蒙特卡羅方法之罕見事件及其應用於電路模擬

The simulation of rare events in Monte Carlo is of practical importance across many disciplines including circuit simulation, finance, and meterology to name a few. To make inferences about the behavior of distributions in systems which have a likelihood of 1 in billions, simulation by traditional Monte Carlo becomes impractical. Intuitively, Monte Carlo samples which are likely (i.e. samples drawn from regions of higher probability density) have little influence on the tail distribution which is associated with rare events. This work provides a novel methodology to directly sample rare events in input multivariate random vector space as a means to efficiently learn about the distribution tail in the output space. In addition, the true form of the Monte Carlo simulation is modeled by first linear and then quadratic forms. A systematic procedure is developed which traces the flow from the linear or quadratic modeling to the computation of distribution statistics such as moments and quantiles directly from the modeling form itself. Next, a general moment calculation method is derived based on the distribution quantiles where no underlying linear or quadratic model is assumed. Finally, each of the proposed methods is grounded in practical circuit simulation examples. Overall, the thesis provides several new methods and approaches to tackle some challenging problems in Monte Carlo simulation, probability distribution modeling, and statistical analysis.

並列關鍵字

monte carlo ； statistics ； rare events ； high sigma

參考文獻

[1] M. Abramowitz and I. A. Stegun. Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. Dover, New York, 1964.

Google Scholar

[2] J. Aitchison and J. A. Brown. The lognormal distribution with special reference to its uses in economics. University of Cambridge Department of Applied Ecnomics Monograph 5, page 176, 1957.