We propose and compare methods for the analysis of extreme events in complex systems governed by PDEs that involve random parameters, in situations where we are interested in quantifying the probability that a scalar function of the system's solution is above a threshold. If the threshold is large, this probability is small and its accurate estimation is challenging. To tackle this difficulty, we blend theoretical results from large deviation theory (LDT) with numerical tools from PDE-constrained optimization. Our methods first compute parameters that minimize the LDT-rate function over the set of parameters leading to extreme events, using adjoint methods to compute the gradient of this rate function. The minimizers give information about the mechanism of the extreme events as well as estimates of their probability. We then propose a series of methods to refine these estimates, either via importance sampling or geometric approximation of the extreme event sets. Results are formulated for general parameter distributions and detailed expressions are provided when Gaussian distributions. We give theoretical and numerical arguments showing that the performance of our methods is insensitive to the extremeness of the events we are interested in. We illustrate the application of our approach to quantify the probability of extreme tsunami events on shore. Tsunamis are typically caused by a sudden, unpredictable change of the ocean floor elevation during an earthquake. We model this change as a random process, which takes into account the underlying physics. We use the one-dimensional shallow water equation to model tsunamis numerically. In the context of this example, we present a comparison of our methods for extreme event probability estimation, and find which type of ocean floor elevation change leads to the largest tsunamis on shore.

翻译：本文提出并比较了在复杂系统中由含随机参数的偏微分方程（PDE）支配的极端事件分析方法，研究目标为系统解的标量函数超过某一阈值时的概率量化问题。当阈值较高时，该概率极小，精确估计极具挑战。为解决这一难题，我们融合了大偏差理论的数学结果与PDE约束优化的数值手段。首先，我们利用伴随方法计算大偏差率函数的梯度，在导致极端事件的参数集合中搜索使该率函数最小的参数值。这些极小值点既揭示了极端事件的触发机制，也提供了其概率的初步估计。随后，我们提出一系列改进估计的方法，包括重要性采样与极端事件集合的几何近似。本文针对一般参数分布给出了理论结果，并针对高斯分布给出了详细表达式。理论分析与数值实验表明，所提方法的性能对极端事件的极端程度不敏感。我们以海岸极端海啸事件的概率量化为例展示了方法的应用。海啸通常由地震期间海底地形的突然、不可预测变化引发。我们将这种变化建模为随机过程，并纳入物理约束。采用一维浅水方程进行数值模拟。基于此实例，我们比较了所提出的极端事件概率估计方法，并确定了导致海岸最大海啸的海底地形变化类型。