  We consider a dynamical system with two sources of uncertainties: (1)
parameterized input with a known probability distribution and (2) stochastic
input-to-response (ItR) function with heteroscedastic randomness. Our purpose
is to efficiently quantify the extreme response probability when the ItR
function is expensive to evaluate. The problem setup arises often in physics
and engineering problems, with randomness in ItR coming from either intrinsic
uncertainties (say, as a solution to a stochastic equation) or additional
(critical) uncertainties that are not incorporated in the input parameter
space. To reduce the required sampling numbers, we develop a sequential
Bayesian experimental design method leveraging the variational heteroscedastic
Gaussian process regression (VHGPR) to account for the stochastic ItR, along
with a new criterion to select the next-best samples sequentially. The validity
of our new method is first tested in two synthetic problems with the stochastic
ItR functions defined artificially. Finally, we demonstrate the application of
our method to an engineering problem of estimating the extreme ship motion
probability in ensemble of wave groups, where the uncertainty in ItR naturally
originates from the uncertain initial condition of ship motion in each wave
group.
