A Computational Framework for Quantifying Extreme-Event Statistics in Nonlinear Systems with Stochastic Input

Abstract

Extreme events happen in many stochastic natural and engineering systems. Although these events occur with a low probability, they are often associated with catastrophic consequences, making the quantification of their statistics vitally important. In this dissertation, we aim to build efficient and accurate methods for the resolution of extreme-event statistics, using a master example of extreme ship motions in random seas.

A direct computation of the extreme ship motion statistics requires running numerical ship simulations to a long wave signal covering all wave conditions. Its computational cost, however, is prohibitively high considering the high complexity of the random wave field, the rareness of the extreme motions, and the expensiveness of the numerical simulation. One critical effort to reduce the computational cost is reducing the complexity of the random sources, i.e., parameterizing the wave field. The original problem then becomes a standard uncertainty quantification task to quantify the extreme response statistics given an input-to-response (ItR) function (that needs to be learned) with known input probability. Many methods have been proposed to address such problems, with one method we are particularly interested in---surrogate modeling trained with active learning. In detail, one can train a surrogate to approximate the ItR function. The training samples are sequentially selected by optimizing an acquisition function based on the existing samples to facilitate the convergence of the extreme-event statistics.

In this dissertation, we design a set of methods to resolve extreme-event statistics in various forms. Regarding ship motions in random waves, we first introduce a basic framework following existing methods in wave group parameterization and sequential sampling. In addition to some algorithmic improvements on these two components, we also enrich the framework by considering complete system dynamics through nonlinear wave simulation and ship-wave interaction CFD simulation. In this basic framework, the ship response statistics are defined in terms of the maximum motion in each wave group, which is easy to implement but not straightforward to interpret. We next adapt the framework to quantify a more robust measure, the temporal exceeding probability as the fraction of time that responses exceed a given threshold. While group parameterization significantly speeds up the computation in the above two works, the uncertainties introduced by reduced complexities have not been quantified. To incorporate the lost information, we also consider systems characterized by a stochastic ItR with heteroscedastic randomness due to dimension reduction.

In addition, we develop a multi-fidelity method to further reduce the computational cost. The key idea here is to leverage low-fidelity models whose cost is only a certain fraction of their high-fidelity counterparts. In particular, we employ the multi-fidelity Gaussian process as a surrogate model and design a new acquisition function to select both the location and fidelity of the next sample. We further adapt the multi-fidelity framework to quantify exceeding/failure probability over a threshold in the context of reliability analysis of connected and autonomous vehicles (CAV). Our acquisition is formulated through information-theoretic consideration which is not only desired to reduce the cost of CAV evaluation but also valuable to the general field of reliability analysis.

We next improve a likelihood-weighted acquisition (algorithm) initially designed for rare-event statistics and later extended to many other applications. In the final part of this dissertation, we present an ongoing work on batch sampling and conclude with a discussion of limitations and future research.