Probabilistic reachability: a review of estimation methods

We evaluate numerically-precise Monte Carlo (MC), Quasi-Monte Carlo (QMC) and Randomised Quasi-Monte Carlo (RQMC) methods for computing probabilistic reachability in hybrid systems with random parameters. Computing reachability probability amounts to computing (multidimensional) integrals. In particular, we pay attention to QMC methods due to their theoretical benefits in convergence speed with respect to the MC method. The Koksma-Hlawka inequality is a standard result that bounds the approximation error of an integral by QMC techniques. However, it is not useful in practice because it depends on the variation of the integrand function, which is in general difficult to compute. The question arises whether it is possible to apply statistical or empirical methods for estimating the approximation error. In this research we compare a number of interval estimation techniques based on the Central Limit Theorem (CLT), and we also introduce a new approach based on the CLT for computing confidence intervals for probability near the borders of the [0,1] interval.