |isbn= |page= |pages= |url= | ref=harvnb}}
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| *{{cite book|first=Malvin H.|last=Kalos|first2=Paula A.|last2=Whitlock|title=Monte Carlo Methods| publisher=[[Wiley-VCH]]|year=2008|isbn=978-3-527-40760-6|ref=harvnb}}
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| *{{Cite book |title=Handbook of Monte Carlo Methods|last=Kroese |first=D. P. |authorlink= |coauthors=Taimre, T.; Botev, Z.I. |year=2011 |publisher=[[John Wiley & Sons]] |location=New York |isbn=0-470-17793-4 |page=772 |url=http://www.montecarlohandbook.org | ref=harvnb}}
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| *{{Cite journal|first= H. T.|last= MacGillivray|first2= R. J. |last2=Dodd |title=Monte-Carlo simulations of galaxy systems |journal=[[Astrophysics and Space Science]]|volume= 86|issue= 2|year= 1982 | publisher=[[Springer Netherlands]] |url=http://www.springerlink.com/content/rp3g1q05j176r108/fulltext.pdf | ref=harvnb}}
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| *{{Cite book |title=Stochastic Simulation in Physics |last=MacKeown |first=P. Kevin |authorlink= |coauthors= |year=1997 |publisher=Springer |location=New York |isbn=981-3083-26-3 |page= |pages= |url= | ref=harvnb}}
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| *{{Cite journal|last=Metropolis |first=N. |authorlink=Nicholas Metropolis|url=http://library.lanl.gov/la-pubs/00326866.pdf|title=The beginning of the Monte Carlo method|journal=Los Alamos Science|issue=1987 Special Issue dedicated to Stanislaw Ulam|pages=125–130|year=1987 | ref=harvnb}}
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| *{{Cite journal |last=Metropolis |first=Nicholas |authorlink=Nicholas Metropolis |coauthors=Rosenbluth, Arianna W.; Rosenbluth, Marshall N.; Teller, Augusta H.; Teller, Edward |year=1953 |month= |title=[[Equation of State Calculations by Fast Computing Machines]] |journal=Journal of Chemical Physics |volume=21 |issue=6 |page=1087 |doi=10.1063/1.1699114 |url= |accessdate= |quote= | ref=harvnb|bibcode = 1953JChPh..21.1087M }}
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| *{{Cite journal |last=Metropolis |first=N. |authorlink=Nicholas Metropolis |last2=Ulam|first2 = S. |authorlink2=Stanislaw Ulam |year=1949 |month= |title=The Monte Carlo Method |journal=Journal of the American Statistical Association |volume=44 |issue=247 |pages=335–341 |doi=10.2307/2280232|accessdate= |quote= |pmid=18139350 |publisher=American Statistical Association | ref=harvnb |jstor=2280232}}
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| *{{cite journal | doi = 10.1002/prot.340150104 | title = Insertion of peptide chains into lipid membranes: an off-lattice Monte Carlo dynamics model | author = M. Milik and J. Skolnick| journal = Proteins | volume = 15 | issue = 1 | pages = 10–25 |date=Jan 1993 | pmid = 8451235 | ref=harvnb}}
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| *{{Cite journal |last=Mosegaard |first=Klaus |authorlink= |coauthors=Tarantola, Albert |year=1995 |month= |title=Monte Carlo sampling of solutions to inverse problems |journal=J. Geophys. Res. |volume=100 |issue=B7 |pages=12431–12447 |doi=10.1029/94JB03097 |url= |accessdate= |quote= | ref=harvnb|bibcode = 1995JGR...10012431M }}
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| *{{cite journal | title = Monte Carlo Simulations of Proteins in Cages: Influence of Confinement on the Stability of Intermediate States | author = P. Ojeda and M. Garcia and A. Londono and N.Y. Chen | journal = Biophys. Jour. | volume = 96 | issue = 3 | pages = 1076–1082 |date=Feb 2009 | doi = 10.1529/biophysj.107.125369 | publisher = Biophysical Society | ref=harvnb|bibcode = 2009BpJ....96.1076O }}
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| *{{cite journal |doi=10.1504/IJVD.2001.001963 |author=Int Panis L |coauthors= De Nocker L, De Vlieger I, Torfs R |year=2001 |title=Trends and uncertainty in air pollution impacts and external costs of Belgian passenger car traffic International |journal=Journal of Vehicle Design |volume=27 |issue=1–4 |pages=183–194 | ref=harvnb}}
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| *{{cite journal |author=Int Panis L, Rabl A, De Nocker L, Torfs R |year=2002 |title=Diesel or Petrol ? An environmental comparison hampered by uncertainty |journal=Mitteilungen Institut für Verbrennungskraftmaschinen und Thermodynamik |editor=P. Sturm |publisher=Technische Universität Graz Austria |volume=Heft 81 Vol 1 |pages=48–54 | ref=harvnb}}
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| *{{cite book |last1=Press |first1= William H.|authorlink1= |last2= Teukolsky|first2=Saul A. |last3= Vetterling|first3=William T. |last4= Flannery|first4=Brian P. |title= Numerical Recipes in Fortran 77: The Art of Scientific Computing |edition= Second |series= Fortran Numerical Recipes|volume= 1 |year=1996 |origyear= 1986|publisher= [[Cambridge University Press]]|isbn=0-521-43064-X |ref=harvnb}}
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| *{{Cite book| last=Ripley|first=B. D.|title=Stochastic Simulation|publisher = [[Wiley & Sons]]|year=1987}}
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| *{{Cite book |title=Monte Carlo Statistical Methods |last=Robert |first=C. P. |authorlink= |coauthors=Casella, G. |year=2004 |edition=2nd |publisher=Springer |location=New York |isbn=0-387-21239-6 |page= |pages= |url= | ref=harvnb}}
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| *{{Cite book |title=Simulation and the Monte Carlo Method |last=Rubinstein |first=R. Y. |authorlink= |coauthors=Kroese, D. P. |year=2007 |edition=2nd |publisher=John Wiley & Sons |location=New York |isbn=978-0-470-17793-8 |page= |pages= |url= | ref=harvnb}}
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| *{{Cite journal|last= Savvides|first= Savvakis C.|title=Risk Analysis in Investment Appraisal|journal= Project Appraisal Journal|volume= 9|year= 1994|doi=10.2139/ssrn.265905|ref=harvnb|issue= 1}}
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| *{{cite book |last1=Sawilowsky |first1=Shlomo S. |last2=Fahoome |first2=Gail C. |year=2003 |title=Statistics via Monte Carlo Simulation with Fortran |publisher=Rochester Hills, MI: JMASM |isbn=0-9740236-0-4 | ref=harvnb}}
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| *{{cite journal|last1=Sawilowsky |first1=Shlomo S. |title=You think you've got trivials? |journal =[[Journal of Modern Applied Statistical Methods]] | volume=2|issue=1|pages = 218–225|year=2003 | url = http://education.wayne.edu/jmasm/sawilowsky_effect_size_debate.pdf| ref=harvnb}}
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| *{{Cite book | last =Silver | first =David | last2 =Veness | first2 =Joel | year =2010 | contribution =Monte-Carlo Planning in Large POMDPs | contribution-url =http://books.nips.cc/papers/files/nips23/NIPS2010_0740.pdf | editor-last =Lafferty | editor-first =J. | editor2-last =Williams | editor2-first =C. K. I. | editor3-last =Shawe-Taylor | editor3-first =J. | editor4-last =Zemel | editor4-first =R. S. | editor5-last =Culotta | editor5-first =A. | title =Advances in Neural Information Processing Systems 23 | publisher =Neural Information Processing Systems Foundation | volume = | pages = | isbn = | doi = | ref =harv}}
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| *{{cite book| first=László |last=Szirmay-Kalos|title=Monte Carlo Methods in Global Illumination - Photo-realistic Rendering with Randomization| publisher = VDM Verlag Dr. Mueller e.K.|year = 2008|isbn=978-3-8364-7919-6}}
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| *{{Cite book |title=Inverse Problem Theory |last=Tarantola |first=Albert |authorlink=Albert Tarantola |coauthors= |year=2005 |publisher=Society for Industrial and Applied Mathematics |location=Philadelphia |isbn=0-89871-572-5 |page= |pages= |url=http://www.ipgp.jussieu.fr/~tarantola/Files/Professional/SIAM/index.html|ref=harvnb}}
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| *{{Cite book|first=David |last=Vose |title=Risk Analysis, A Quantitative Guide|edition=Third|publisher=[[John Wiley & Sons]]|year= 2008 | ref=harvnb}}
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| | |
| ==External links==
| |
| {{Commons category|Monte Carlo method}}
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| {{External links|date=October 2013}}
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| * {{springer|title=Monte-Carlo method|id=p/m064870}}
| |
| * [http://mathworld.wolfram.com/MonteCarloMethod.html Overview and reference list], Mathworld
| |
| * [http://www.math.u-bordeaux1.fr/~delmoral/simulinks.html Feynman-Kac models and particle Monte Carlo algorithms]
| |
| *[http://www.phy.ornl.gov/csep/CSEP/MC/MC.html Introduction to Monte Carlo Methods], Computational Science Education Project
| |
| *[http://www.chem.unl.edu/zeng/joy/mclab/mcintro.html The Basics of Monte Carlo Simulations], [[University of Nebraska-Lincoln]]
| |
| *[http://office.microsoft.com/en-us/excel-help/introduction-to-monte-carlo-simulation-HA010282777.aspx Introduction to Monte Carlo simulation] (for [[Microsoft Excel]]), Wayne L. Winston
| |
| *[http://www.mathworks.com/discovery/monte-carlo-simulation.html Monte Carlo Simulation for MATLAB and Simulink]
| |
| *[http://www.brighton-webs.co.uk/montecarlo/concept.htm Monte Carlo Methods – Overview and Concept], brighton-webs.co.uk
| |
| *[http://personal-pages.ps.ic.ac.uk/~achremos/Applet1-page.htm Monte Carlo techniques applied in physics]
| |
| * [http://orcik.net/programming/approximate-and-double-check-probability-problems-using-monte-carlo-method/ Approximate And Double Check Probability Problems Using Monte Carlo method] at Orcik Dot Net
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| {{Statistics}}
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