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# Monte-Carlo Simulation

• Besides the traditional ways of data acquisition in laboratory experiments and field tests the generation of so-called synthetic data via computer simulation has gained increasing importance.
• There is a variety of reasons for the increased benefit drawn from computer simulation used to investigate a wide range of issues, objects and processes:
• The most prominent reason is the rapidly growing performance of modern computer systems which has extended our computational capabilities in a way that would not have been imaginable even a short time ago.
• Consequently, computer-based data generation is often considerably cheaper and less time-consuming than traditional data acquisition in laboratory experiments and field tests.
• Moreover, computer experiments can be repeated under constant conditions as frequently as necessary whereas in traditional scientific experiments the investigated object is often damaged or even destroyed.
• A further reason for the value of computer simulations is the fact
• that volume and structure of the analyzed data is often very complex
• and that in this case data processing and evaluation is typically based on mathematical models whose characteristics cannot be (completely) described by analytical formulae.
• Thus, computer simulations of the considered models present a valuable alternative tool for analysis.

• Computer experiments for the investigation of the issues, objects and processes of scientific interest are based on stochastic simulation algorithms. In this context one also uses the term Monte-Carlo simulation summarizing a huge variety of simulation algorithms.

1. Random number generators are the basis for Monte-Carlo simulation of single features, quantities and variables.
• By these algorithms realizations of random variables can be generated via the computer. Those are called pseudo-random numbers.
• The simulation of random variables is based on so-called standard random number generators providing realizations of random variables that are uniformly distributed on the unit interval .
• Certain transformation and rejection methods can be applied to these standard pseudo-random numbers in order to generate pseudo-random numbers for other (more complex) random variables having e.g. binomial, Poisson or normal distributions.

2. Computer experiments designed to investigate high-dimensional random vectors or the evolution of certain objects in time are based on more sophisticated algorithms from so-called dynamic Monte-Carlo simulation.
• In this context Markov-Chain-Monte-Carlo-Simulation (MCMC simulation) is a construction principle for algorithms that are particularly appropriate to simulate time stationary equilibria of objects or processes.
• Another example for the application of MCMC simulation is statistical image analysis.
• An active field of research that resulted in numerous publications during the last years are so-called coupling algorithms for perfect MCMC simulation.
• These coupling algorithms enable us to simulate time-stationary equilibria of objects and processes in a way that does not only allow approximations but simulations that are ,,perfect'' in a certain sense.

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Ursa Pantle 2006-07-20