  We develop a method for the evaluation of extreme event statistics associated
with nonlinear dynamical systems, using a small number of samples. From an
initial dataset of design points, we formulate a sequential strategy that
provides the 'next-best' data point (set of parameters) that when evaluated
results in improved estimates of the probability density function (pdf) for a
scalar quantity of interest. The approach utilizes Gaussian process regression
to perform Bayesian inference on the parameter-to-observation map describing
the quantity of interest. We then approximate the desired pdf along with
uncertainty bounds utilizing the posterior distribution of the inferred map.
The 'next-best' design point is sequentially determined through an optimization
procedure that selects the point in parameter space that maximally reduces
uncertainty between the estimated bounds of the pdf prediction. Since the
optimization process utilizes only information from the inferred map it has
minimal computational cost. Moreover, the special form of the metric emphasizes
the tails of the pdf. The method is practical for systems where the
dimensionality of the parameter space is of moderate size, i.e. order O(10). We
apply the method to estimate the extreme event statistics for a very
high-dimensional system with millions of degrees of freedom: an offshore
platform subjected to three-dimensional irregular waves. It is demonstrated
that the developed approach can accurately determine the extreme event
statistics using limited number of samples.
