A Sampling and Transformation Approach to Solving Random Differential Equations

This research explores an innovative sampling method used to conduct uncertainty analysis on a system with one random input. Given the distribution of the random input, X, we seek to find the distribution of the output random variable Y. When the functional form of the transformation Y=g(X) is not explicitly known, complicated procedures, such as stochastic projection or Monte Carlo simulation must be employed. The main focus of this research is determining the distribution of the random variable Y=g(X) where g(X) is the solution to an ordinary differential equation and X is a random parameter. Here, y=g(X) is approximated by constructing a sample {Xi, Yi} where the Xi are not random, but chosen to be evenly spaced on the interval [a, b] and Yi=g(Xi). Using this data, an efficient approximation g(X) g(X) is constructed. Then the transformation method, in conjunction with g(X), is used to find the probability density function of the random variable Y. This uniform sampling method and transformation method will be compared to the stochastic projection and Monte Carlo methods currently being used in uncertainty analysis. It will be demonstrated, through several examples, that the proposed uniform sampling method and transformation method can work faster and more efficiently than the methods mentioned. The original document contains color images.