  The average time between two occurrences of the same event, referred to as
its return time (or return period), is a useful statistical concept for
practical applications. For instance insurances or public agency may be
interested by the return time of a 10m flood of the Seine river in Paris.
However, due to their scarcity, reliably estimating return times for rare
events is very difficult using either observational data or direct numerical
simulations. For rare events, an estimator for return times can be built from
the extrema of the observable on trajectory blocks. Here, we show that this
estimator can be improved to remain accurate for return times of the order of
the block size. More importantly, we show that this approach can be generalised
to estimate return times from numerical algorithms specifically designed to
sample rare events. So far those algorithms often compute probabilities, rather
than return times. The approach we propose provides a computationally extremely
efficient way to estimate numerically the return times of rare events for a
dynamical system, gaining several orders of magnitude of computational costs.
We illustrate the method on two kinds of observables, instantaneous and
time-averaged, using two different rare event algorithms, for a simple
stochastic process, the Ornstein-Uhlenbeck process. As an example of realistic
applications to complex systems, we finally discuss extreme values of the drag
on an object in a turbulent flow.
