Stochastic simulation probability theory

Over the last two decades, there has been increasing interest in probabilistic, or stochastic, robust control theory. Monte Carlo simulation methods have been used to synthesize and analyze controllers for uncertain systems [170,255], First- and second-order reliability methods were incorporated to compute the probable performance of linear-quadratic-regulator... 

We now consider probability theory, and its applications in stochastic simulation. First, we define some basic probabihstic concepts, and demonstrate how they may be used to model physical phenomena. Next, we derive some important probability distributions, in particular, the Gaussian (normal) and Poisson distributions. Following this is a treatment of stochastic calculus, with a particular focus upon Brownian dynamics. Monte Carlo methods are then presented, with apphcations in statistical physics, integration, and global minimization (simulated annealing). Finally, genetic optimization is discussed. This chapter serves as a prelude to the discussion of statistics and parameter estimation, in which the Monte Carlo method will prove highly usefiil in Bayesian analysis. 

Next follows a detailed discussion of probability theory, stochastic simulation, statistics, and parameter estimation. As engineering becomes more focused upon the molecular level, stochastic simulation techniques gain in importance. Particular attention is paid to Brownian dynamics, stochastic calculus, and Monte Carlo simulation. Statistics and parameter estimation are addressed from a Bayesian viewpoint, in which Monte Carlo simulation proves a powerful and general tool for making inferences and testing hypotheses from experimental data. 

Chapter 7 discusses a variety of topics all of which are related to the class of probabilistic CA (PCA) i.e. CA that involve some elements of probability in their state and/or time-evolution. The chapter begins with a physicist s overview of critical phenomena. Later sections include discussions of the equivalence between PCA and spin models, the critical behavior of PCA, mean-field theory, CA simulation of conventional spin models and a stochastic version of Conway s Life rule. 

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