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Unformatted text preview: Stochastic Simulation and Randomness Random Number Generators Quasi-Random Sequences Scientific Computing: An Introductory Survey Chapter 13 Random Numbers and Stochastic Simulation Prof. Michael T. Heath Department of Computer Science University of Illinois at Urbana-Champaign Copyright c 2002. Reproduction permitted for noncommercial, educational use only. Michael T. Heath Scientific Computing 1 / 17 Stochastic Simulation and Randomness Random Number Generators Quasi-Random Sequences Stochastic Simulation Stochastic simulation mimics or replicates behavior of system by exploiting randomness to obtain statistical sample of possible outcomes Because of randomness involved, simulation methods are also known as Monte Carlo methods Such methods are useful for studying Nondeterministic (stochastic) processes Deterministic systems that are too complicated to model analytically Deterministic problems whose high dimensionality makes standard discretizations infeasible (e.g., Monte Carlo integration) < interactive example > < interactive example > Michael T. Heath Scientific Computing 2 / 17 Stochastic Simulation and Randomness Random Number Generators Quasi-Random Sequences Stochastic Simulation, continued Two main requirements for using stochastic simulation methods are Knowledge of relevant probability distributions Supply of random numbers for making random choices Knowledge of relevant probability distributions depends on theoretical or empirical information about physical system being simulated By simulating large number of trials, probability distribution of overall results can be approximated, with accuracy attained increasing with number of trials < interactive example > < interactive example > Michael T. Heath Scientific Computing 3 / 17 Stochastic Simulation and Randomness Random Number Generators Quasi-Random Sequences Randomness Randomness is somewhat difficult to define, but we usually associate randomness with unpredictability One definition is that sequence of numbers is random if it has no shorter description than itself Physical processes, such as flipping coin or tossing dice, are deterministic if enough is known about equations governing their motion and appropriate initial conditions Even for deterministic systems, extreme sensitivity to initial conditions can make their chaotic behavior unpredictable in practice Wheter deterministic or not, highly complicated systems are often tractable only by stochastic simulation methods...
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