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.
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 this 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 these distribution are 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. 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.
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