Computing Failure Probabilities for PDEs with Random Data

Abstract

We deal with partial differential equations with random data and in particular Poisson’s equation with random data. This equation has a unique solution. The failure probability is the probability that a functional of that solution is less (or greater) than a given value. Algorithms for approximating failure probabilities are studied and tested and a new iterative method of approximating the failure probability is presented and examined in numerical experiments. As the thesis involves both random variables and partial differential equations, both probabilistic problems and problems with partial differential equations are studied along the way. The results of the numerical experiments show that the method performed well, with respect to computational cost, in comparison with a basic Monte Carlo simulation.