A number of methods have been developed for unbiased and efficient approximation of small probabilities and expected values that depend heavily on tail events. Examples include importance sampling and particle splitting methods. However, successfully implementing these methods can require some care. Traditional diagnostics one might use to assess algorithm performance can be misleading, and may suggest the method is working well when in fact it is not. As a consequence, methods that combine design with rigorous performance analysis are particularly useful.
This talk will give an overview of how subsolutions to a related Hamilton-Jacobi- Bellman equation (a nonlinear PDE) can be used for the design and analysis of the types of numerical methods mentioned previously. We first review the construction of both importance sampling and splitting schemes, and recall problems that one might encounter in the rare event setting. Then we outline how designs based on subsolutions avoid these problems, and in fact give what are in some sense necessary and sufficient conditions for good performance as the probability of interest tends to zero. Time permitting we describe more recent results, such as qualitative differences between the two methods when computing escape probabilities near a rest point and non-asymptotic bounds on variance.