  We propose and analyze a method for computing failure probabilities of
systems modeled as numerical deterministic models (e.g., PDEs) with uncertain
input data. A failure occurs when a functional of the solution to the model is
below (or above) some critical value. By combining recent results on quantile
estimation and the multilevel Monte Carlo method we develop a method which
reduces computational cost without loss of accuracy. We show how the
computational cost of the method relates to error tolerance of the failure
probability. For a wide and common class of problems, the computational cost is
asymptotically proportional to solving a single accurate realization of the
numerical model, i.e., independent of the number of samples. Significant
reductions in computational cost are also observed in numerical experiments.
