Computing Failure Probabilities for PDEs with Random Data
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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.
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PDE with random data, failure probability, Monte Carlo method, finite element method, selective refinement