These flows of probability distributions can always be interpreted as the distributions of the random states of a Markov process whose transition probabilities depend on the distributions of the current random states (see McKean–Vlasov processes, nonlinear filtering equation). In other problems, the objective is generating draws from a sequence of probability distributions satisfying a nonlinear evolution equation. By the ergodic theorem, the stationary distribution is approximated by the empirical measures of the random states of the MCMC sampler. That is, in the limit, the samples being generated by the MCMC method will be samples from the desired (target) distribution. The central idea is to design a judicious Markov chain model with a prescribed stationary probability distribution. When the probability distribution of the variable is parametrized, mathematicians often use a Markov chain Monte Carlo (MCMC) sampler. the sample mean) of independent samples of the variable. By the law of large numbers, integrals described by the expected value of some random variable can be approximated by taking the empirical mean (a.k.a. In principle, Monte Carlo methods can be used to solve any problem having a probabilistic interpretation.
In application to systems engineering problems (space, oil exploration, aircraft design, etc.), Monte Carlo–based predictions of failure, cost overruns and schedule overruns are routinely better than human intuition or alternative "soft" methods. Other examples include modeling phenomena with significant uncertainty in inputs such as the calculation of risk in business and, in mathematics, evaluation of multidimensional definite integrals with complicated boundary conditions. In physics-related problems, Monte Carlo methods are 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, interacting particle systems, McKean–Vlasov processes, kinetic models of gases). Monte Carlo methods are mainly used in three problem classes: optimization, numerical integration, and generating draws from a probability distribution. They are often used in physical and mathematical problems and are most useful when it is difficult or impossible to use other approaches. The underlying concept is to use randomness to solve problems that might be deterministic in principle.
It is argued that special-purpose simulators will be constructed as a general-purpose simulator provided with a special-purpose simulation environment rather than as specially hardware designed simulators.Monte Carlo methods, or Monte Carlo experiments, are a broad class of computational algorithms that rely on repeated random sampling to obtain numerical results.
Implications with respect to hardware and software are discussed. An outline of a simulation environment is given.
Several of these tasks ask for powerful and fast computers. It is made clear that a simulation environment has to perform extra tasks in comparison to a normal programming environment of a computer: description, modelling, experimentation, knowledge handling, reporting. The tasks a simulator has to support are depicted by a discussion about simulation studies in Research & Development. the experimenter) and the system to be experimentally studied. A simulation environment is defined as a programming environment of a computer, that is dedicated to systems simulation and that takes care for a flexible and intelligent interfacing between a user (i.e.
a programming environment dedicated to simulation. In the paper it is discussed that in systems simulation nowadays a simulator is becoming a synonym for a computer provided with a simulation environment, i.e.
Book series (ADVS.SIMULATION, volume 2) Abstract