Extreme event probability estimation using PDE-constrained optimization and large deviation theory, with application to tsunamis
Extreme event probability estimation using PDE-constrained optimization and large deviation theory, with application to tsunamis
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DOI：
10.2140/camcos.2021.16.181
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发表时间：
2020-07
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通讯作者：
Shanyin Tong;E. Vanden-Eijnden;G. Stadler
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作者：
Shanyin Tong;E. Vanden-Eijnden;G. Stadler
我们提出并比较了估计偏微分方程所管辖的复杂系统中的极端事件概率的方法。我们的方法是指导思想从大偏差理论（LDT）和借用工具，从偏微分方程约束优化。所考虑的系统涉及随机参数，我们感兴趣的是量化的概率，系统状态解的标量函数是在或高于阈值。如果该阈值较大，则这些概率较小，并且其准确估计具有挑战性。我们的方法首先计算参数，最小化的LDT率函数的参数集导致极端事件。这些解提供了关于小概率事件的渐近信息。我们提出了一系列的方法来完善这些估计，即基于重要性抽样和几何近似的极端事件集的方法。结果制定一般的参数分布和详细的表达式时，基本分布是高斯。我们给出的理论和数值参数表明，我们的方法的性能是不敏感的极值事件，我们感兴趣的。我们说明了我们的方法来量化的概率极端海啸事件的海岸的应用。海啸通常是由地震期间海底高度的突然、不可预测的变化引起的。我们将这种变化建模为一个随机过程，考虑到了潜在的物理学。我们使用一维浅水方程来模拟海啸的数值。伴随方法用于计算求解LDT优化问题所需的导数。我们提出了一个极端事件概率估计方法的比较，并找到哪种类型的海底高程变化导致最大的海啸海岸。
We propose and compare methods for the estimation of extreme event probabilities in complex systems governed by PDEs. Our approach is guided by ideas from large deviation theory (LDT) and borrows tools from PDE-constrained optimization. The systems under consideration involve random parameters and we are interested in quantifying the probability that a scalar function of the system state solution is at or above a threshold. If this threshold is large, these probabilities are small and their accurate estimation is challenging. Our methods first compute parameters that minimize the LDT-rate function over the set of parameters leading to extreme events. These solutions provide asymptotic information about small probability events. We propose a series of methods to refine these estimates, namely methods based on importance sampling and on geometric approximation of the extreme event sets. Results are formulated for general parameter distributions and detailed expressions are provided when the underlying distribution is Gaussian. 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. Adjoint methods are used to compute the derivatives needed to solve the LDT-optimization problem. We present a comparison of the methods for extreme event probability estimation, and find which type of ocean floor elevation change leads to the largest tsunamis on shore.