A Sampling and Transformation Approach to Solving Random Differential Equations
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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 transformationY=g(X) is not explicitly known, complicated procedures, such as stochastic projection or Monte Carlo simulation must be employed. The main focus of thisresearch 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 randomparameter. He...
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 transformationY=g(X) is not explicitly known, complicated procedures, such as stochastic projection or Monte Carlo simulation must be employed. The main focus of thisresearch 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 randomparameter. 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 "(X) g(X) is constructed. Then the transformation method, in conjunction with "(X), is used tofind the probability density function of the random variable Y. This uniform sampling method and transformation method will be compared to the stochasticprojection and Monte Carlo methods currently being used in uncertainty analysis. It will be demonstrated, through several examples, that the proposed uniformsampling method and transformation method can work faster and more efficiently than the methods mentioned. This work has been selected by scholars as being culturally important, and is part of the knowledge base of civilization as we know it. This work was reproduced from the original artifact, and remains as true to the original work as possible. Therefore, you will see the original copyright references, library stamps (as most of these works have been housed in our most important libraries around the world), and other notations in the work. This work is in the public domain in the United States of America, and possibly other nations. Within the United States, you may freely copy and distribute this work, as no entity (individual or corporate) has a copyright on the body of the work. As a reproduction of a historical artifact, this work may contain missing or blurred pages, poor pictures, errant marks, etc. Scholars believe, and we concur, that this work is important enough to be preserved, reproduced, and made generally available to the public. We appreciate your support of the preservation process, and thank you for being an important part of keeping this knowledge alive and relevant.
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