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
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. 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 and the transmission of microwave radiation through a rough planetary surface.
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.
§ in wind energy yield analysis, the predicted energy output of a wind farm during its lifetime is calculated giving different levels of uncertainty (P90, P50, etc.)
§ impacts of pollution are simulated and diesel compared with petrol.
§ 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 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 membranes, images of cancer, 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.
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
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.
One of the main problems this approach has in game playing is that it sometimes misses an isolated good move. These approaches are often strong strategically, but weak tactically, as tactical decisions tend to rely on a small number of crucial moves that the randomly searching Monte Carlo algorithm easily misses.
Design and visuals
Monte Carlo methods are also efficient in solving coupled integral differential equations of radiation fields and energy transport, and thus these methods have been used in global illumination computations that produce photo-realistic images of virtual 3D models, with applications in video games,architecture, design, computer generated films, and cinematic special effects.
Finance and business
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.
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