  We consider an input-to-response (ItR) system characterized by (1)
parameterized input with a known probability distribution and (2) stochastic
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 a low-dimensional input parameter space (as a result of
dimension reduction applied to the original high-dimensional input 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 irregular waves, where the uncertainty in ItR naturally
originates from standard wave group parameterization, which reduces the
original high-dimensional wave field into a two-dimensional parameter space.
