Extreme event probability estimation using PDE-constrained optimization and large deviation theory, with application to tsunamis

TL;DR

This paper develops and compares PDE-constrained optimization methods combined with large deviation theory to efficiently estimate the probabilities of rare, extreme events such as tsunamis, by identifying likely mechanisms and refining probability estimates.

Contribution

It introduces a novel approach integrating large deviation theory with PDE optimization and importance sampling for accurate rare event probability estimation in complex systems.

Findings

Methods accurately estimate small probabilities of extreme events.

Approach is insensitive to the extremeness of the events.

Application to tsunami modeling identifies key ocean floor changes leading to large tsunamis.

Abstract

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.