  We model shallow-water waves using a one-dimensional Korteweg-de Vries
equation with the wave generation parameterized by random wave amplitudes for a
predefined sea state. These wave amplitudes define the high-dimensional
stochastic input vector for which we estimate the short-term wave crest
exceedance probability at a reference point. For this high-dimensional and
complex problem, most reliability methods fail, while Monte Carlo methods
become impractical due to the slow convergence rate. Therefore, first within
offshore applications, we employ the dimensionality reduction method called
\textit{Active-Subspace Analysis}. This method identifies a low-dimensional
subspace of the input space that is most significant to the input-output
variability. We exploit this to efficiently train a Gaussian process that
models the maximum 10-minute crest elevation at the reference point, and to
thereby efficiently estimate the short-term wave crest exceedance probability.
The active low-dimensional subspace for the Korteweg-de Vries model also
exposes the expected incident wave groups associated with extreme waves and
loads. Our results show the advantages and the effectiveness of the
active-subspace analysis against the Monte Carlo implementation for offshore
applications.

Fluid Dynamics;Data Analysis, Statistics and Probability