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 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 (i.e., a kriging model) that models the maximum 10-min crest elevation at the reference point, and to thereby efficiently estimate the short-term wave crest exceedance probability function. 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.

中文翻译：

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浅水波浪超越概率的主动子空间分析

我们使用一维 Korteweg-de Vries 方程模拟浅水波，其中波浪生成由预定义海况的随机波振幅参数化。这些波幅定义了高维随机输入向量，我们估计参考点的短期波峰超越概率。对于这个高维复杂的问题，大多数可靠性方法都失败了，而蒙特卡罗方法由于收敛速度慢而变得不切实际。因此，首先在离岸应用中，我们采用称为主动子空间分析的降维方法。该方法识别输入空间的低维子空间，该子空间对输入-输出可变性最重要。我们利用这一点来有效地训练高斯过程（即，克里金模型）模拟参考点的最大 10 分钟波峰高程，从而有效地估计短期波峰超越概率函数。Korteweg-de Vries 模型的活动低维子空间也暴露了与极端波浪和载荷相关的预期入射波群。我们的结果显示了主动子空间分析相对于离岸应用的蒙特卡罗实现的优势和有效性。