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.
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 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 Localization 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.
Computer Graphics
Path Tracing, occasionally referred to as Monte Carlo Ray
Tracing, renders a 3D scene by randomly tracing samples of possible light
paths. Repeated sampling of any given pixel will eventually cause the average
of the samples to converge on the correct solution of the rendering equation,
making it one of the most physically accurate 3D graphics rendering methods in
existence.
Applied statistics
In applied statistics, Monte Carlo methods are generally
used for two purposes:
To compare competing statistics for small samples under
realistic data conditions. Although Type I error and power properties of
statistics can be calculated for data drawn from classical theoretical
distributions (e.g., normal curve, Cauchy distribution) for asymptotic
conditions (i. e, infinite sample size and infinitesimally small treatment
effect), real data often do not have such distributions.
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).
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.
