Visualization is an essential operation when assessing the risk of rare events such as coastal or river floodings. The goal is to display a few prototype events that best represent the probability law of the observed phenomenon, a task known as quantization. It becomes a challenge when data is expensive to generate and critical events are scarce, like extreme natural hazard. In the case of floodings, each event relies on an expensive-to-evaluate hydraulic simulator which takes as inputs offshore meteo-oceanic conditions and dyke breach parameters to compute the water level map. In this article, Lloyd's algorithm, which classically serves to quantize data, is adapted to the context of rare and costly-to-observe events. Low probability is treated through importance sampling, while Functional Principal Component Analysis combined with a Gaussian process deal with the costly hydraulic simulations. The calculated prototype maps represent the probability distribution of the flooding events in a minimal expected distance sense, and each is associated to a probability mass. The method is first validated using a 2D analytical model and then applied to a real coastal flooding scenario. The two sources of error, the metamodel and the importance sampling, are evaluated to quantify the precision of the method.

翻译：可视化是评估洪水（如沿海或河流泛滥）等罕见事件风险时的关键操作。其目标在于展示最能代表观测现象概率分布的若干典型事件，这一任务被称为"量化"（quantization）。当数据生成成本高昂且关键事件（如极端自然灾害）稀缺时，该任务极具挑战性。以洪水为例，每个事件依赖于一个计算成本高昂的水力模拟器——该模拟器需输入海上气象-海洋条件及堤坝溃口参数，以计算水位分布图。本文对经典用于数据量化的劳埃德（Lloyd）算法进行了改进，使其适用于罕见且观测成本高昂的事件场景。通过重要性采样（importance sampling）处理低概率问题，并结合函数主成分分析（Functional Principal Component Analysis）与高斯过程（Gaussian process）处理高成本水力模拟。计算所得的典型分布图在最小期望距离意义下表征了洪水事件的概率分布，且每个分布图关联一个概率质量。该方法首先通过二维解析模型进行验证，随后应用于真实沿海洪水场景。本文对元模型（metamodel）与重要性采样两类误差源进行了评估，以量化方法精度。