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Monte Carlo methodFrom Wikipedia, the free encyclopedia

Monte Carlo methods (or Monte Carlo experiments) are a class of computational algorithms that relyon repeated random sampling to compute their results. Monte Carlo methods are often used in computersimulations of physical and mathematical systems. These methods are most suited to calculation by acomputer and tend to be used when it is infeasible to compute an exact result with a deterministicalgorithm.[1] This method is also used to complement theoretical derivations.

Monte Carlo methods are especially useful for simulating systems with many coupled degrees of freedom,such as fluids, disordered materials, strongly coupled solids, and cellular structures (see cellular Pottsmodel). They are used to model phenomena with significant uncertainty in inputs, such as the calculationof risk in business. They are widely used in mathematics, for example to evaluate multidimensional definiteintegrals with complicated boundary conditions. When Monte Carlo simulations have been applied inspace exploration and oil exploration, their predictions of failures, cost overruns and schedule overruns areroutinely better than human intuition or alternative "soft" methods.[2]

The Monte Carlo method was coined in the 1940s by John von Neumann, Stanislaw Ulam and NicholasMetropolis, while they were working on nuclear weapon projects (Manhattan Project) in the Los AlamosNational Laboratory. It was named after the Monte Carlo Casino, a famous casino where Ulam's uncleoften gambled away his money.[3]

Contents1 Introduction2 History3 Definitions

3.1 Monte Carlo and random numbers3.2 Monte Carlo simulation versus "what if" scenarios

4 Applications4.1 Physical sciences4.2 Engineering4.3 Computational biology4.4 Computer Graphics4.5 Applied statistics4.6 Games4.7 Design and visuals4.8 Finance and business4.9 Telecommunications

5 Use in mathematics5.1 Integration5.2 Simulation - Optimization5.3 Inverse problems5.4 Computational mathematics

6 See also7 Notes8 References9 External links

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Monte Carlo method applied toapproximating the value of π. Afterplacing 30000 random points, theestimate for π is within 0.07% of theactual value. This happens with anapproximate probability of 20%.After 30000 points it is within 7%.[needs reference or/and verification].

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

1. Define a domain of possible inputs.2. Generate inputs randomly from a probability distribution

over the domain.3. Perform a deterministic computation on the inputs.4. Aggregate the results.

For example, consider a circle inscribed in a unit square. Giventhat the circle and the square have a ratio of areas that is π/4, thevalue of π can be approximated using a Monte Carlo method:[4]

1. Draw a square on the ground, then inscribe a circle withinit.

2. Uniformly scatter some objects of uniform size (grains ofrice or sand) over the square.

3. Count the number of objects inside the circle and the totalnumber of objects.

4. The ratio of the two counts is an estimate of the ratio of thetwo areas, which is π/4. Multiply the result by 4 to estimateπ.

In this procedure the domain of inputs is the square thatcircumscribes our circle. We generate random inputs by scattering grains over the square then perform acomputation on each input (test whether it falls within the circle). Finally, we aggregate the results toobtain our final result, the approximation of π.

If grains are purposefully dropped into only the center of the circle, they are not uniformly distributed, soour approximation is poor. Second, there should be a large number of inputs. The approximation isgenerally poor if only a few grains are randomly dropped into the whole square. On average, theapproximation improves as more grains are dropped.

HistoryBefore the Monte Carlo method was developed, simulations tested a previously understood deterministicproblem and statistical sampling was used to estimate uncertainties in the simulations. Monte Carlosimulations invert this approach, solving deterministic problems using a probabilistic analog (see Simulatedannealing).

An early variant of the Monte Carlo method can be seen in the Buffon's needle experiment, in which π canbe estimated by dropping needles on a floor made of parallel strips of wood. In the 1930s, Enrico Fermifirst experimented with the Monte Carlo method while studying neutron diffusion, but did not publishanything on it.[3]

In 1946, physicists at Los Alamos Scientific Laboratory were investigating radiation shielding and the

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distance that neutrons would likely travel through various materials. Despite having most of the necessarydata, such as the average distance a neutron would travel in a substance before it collided with an atomicnucleus, and how much energy the neutron was likely to give off following a collision, the Los Alamosphysicists were unable to solve the problem using conventional, deterministic mathematical methods.Stanisław Ulam had the idea of using random experiments. He recounts his inspiration as follows:

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

–Stanisław Ulam[5]

Being secret, the work of von Neumann and Ulam required a code name. Von Neumann chose the nameMonte Carlo. The name refers to the Monte Carlo Casino in Monaco where Ulam's uncle would borrowmoney to gamble.[1][6][7] Using lists of "truly" random random numbers was extremely slow, but vonNeumann developed a way to calculate pseudorandom numbers, using the middle-square method. Thoughthis method has been criticized as crude, von Neumann was aware of this: he justified it as being fasterthan any other method at his disposal, and also noted that when it went awry it did so obviously, unlikemethods that could be subtly incorrect.

Monte Carlo methods were central to the simulations required for the Manhattan Project, though severelylimited by the computational tools at the time. In the 1950s they were used at Los Alamos for early workrelating 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 ofthe major organizations responsible for funding and disseminating information on Monte Carlo methodsduring this time, and they began to find a wide application in many different fields.

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

Definitions

There is no consensus on how Monte Carlo should be defined. For example, Ripley[8] defines mostprobabilistic modeling as stochastic simulation, with Monte Carlo being reserved for Monte Carlointegration and Monte Carlo statistical tests. Sawilowsky[9] distinguishes between a simulation, a MonteCarlo method, and a Monte Carlo simulation: a simulation is a fictitious representation of reality, a MonteCarlo method is a technique that can be used to solve a mathematical or statistical problem, and a MonteCarlo 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 tosimulate the tossing of a coin: If the value is less than or equal to 0.50 designate the outcome asheads, but if the value is greater than 0.50 designate the outcome as tails. This is a simulation, butnot a Monte Carlo simulation.Monte Carlo method: The area of an irregular figure inscribed in a unit square can be determined bythrowing darts at the square and computing the ratio of hits within the irregular figure to the totalnumber of darts thrown. This is a Monte Carlo method of determining area, but not a simulation.Monte Carlo simulation: Drawing a large number of pseudo-random uniform variables from theinterval (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.

Kalos and Whitlock[4] point out that such distinctions are not always easy to maintain. For example, theemission of radiation from atoms is a natural stochastic process. It can be simulated directly, or its averagebehavior can be described by stochastic equations that can themselves be solved using Monte Carlomethods. "Indeed, the same computer code can be viewed simultaneously as a 'natural simulation' or as asolution of the equations by natural sampling."

Monte Carlo and random numbers

Monte Carlo simulation methods do not always require truly random numbers to be useful — while forsome applications, such as primality testing, unpredictability is vital.[10] Many of the most usefultechniques use deterministic, pseudorandom sequences, making it easy to test and re-run simulations. Theonly quality usually necessary to make good simulations is for the pseudo-random sequence to appear"random enough" in a certain sense.

What this means depends on the application, but typically they should pass a series of statistical tests.Testing that the numbers are uniformly distributed or follow another desired distribution when a largeenough number of elements of the sequence are considered is one of the simplest, and most common ones.

Sawilowsky lists the characteristics of a high quality Monte Carlo simulation:[9]

the (pseudo-random) number generator has certain characteristics (e.g., a long “period” before thesequence repeats)the (pseudo-random) number generator produces values that pass tests for randomnessthere are enough samples to ensure accurate resultsthe proper sampling technique is usedthe algorithm used is valid for what is being modeledit simulates the phenomenon in question.

Pseudo-random number sampling algorithms are used to transform uniformly distributed pseudo-randomnumbers into numbers that are distributed according to a given probability distribution.

Low-discrepancy sequences are often used instead of random sampling from a space as they ensure evencoverage and normally have a faster order of convergence than Monte Carlo simulations using random orpseudorandom sequences. Methods based on their use are called quasi-Monte Carlo methods.

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

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chosen and the results recorded.[11]

By contrast, Monte Carlo simulations sample probability distribution for each variable to produce hundredsor thousands of possible outcomes. The results are analyzed to get probabilities of different outcomesoccurring.[12] 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 probabilitydistributions shows that the Monte Carlo analysis has a narrower range than the “what if” analysis. This isbecause the “what if” analysis gives equal weight to all scenarios (see quantifying uncertainty in corporatefinance).

ApplicationsMonte Carlo methods are especially useful for simulating phenomena with significant uncertainty in inputsand systems with a large number of coupled degrees of freedom. Areas of application include:

Physical sciences

See also: Monte Carlo method in statistical physics

Monte Carlo methods are very important in computational physics, physical chemistry, and related appliedfields, and have diverse applications from complicated quantum chromodynamics calculations to designingheat shields and aerodynamic forms. In statistical physics Monte Carlo molecular modeling is an alternativeto computational molecular dynamics, and Monte Carlo methods are used to compute statistical fieldtheories of simple particle and polymer systems.[13] Quantum Monte Carlo methods solve the many-bodyproblem for quantum systems. In experimental particle physics, Monte Carlo methods are used fordesigning detectors, understanding their behavior and comparing experimental data to theory. Inastrophysics, they are used in such diverse manners as to model both the evolution of galaxies[14] and thetransmission of microwave radiation through a rough planetary surface.[15]

Monte Carlo methods are also used in the ensemble models that form the basis of modern weatherforecasting.

Engineering

Monte Carlo methods are widely used in engineering for sensitivity analysis and quantitative probabilisticanalysis in process design. The need arises from the interactive, co-linear and non-linear behavior oftypical process simulations. For example,

in microelectronics engineering, Monte Carlo methods are applied to analyze correlated anduncorrelated variations in analog and digital integrated circuits.in geostatistics and geometallurgy, Monte Carlo methods underpin the design of mineral processingflowsheets and contribute to quantitative risk analysis.in wind energy yield analysis, the predicted energy output of a wind farm during its lifetime iscalculated giving different levels of uncertainty (P90, P50, etc.)impacts of pollution are simulated[16] and diesel compared with petrol.[17]

In autonomous robotics, Monte Carlo localization can determine the position of a robot. It is oftenapplied to stochastic filters such as the Kalman filter or Particle filter that forms the heart of theSLAM (Simultaneous Localization and Mapping) algorithm.

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Computational biology

Monte Carlo methods are used in computational biology, such for as Bayesian inference in phylogeny.

Biological systems such as proteins[18] membranes,[19] images of cancer,[20] are being studied by meansof computer simulations.

The systems can be studied in the coarse-grained or ab initio frameworks depending on the desiredaccuracy. Computer simulations allow us to monitor the local environment of a particular molecule to see ifsome chemical reaction is happening for instance. We can also conduct thought experiments when thephysical experiments are not feasible, 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 randomlytracing samples of possible light paths. Repeated sampling of any given pixel will eventually cause theaverage of the samples to converge on the correct solution of the rendering equation, making it one of themost physically accurate 3D graphics rendering methods in existence.

Applied statistics

In applied statistics, Monte Carlo methods are generally used for two purposes:

1. To compare competing statistics for small samples under realistic data conditions. Although Type Ierror and power properties of statistics can be calculated for data drawn from classical theoreticaldistributions (e.g., normal curve, Cauchy distribution) for asymptotic conditions (i. e, infinite samplesize and infinitesimally small treatment effect), real data often do not have such distributions.[21]

2. To provide implementations of hypothesis tests that are more efficient than exact tests such aspermutation tests (which are often impossible to compute) while being more accurate than criticalvalues for asymptotic distributions.

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 entailspotentially enormous housekeeping of which permutations have been considered). The Monte Carloapproach is based on a specified number of randomly drawn permutations (exchanging a minor loss inprecision if a permutation is drawn twice – or more frequently—for the efficiency of not having to trackwhich permutations have already been selected).

Games

See also: Computer Go

Monte Carlo methods have recently been incorporated in algorithms for playing games that haveoutperformed previous algorithms in games such as Go, Tantrix, Battleship, and Havannah. Thesealgorithms employ Monte Carlo tree search. Possible moves are organized in a tree and a large number ofrandom simulations are used to estimate the long-term potential of each move. A black box simulatorrepresents the opponent's moves.

In November 2011, a Tantrix playing robot named FullMonte, which employs the Monte Carlo method,

played and beat the previous world champion Tantrix robot (Goodbot) quite easily. In a 200 game match

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Monte Carlo tree search applied to agame of Battleship. Initially thealgorithm takes random shots, but aspossible states are eliminated, theshots can be more selective. As acrude example, if a ship is hit (figureA), then adjacent squares becomemuch higher priorities (figures B andC).

played and beat the previous world champion Tantrix robot (Goodbot) quite easily. In a 200 game matchFullMonte won 58.5%, lost 36%, and drew 5.5% without ever running over the fifteen minute time limit.

In games like Battleship, where there is only limited knowledge of the state of the system (i.e., thepositions of the ships), a belief state is constructed consisting of probabilities for each state and then initialstates are sampled for running simulations. The belief state is updated as the game proceeds, as in thefigure. On a 10 x 10 grid, in which the total possible number of moves is 100, one algorithm sank all the

ships 50 moves faster, on average, than random play.[22]

Design and visuals

Monte Carlo methods are also efficient in solving coupled integraldifferential equations of radiation fields and energy transport, andthus these methods have been used in global illuminationcomputations that produce photo-realistic images of virtual 3Dmodels, with applications in video games, architecture, design,computer generated films, and cinematic special effects.[23]

Finance and business

See also: Monte Carlo methods for option pricing, Stochasticmodelling (insurance) , and Stochastic asset model

Monte Carlo methods in finance are often used to calculate thevalue of companies, to evaluate investments in projects at abusiness unit or corporate level, or to evaluate financialderivatives. They can be used to model project schedules, wheresimulations aggregate estimates for worst-case, best-case, and mostlikely durations for each task to determine outcomes for the overallproject.

Telecommunications

When planning a wireless network, design must be proved towork for a wide variety of scenarios that depend mainly on thenumber of users, their locations and the services they want to use.Monte Carlo methods are typically used to generate these usersand their states. The network performance is then evaluated and, if

results are not satisfactory, the network design goes through an optimization process.

Use in mathematicsIn general, Monte Carlo methods are used in mathematics to solve various problems by generating suitablerandom numbers and observing that fraction of the numbers that obeys some property or properties. Themethod is useful for obtaining numerical solutions to problems too complicated to solve analytically. Themost common application of the Monte Carlo method is Monte Carlo integration.

Integration

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Monte-Carlo integration works bycomparing random points with thevalue of the function

Errors reduce by a factor of

Main article: Monte Carlo integration

Deterministic numerical integration algorithms work well in asmall number of dimensions, but encounter two problems whenthe functions have many variables. First, the number of functionevaluations needed increases rapidly with the number ofdimensions. For example, if 10 evaluations provide adequateaccuracy in one dimension, then 10100 points are needed for 100dimensions—far too many to be computed. This is called the curseof dimensionality. Second, the boundary of a multidimensionalregion may be very complicated, so it may not be feasible toreduce the problem to a series of nested one-dimensionalintegrals.[24] 100 dimensions is by no means unusual, since inmany physical problems, a "dimension" is equivalent to a degreeof freedom.

Monte Carlo methods provide a way out of this exponentialincrease in computation time. As long as the function in questionis reasonably well-behaved, it can be estimated by randomlyselecting points in 100-dimensional space, and taking some kindof average of the function values at these points. By the centrallimit theorem, this method displays convergence—i.e.,quadrupling the number of sampled points halves the error,regardless of the number of dimensions.[24]

A refinement of this method, known as importance sampling instatistics, involves sampling the points randomly, but morefrequently where the integrand is large. To do this precisely onewould have to already know the integral, but one can approximatethe integral by an integral of a similar function or use adaptiveroutines such as stratified sampling, recursive stratified sampling,adaptive umbrella sampling[25][26] or the VEGAS algorithm.

A similar approach, the quasi-Monte Carlo method, uses low-discrepancy sequences. These sequences"fill" the area better and sample the most important points more frequently, so quasi-Monte Carlo methodscan often converge on the integral more quickly.

Another class of methods for sampling points in a volume is to simulate random walks over it (Markovchain Monte Carlo). Such methods include the Metropolis-Hastings algorithm, Gibbs sampling and theWang and Landau algorithm.

Simulation - 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 hasa large number of dimensions. Many problems can be phrased in this way: for example, a computer chessprogram could be seen as trying to find the set of, say, 10 moves that produces the best evaluation functionat the end. In the traveling salesman problem the goal is to minimize distance traveled. There are alsoapplications to engineering design, such as multidisciplinary design optimization.

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applications to engineering design, such as multidisciplinary design optimization.

The traveling salesman problem is what is called a conventional optimization problem. That is, all the facts(distances between each destination point) needed to determine the optimal path to follow are known withcertainty and the goal is to run through the possible travel choices to come up with the one with the lowesttotal distance. However, let's assume that instead of wanting to minimize the total distance traveled to visiteach desired destination, we wanted to minimize the total time needed to reach each destination. This goesbeyond 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 firstunderstand the range of potential times it could take to go from one point to another (represented by aprobability distribution in this case rather than a specific distance) and then optimize our travel decisions toidentify the best path to follow taking that uncertainty into account.

Inverse problems

Probabilistic formulation of inverse problems leads to the definition of a probability distribution in themodel space. This probability distribution combines prior information with new information obtained bymeasuring some observable parameters (data). As, in the general case, the theory linking data with modelparameters is nonlinear, the posterior probability in the model space may not be easy to describe (it may bemultimodal, some moments may not be defined, etc.).

When analyzing an inverse problem, obtaining a maximum likelihood model is usually not sufficient, aswe normally also wish to have information on the resolution power of the data. In the general case we mayhave a large number of model parameters, and an inspection of the marginal probability densities of interestmay be impractical, or even useless. But it is possible to pseudorandomly generate a large collection ofmodels according to the posterior probability distribution and to analyze and display the models in such away that information on the relative likelihoods of model properties is conveyed to the spectator. This canbe accomplished by means of an efficient Monte Carlo method, even in cases where no explicit formulafor the a priori distribution is available.

The best-known importance sampling method, the Metropolis algorithm, can be generalized, and this givesa method that allows analysis of (possibly highly nonlinear) inverse problems with complex a prioriinformation and data with an arbitrary noise distribution.[27][28]

Computational mathematics

Monte Carlo methods are useful in many areas of computational mathematics, where a "lucky choice" canfind the correct result. A classic example is Rabin's algorithm for primality testing: for any n that is notprime, a random x has at least a 75% chance of proving that n is not prime. Hence, if n is not prime, but xsays that it might be, we have observed at most a 1-in-4 event. If 10 different random x say that "n isprobably prime" when it is not, we have observed a one-in-a-million event. In general a Monte Carloalgorithm of this kind produces one correct answer with a guarantee n is composite, and x proves it so,but another one without, but with a guarantee of not getting this answer when it is wrong too often—inthis case at most 25% of the time. See also Las Vegas algorithm for a related, but different, idea.

See alsoAuxiliary field Monte CarloBiology Monte Carlo method

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Comparison of risk analysis Microsoft Excel add-insDirect simulation Monte CarloDynamic Monte Carlo methodKinetic Monte CarloList of software for Monte Carlo molecular modelingMonte Carlo method for photon transportMonte Carlo methods for electron transportMorris method

Notes

1. ^ a b Hubbart 20072. ^ Hubbard 20093. ^ a b Metropolis 19874. ^ a b Kalos & Whitlock 20085. ^ Eckhardt 19876. ^ Grinstead & Snell 19977. ^ Anderson 19868. ^ Ripley 19879. ^ a b Sawilowsky 2003

10. ^ Davenport 199211. ^ Vose 2000, p. 1312. ^ Vose 2000, p. 1613. ^ Baeurle 200914. ^ MacGillivray & Dodd 198215. ^ Golden 197916. ^ Int Panis et al. 200117. ^ Int Panis et al. 200218. ^ Ojeda & et al. 2009,19. ^ Milik & Skolnick 199320. ^ Forastero et al. 201021. ^ Sawilowsky & Fahoome 200322. ^ Silver & Veness 201023. ^ Szirmay-Kalos 200824. ^ a b Press et al. 199625. ^ MEZEI, M (31 December 1986). "Adaptive umbrella sampling: Self-consistent determination of the non-

Boltzmann bias". Journal of Computational Physics 68 (1): 237–248. Bibcode 1987JCoPh..68..237M(http://adsabs.harvard.edu/abs/1987JCoPh..68..237M) . doi:10.1016/0021-9991(87)90054-4(http://dx.doi.org/10.1016%2F0021-9991%2887%2990054-4) .

26. ^ Bartels, Christian; Karplus, Martin (31 December 1997). "Probability Distributions for Complex Systems:Adaptive Umbrella Sampling of the Potential Energy". The Journal of Physical Chemistry B 102 (5): 865–880.doi:10.1021/jp972280j (http://dx.doi.org/10.1021%2Fjp972280j) .

27. ^ Mosegaard & Tarantola 199528. ^ Tarantola & 2005 http://www.ipgp.fr/~tarantola/Files/Professional/Books/index.html

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methodologies: A survey about recent developments". Journal of Mathematical Chemistry 46 (2):

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methodologies: A survey about recent developments". Journal of Mathematical Chemistry 46 (2):363–426. doi:10.1007/s10910-008-9467-3 (http://dx.doi.org/10.1007%2Fs10910-008-9467-3) .Berg, Bernd A. (2004). Markov Chain Monte Carlo Simulations and Their Statistical Analysis (WithWeb-Based Fortran Code). Hackensack, NJ: World Scientific. ISBN 981-238-935-0.Binder, Kurt (1995). The Monte Carlo Method in Condensed Matter Physics. New York: Springer.ISBN 0-387-54369-4.Caflisch, R. E. (1998). Monte Carlo and quasi-Monte Carlo methods. Acta Numerica. 7.Cambridge University Press. pp. 1–49.Davenport, J. H.. "Primality testing revisited". Proceeding ISSAC '92 Papers from the internationalsymposium on Symbolic and algebraic computation: 123 129. doi:10.1145/143242.143290(http://dx.doi.org/10.1145%2F143242.143290) . ISBN 0-89791-489-9.Doucet, Arnaud; Freitas, Nando de; Gordon, Neil (2001). Sequential Monte Carlo methods inpractice. New York: Springer. ISBN 0-387-95146-6.Eckhardt, Roger (1987). "Stan Ulam, John von Neumann, and the Monte Carlo method"(http://www.lanl.gov/history/admin/files/Stan_Ulam_John_von_Neumann_and_the_Monte_Carlo_Method.pdf) . Los Alamos Science, Special Issue (15): 131–137.http://www.lanl.gov/history/admin/files/Stan_Ulam_John_von_Neumann_and_the_Monte_Carlo_Method.pdf.Fishman, G. S. (1995). Monte Carlo: Concepts, Algorithms, and Applications. New York: Springer.ISBN 0-387-94527-X.C. Forastero and L. Zamora and D. Guirado and A. Lallena (2010). "A Monte Carlo tool tosimulate breast cancer screening programmes". Phys. In Med. And Biol. 55 (17): 5213. Bibcode2010PMB....55.5213F (http://adsabs.harvard.edu/abs/2010PMB....55.5213F) . doi:10.1088/0031-9155/55/17/021 (http://dx.doi.org/10.1088%2F0031-9155%2F55%2F17%2F021) .Golden, Leslie M. (1979). "The Effect of Surface Roughness on the Transmission of MicrowaveRadiation Through a Planetary Surface". Icarus 38 (3): 451. Bibcode 1979Icar...38..451G(http://adsabs.harvard.edu/abs/1979Icar...38..451G) . doi:10.1016/0019-1035(79)90199-4(http://dx.doi.org/10.1016%2F0019-1035%2879%2990199-4) .Gould, Harvey; Tobochnik, Jan (1988). An Introduction to Computer Simulation Methods, Part 2,Applications to Physical Systems. Reading: Addison-Wesley. ISBN 0-201-16504-X.Grinstead, Charles; Snell, J. Laurie (1997). Introduction to Probability. American MathematicalSociety. pp. 10–11.Hammersley, J. M.; Handscomb, D. C. (1975). Monte Carlo Methods. London: Methuen. ISBN 0-416-52340-4.Hartmann, A.K. (2009). Practical Guide to Computer Simulations(http://www.worldscibooks.com/physics/6988.html) . World Scientific. ISBN 978-981-283-415-7.http://www.worldscibooks.com/physics/6988.html.Hubbard, Douglas (2007). How to Measure Anything: Finding the Value of Intangibles in Business.John Wiley & Sons. p. 46.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. ISBN 978-3-527-40760-6.Kroese, D. P.; Taimre, T.; Botev, Z.I. (2011). Handbook of Monte Carlo Methods(http://www.montecarlohandbook.org) . New York: John Wiley & Sons. p. 772. ISBN 0-470-17793-4. http://www.montecarlohandbook.org.MacGillivray, H. T.; Dodd, R. J. (1982). "Monte-Carlo simulations of galaxy systems"(http://www.springerlink.com/content/rp3g1q05j176r108/fulltext.pdf) . Astrophysics and SpaceScience (Springer Netherlands) 86 (2).http://www.springerlink.com/content/rp3g1q05j176r108/fulltext.pdf.

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MacKeown, P. Kevin (1997). Stochastic Simulation in Physics. New York: Springer. ISBN 981-3083-26-3.Metropolis, N. (1987). "The beginning of the Monte Carlo method" (http://library.lanl.gov/la-pubs/00326866.pdf) . Los Alamos Science (1987 Special Issue dedicated to Stanisław Ulam): 125–130. http://library.lanl.gov/la-pubs/00326866.pdf.Metropolis, Nicholas; Rosenbluth, Arianna W.; Rosenbluth, Marshall N.; Teller, Augusta H.; Teller,Edward (1953). "Equation of State Calculations by Fast Computing Machines". Journal ofChemical Physics 21 (6): 1087. Bibcode 1953JChPh..21.1087M(http://adsabs.harvard.edu/abs/1953JChPh..21.1087M) . doi:10.1063/1.1699114(http://dx.doi.org/10.1063%2F1.1699114) .Metropolis, N.; Ulam, S. (1949). "The Monte Carlo Method". Journal of the American StatisticalAssociation (American Statistical Association) 44 (247): 335–341. doi:10.2307/2280232(http://dx.doi.org/10.2307%2F2280232) . JSTOR 2280232 (http://www.jstor.org/stable/2280232) .PMID 18139350 (//www.ncbi.nlm.nih.gov/pubmed/18139350) .M. Milik and J. Skolnick (Jan 1993). "Insertion of peptide chains into lipid membranes: an off-latticeMonte Carlo dynamics model". Proteins 15 (1): 10–25. doi:10.1002/prot.340150104(http://dx.doi.org/10.1002%2Fprot.340150104) . PMID 8451235(//www.ncbi.nlm.nih.gov/pubmed/8451235) .Mosegaard, Klaus; Tarantola, Albert (1995). "Monte Carlo sampling of solutions to inverseproblems". J. Geophys. Res. 100 (B7): 12431–12447. Bibcode 1995JGR...10012431M(http://adsabs.harvard.edu/abs/1995JGR...10012431M) . doi:10.1029/94JB03097(http://dx.doi.org/10.1029%2F94JB03097) .P. Ojeda and M. Garcia and A. Londono and N.Y. Chen (Feb 2009). "Monte Carlo Simulations ofProteins in Cages: Influence of Confinement on the Stability of Intermediate States". Biophys. Jour.(Biophysical Society) 96 (3): 1076–1082. Bibcode 2009BpJ....96.1076O(http://adsabs.harvard.edu/abs/2009BpJ....96.1076O) . doi:10.1529/biophysj.107.125369(http://dx.doi.org/10.1529%2Fbiophysj.107.125369) .Int Panis L; De Nocker L, De Vlieger I, Torfs R (2001). "Trends and uncertainty in air pollutionimpacts and external costs of Belgian passenger car traffic International". Journal of Vehicle Design27 (1–4): 183–194. doi:10.1504/IJVD.2001.001963(http://dx.doi.org/10.1504%2FIJVD.2001.001963) .Int Panis L, Rabl A, De Nocker L, Torfs R (2002). P. Sturm. ed. "Diesel or Petrol ? Anenvironmental comparison hampered by uncertainty". Mitteilungen Institut fürVerbrennungskraftmaschinen und Thermodynamik (Technische Universität Graz Austria) Heft 81Vol 1: 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. 1(Second ed.). Cambridge University Press. ISBN 0-521-43064-X.Ripley, B. D. (1987). Stochastic Simulation. Wiley & Sons.Robert, C. P.; Casella, G. (2004). Monte Carlo Statistical Methods (2nd ed.). New York: Springer.ISBN 0-387-21239-6.Rubinstein, R. Y.; Kroese, D. P. (2007). Simulation and the Monte Carlo Method (2nd ed.). NewYork: John Wiley & Sons. ISBN 978-0-470-17793-8.Savvides, Savvakis C. (1994). "Risk Analysis in Investment Appraisal". Project Appraisal Journal9 (1). doi:10.2139/ssrn.265905 (http://dx.doi.org/10.2139%2Fssrn.265905) .Sawilowsky, Shlomo S.; Fahoome, Gail C. (2003). Statistics via Monte Carlo Simulation withFortran. Rochester Hills, MI: JMASM. ISBN 0-9740236-0-4.Sawilowsky, Shlomo S. (2003). "You think you've got trivials?"(http://education.wayne.edu/jmasm/sawilowsky_effect_size_debate.pdf) . Journal of ModernApplied Statistical Methods 2 (1): 218–225.http://education.wayne.edu/jmasm/sawilowsky_effect_size_debate.pdf.Silver, David; Veness, Joel (2010). "Monte-Carlo Planning in Large POMDPs"

12/6/12 Monte Carlo method ‑ Wikipedia, the free encyclopedia

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(http://books.nips.cc/papers/files/nips23/NIPS2010_0740.pdf) . In Lafferty, J.; Williams, C. K. I.;Shawe-Taylor, J. et al.. Advances in Neural Information Processing Systems 23. Neural InformationProcessing Systems Foundation. http://books.nips.cc/papers/files/nips23/NIPS2010_0740.pdf.Szirmay-Kalos, László (2008). Monte Carlo Methods in Global Illumination - Photo-realisticRendering with Randomization. VDM Verlag Dr. Mueller e.K.. ISBN 978-3-8364-7919-6.Tarantola, Albert (2005). Inverse Problem Theory(http://www.ipgp.jussieu.fr/~tarantola/Files/Professional/SIAM/index.html) . Philadelphia: Societyfor Industrial and Applied Mathematics. ISBN 0-89871-572-5.http://www.ipgp.jussieu.fr/~tarantola/Files/Professional/SIAM/index.html.Vose, David (2008). Risk Analysis, A Quantitative Guide (Third ed.). John Wiley & Sons.

External linksOverview and reference list (http://mathworld.wolfram.com/MonteCarloMethod.html) , Mathworld"Café math : Monte Carlo Integration" (http://cafemath.kegtux.org/mathblog/article.php?page=MonteCarlo.php) : A blog article describing Monte Carlo integration (principle, hypothesis,confidence interval)Feynman-Kac models and particle Monte Carlo algorithms (http://www.math.u-bordeaux1.fr/~delmoral/simulinks.html) Website on the applications of particle Monte Carlomethods in

signal processing, rare event simulation, molecular dynamics, financial mathematics, optimal control,computational physics, and biology.

Introduction to Monte Carlo Methods (http://www.phy.ornl.gov/csep/CSEP/MC/MC.html) ,Computational Science Education ProjectThe Basics of Monte Carlo Simulations (http://www.chem.unl.edu/zeng/joy/mclab/mcintro.html) ,University of Nebraska-LincolnIntroduction to Monte Carlo simulation (http://office.microsoft.com/en-us/excel-help/introduction-to-monte-carlo-simulation-HA010282777.aspx) (for Microsoft Excel), Wayne L. WinstonMonte Carlo Simulation for MATLAB and Simulink(http://www.mathworks.com/discovery/monte-carlo-simulation.html)Monte Carlo Methods – Overview and Concept (http://www.brighton-webs.co.uk/montecarlo/concept.htm) , brighton-webs.co.ukMolecular Monte Carlo Intro (http://www.cooper.edu/engineering/chemechem/monte.html) , CooperUnionMonte Carlo techniques applied in physics (http://personal-pages.ps.ic.ac.uk/~achremos/Applet1-page.htm)Monte Carlo Method Example (http://waqqasfarooq.com/waqqasfarooq/index.php?option=com_content&view=article&id=47:monte-carlo&catid=34:statistics&Itemid=53) , A step-by-step guide to creating a monte carlo excel spreadsheetApproximate And Double Check Probability Problems Using Monte Carlo method(http://orcik.net/programming/approximate-and-double-check-probability-problems-using-monte-carlo-method/) at Orcik Dot Net

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