Monte Carlo method

Not to be confused with Monte Carlo algorithm.

Computational physics
Numerical analysis · Simulation Data analysis · Visualization
Potentials Lennard-Jones potential · Yukawa potential · Morse potential
Fluid dynamics Finite element · Riemann solver Smoothed particle hydrodynamics
Monte Carlo methods Integration · Gibbs sampling · Metropolis algorithm
Particle N-body · Particle-in-cell Molecular dynamics
Scientists von Neumann · Godunov
v · d · e

Monte Carlo methods (or Monte Carlo experiments) are a class of computational algorithms that rely on repeated random sampling to compute their results. Monte Carlo methods are often used in computer simulations of physical and mathematical systems. These methods are most suited to calculation by a computer and tend to be used when it is infeasible to compute an exact result with a deterministic algorithm.^[1] This method is also used to complement the 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 Potts model). They are used to model phenomena with significant uncertainty in inputs, such as the calculation of risk in business. They are widely used in mathematics, for example to evaluate multidimensional definite integrals with complicated boundary conditions. When Monte Carlo simulations have been applied in space exploration and oil exploration, their predictions of failures, cost overruns and schedule overruns are routinely 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 Nicholas Metropolis, while they were working on nuclear weapon projects (Manhattan Project) in the Los Alamos National Laboratory. It was named in homage to the Monte Carlo Casino, a famous casino where Ulam's uncle would often gamble away his money.^[3]

Introduction

Monte Carlo method applied to approximating the value of π. After placing 30000 random points, the estimate for π is within .07% of the actual value.

Monte 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. Given that the circle and the square have a ratio of areas that is π/4, the value of π can be approximated using a Monte Carlo method:^[4]

1. Draw a square on the ground, then inscribe a circle within it.
2. Uniformly scatter some objects of uniform size (grains of rice or sand) over the square.
3. Count the number of objects inside the circle and the total number of objects.
4. The ratio of the two counts is an estimate of the ratio of the two areas, which is π/4. Multiply the result by 4 to estimate π.

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 π.

To get an accurate approximation for π this procedure should have two other common properties of Monte Carlo methods. First, the inputs should truly be random. If grains are purposefully dropped into only the center of the circle, they will not be uniformly distributed, and so our approximation will be poor. Second, there should be a large number of inputs. The approximation will generally be poor if only a few grains are randomly dropped into the whole square. On average, the approximation improves as more grains are dropped.

History

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

An early variant of the Monte Carlo method can be seen in the Buffon's needle experiment, in which π 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.^[3]

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

–Stanisław Ulam^[5]

Being secret, the work of von Neumann and Ulam required a code name. Von Neumann chose the name "Monte Carlo". The name is a reference to the Monte Carlo Casino in Monaco where Ulam's uncle would borrow money to gamble.^[1][6][7] Using lists of "truly" random numbers was extremely slow, von Neumann developed a form of making pseudorandom numbers, using the middle-square method. Though this method has been criticized as crude, von Neumann was aware of this: he justified it as being faster than any other method at his disposal, and also noted that when it went awry it did so obviously, unlike methods which could be subtly incorrect.

Monte Carlo methods were central to the simulations required for the Manhattan Project, though severely limited by the computational tools at the time. In the 1950s they were used at Los Alamos for early work relating to the development of the hydrogen bomb, and became popularized in the fields of physics, physical chemistry, and operations research. The Rand Corporation and the U.S. Air Force were two of the major organizations responsible for funding and disseminating information on Monte Carlo methods during this time, and they began to find a wide application in many different fields.

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

Definitions

There is no consensus on how Monte Carlo should be defined. For example, Ripley^[8] defines most probabilistic modeling as stochastic simulation, with Monte Carlo being reserved for Monte Carlo integration and Monte Carlo statistical tests. Sawilowsky^[9] distinguishes between a simulation, 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:

• 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.
• 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.
• 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.

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

Monte Carlo and random numbers

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

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

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 the sequence repeats)
• the (pseudo-random) number generator produces values that pass tests for randomness
• there are enough samples to ensure accurate results
• the proper sampling technique is used
• the algorithm used is valid for what is being modeled
• it simulates the phenomenon in question.

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

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

Monte Carlo simulation versus "what if" scenarios

There are ways of using probabilities that are definitely not Monte Carlo simulations—for example, deterministic modeling using single-point estimates. Each uncertain variable within a model is assigned a “best guess” estimate. Scenarios (such as best, worst, or most likely case) for each input variable are chosen and the results recorded.^[11]

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.^[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 probability distributions shows that the Monte Carlo analysis has a narrower range than the “what if” analysis. This is because the “what if” analysis gives equal weight to all scenarios (see quantifying uncertainty in corporate finance).

Applications

Monte Carlo methods are especially useful for simulating phenomena with significant uncertainty in inputs and 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 applied fields, and have diverse applications from complicated quantum chromodynamics calculations to designing heat shields and 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 theories of simple particle and polymer systems.^[13] Quantum Monte Carlo methods solve the many-body problem for quantum systems. In experimental particle physics, Monte Carlo methods are used for designing detectors, understanding their behavior and comparing experimental data to theory. In astrophysics, they are used in such diverse manners as to model both the evolution of galaxies^[14] and the transmission 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 weather forecasting.

Engineering

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

• in microelectronics engineering, Monte Carlo methods are applied to analyze correlated and uncorrelated variations in analog and digital integrated circuits.
• in geostatistics and geometallurgy, Monte Carlo methods underpin the design of mineral processing flowsheets and contribute to quantitative risk analysis.
• impacts of pollution are simulated^[16] and diesel compared with petrol.^[17]
• In autonomous robotics, Monte Carlo localization can be used to determine the position of a robot, it is often applied to stochastic filters such as the Kalman filter or Particle filter which form the heart of the SLAM ( simultaneous Localisation and Mapping ) algorithm.

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 means of computer simulations.

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

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 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.^[21]
2. To provide implementations of hypothesis tests that are more efficient than exact tests such as permutation tests (which are often impossible to compute) while being more accurate than critical values for asymptotic distributions.

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).

Games

See also: Computer Go

Monte Carlo tree search applied to a game of Battleship. Initially the algorithm takes random shots, but as possible states are eliminated, the shots can be more selective. As a crude example, if a ship is hit (figure A), then adjacent squares become much higher priorities (figures B and C).

Monte Carlo methods have recently been incorporated in algorithms for playing games that have outperformed previous algorithms in games like Go, Tantrix, and Battleship. These algorithms employ Monte Carlo tree search. Possible algorithms are organized in a 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.

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 FullMonte 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., the positions of the ships), a belief state is constructed consisting of probabilities for each state and then initial states are sampled for running simulations. The belief state is updated as the game proceeds, as in the figure. 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]

One of the main problems that this approach has in game playing is that it sometimes misses an isolated, very good move. These approaches are often strong strategically but weak tactically, as tactical decisions tend to rely on a small number of crucial moves which are easily missed by the randomly searching Monte Carlo algorithm.^{[citation needed]}

Design and visuals

Monte Carlo methods have also proven efficient in solving coupled integral differential equations of radiation fields and energy transport, and thus these methods have been used in global illumination computations which produce photo-realistic images of virtual 3D models, 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 and Stochastic modelling (insurance)

Monte Carlo methods in finance are often used to calculate the value of companies, to evaluate investments in projects at a business unit or corporate level, or to evaluate financial derivatives. They can be used to model project schedules, where simulations aggregate estimates for worst-case, best-case, and most likely durations for each task to determine outcomes for the overall project.

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.

Use in mathematics

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 which obeys some property or properties. The method is useful for obtaining numerical solutions to problems which are too complicated to solve analytically. The most common application of the Monte Carlo method is Monte Carlo integration.

Integration

Main article: Monte Carlo integration

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

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

Deterministic numerical integration algorithms work well in a small number of dimensions, but encounter two problems when the functions have many variables. First, the number of function evaluations needed increases rapidly with the number of dimensions. For example, if 10 evaluations provide adequate accuracy in one dimension, then 10¹⁰⁰ points are needed for 100 dimensions—far too many to be computed. This is called the curse of dimensionality. Second, the boundary of a multidimensional region may be very complicated, so it may not be feasible to reduce the problem to a series of nested one-dimensional integrals.^[24] 100 dimensions is by no means unusual, since in many physical problems, a "dimension" is equivalent to a degree of freedom.

Monte Carlo methods provide a way out of this exponential increase in computation time. As long as the function in question is reasonably well-behaved, it can be estimated by randomly selecting points in 100-dimensional space, and taking some kind of average of the function values at these points. By the law of large numbers, this method will display $\scriptstyle 1/\sqrt{N}$ convergence—i.e., quadrupling the number of sampled points will halve the error, regardless of the number of dimensions.^[24]

A refinement of this method, known as importance sampling in statistics, involves sampling the points randomly, but more frequently where the integrand is large. To do this precisely one would have to already know the integral, but one can approximate the integral by an integral of a similar function or use adaptive routines such as Stratified sampling, recursive stratified sampling, adaptive umbrella sampling^[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 methods can often converge on the integral more quickly.

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

Optimization

Main article: Stochastic optimization

Another powerful and very popular application for random numbers in numerical simulation is in numerical optimization. The problem is to minimize (or maximize) functions of some vector that often has 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.

Most Monte Carlo optimization methods are based on random walks. Essentially, the program moves randomly on a multi-dimensional surface, preferring moves that reduce the function, but sometimes moving "uphill".

Inverse problems

Probabilistic formulation of inverse problems leads to the definition of a probability distribution in the model space. This probability distribution combines a priori 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 a posteriori probability in the model space may not be easy to describe (it may be multimodal, some moments may not be defined, etc.).

When analyzing an inverse problem, obtaining a maximum likelihood model is usually not sufficient, as we normally also wish to have information on the resolution power of the data. In the general case we may have 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.

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.^[27][28]

Computational mathematics

Monte Carlo methods are useful in many areas of computational mathematics, where a "lucky choice" can find the correct result. A classic example is Rabin's algorithm for primality testing: for any n which is not prime, a random x has at least a 75% chance of proving that n is not prime. Hence, if n is not prime, but x says that it might be, we have observed at most a 1-in-4 event. If 10 different random x say that "n is probably prime" when it is not, we have observed a one-in-a-million event. In general a Monte Carlo algorithm 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—in this case at most 25% of the time. See also Las Vegas algorithm for a related, but different, idea.

See also

Statistics portal Application areas Modeling light transport in biological tissue Monte Carlo methods for electron transport Biology Monte Carlo method

Other methods employing Monte Carlo Auxiliary field Monte Carlo Direct simulation Monte Carlo Dynamic Monte Carlo method Kinetic Monte Carlo Morris method List of software for Monte Carlo molecular modeling

Notes

1. ^ ^{a b} Hubbart 2007
2. ^ Hubbard 2009
3. ^ ^{a b} Metropolis 1987
4. ^ ^{a b} Kalos & Whitlock 2008
5. ^ Eckardt 1987
6. ^ Grinstead & Snell 1997
7. ^ Anderson 1986
8. ^ Ripley 1987
9. ^ ^{a b} Sawilowsky 2003
10. ^ Davenport 1992
11. ^ Vose 2000, p. 13
12. ^ Vose 2000, p. 16
13. ^ Baeurle 2009
14. ^ MacGillivray & Dodd 1982
15. ^ Golden 1979
16. ^ Int Panis et al. 2001
17. ^ Int Panis et al. 2002
18. ^ Ojeda & et al. 2009,
19. ^ Milik & Skolnick 1993
20. ^ Forastero et al. 2010
21. ^ Sawilowsky & Fahoome 2003
22. ^ Silver & Veness 2010
23. ^ Szirmay-Kalos 2008
24. ^ ^{a b} Press et al. 1996
25. ^ 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. doi:10.1016/0021-9991(87)90054-4. 
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. 
27. ^ Mosegaard & Tarantola 1995
28. ^ Tarantola 2005

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External links

• Overview and reference list, Mathworld
• Video Overview of Top Excel Monte-Carlo Tools and their Main Functionalities, Eric Torkia
• Introduction to Monte Carlo Methods, Computational Science Education Project
• The Basics of Monte Carlo Simulations, University of Nebraska-Lincoln
• Introduction to Monte Carlo simulation (for Microsoft Excel), Wayne L. Winston
• Monte Carlo Methods – Overview and Concept, brighton-webs.co.uk
• Molecular Monte Carlo Intro, Cooper Union
• Monte Carlo techniques applied in physics
• Monte Carlo Method Example, A step-by-step guide to creating a monte carlo excel spreadsheet
• Pricing using Monte Carlo simulation, a practical example, Prof. Giancarlo Vercellino
• Approximate And Double Check Probability Problems Using Monte Carlo method at Orcik Dot Net