Quantifying extreme events is a central issue for many technological processes and natural phenomena. As extreme events, we consider transient responses that push the system away from its statistical steady state and that correspond to large excursions. Complex systems exhibiting extreme events include dynamical systems found in nature, such as the occurrence of anomalous weather and climate events, turbulence, formation of freak waves in the ocean and optics, and dynamical systems in engineering applications, including mechanical components under environmental loads, ship rolling and capsizing, critical events in power grids, as well as chemical reactions and conformational changes in molecules. It has been recognized that extreme events occur more frequently than Gaussian statistics suggest and thus occur often enough that they have practical consequences, and sometimes catastrophic outcomes, that are important to understand and predict. A hallmark characteristic of extreme events in complex dynamical systems is non-Gaussian statistics (e.g. heavy-tails) in the probability density function (pdf) describing the response of their observables. For engineers and applied mathematicians, a central issue is how to efficiently and accurately describe this non-Gaussian behavior. For random dynamical systems with inherently nonlinear dynamics, expressed through intermittent events, nonlinear energy transfers, broad energy spectra, and large intrinsic dimensionality, it is largely the case that we are limited to (direct) Monte-Carlo sampling, which is too expensive to apply in real-world applications.

To address these challenges, we present both direct and adaptive (sampling based) strategies designed to quantify the probabilistic aspects of extreme events in complex dynamical systems, effectively and efficiently. Specifically, we first develop a direct quantification framework that involves a probabilistic decomposition that separately considers intermittent, extreme events from the background stochastic attractor of the dynamical system. This decomposition requires knowledge of the dynamical mechanisms that are responsible for extreme events and partitions the phase space accordingly. We then apply different uncertainty quantification schemes to the two decomposed dynamical regimes:​ the background attractor and the intermittent, extreme-event component. The background component, describing the 'core' of the pdf, although potentially very high-dimensional, can be efficiently described by uncertainty quantification schemes that resolve low-order statistics. On the other hand, the intermittent component, related to the tails, can be described in terms of a low-dimensional representation by a small number of modes through a reduced order model of the extreme events. The probabilistic information from these two regimes is then synthesized according to a total probability law argument, to effectively approximate the heavy-tailed, non-Gaussian probability distribution function for quantities of interest. The method is demonstrated through numerous applications and examples, including the analytical and semi-analytical quantification of the heavy-tailed statistics in mechanical systems under random impulsive excitations (modeling slamming events in high speed craft motion), oscillators undergoing transient parametric resonances and instabilities (modeling ship rolling in irregular seas and beam bending), and extreme events in nonlinear Schr6dinger based equations (modeling rogue waves in the deep ocean). The proposed algorithm is shown to accurately describe tail statistics in all of these examples and is demonstrated to be many orders of magnitude faster than direct Monte-Carlo simulations.

The second part of this thesis involves the development of adaptive, sampling based strategies that aim to accurately estimate the probability distribution and extreme response statistics of a scalar observable, or quantity of interest, through a minimum number of experiments (numerical simulations). These schemes do not require specialized knowledge of the dynamics, nor understanding of the mechanism that cause or trigger extreme responses. For numerous complex systems it may not be possible or very challenging to analyze and quantify conditions that lead to extreme responses or even to obtain an accurate description of the dynamics of all the processes that are significant. To address this important class of problems, we develop a sequential algorithm that provides the next-best design point (set of experimental parameters) that leads to the largest reduction in the error of the probability density function estimate for the scalar quantity of interest when the adaptively predicted design point is evaluated. The proposed algorithm utilizes Gaussian process regression to infer dynamical properties of the quantity of interest, which is then used to estimate the desired pdf along with uncertainty bounds. We iteratively determine new design points through an optimization procedure that finds the optimal point in parameter space that maximally reduces uncertainty between the estimated bounds of the posterior pdf estimate of the observable. We provide theorems that guarantee convergence of the algorithm and analyze its asymptotic behavior.

The adaptive sampling method is illustrated to an example in ocean engineering. We apply the algorithm to estimate the non-Gaussian statistics describing the loads on an offshore platform in irregular seas. The response of the platform is quantified through three-dimensional smoothed particle hydrodynamics simulations. Because of the extreme computational cost of these numerical models, quantification of the extreme event statistics for such systems has been a formidable challenge. We demonstrate that the adaptive algorithm accurately quantifies the extreme event statistics of the loads on the structure through a small number of numerical experiments, showcasing that the proposed algorithm can realistically account for extreme events in the design and optimization processes for large-scale engineering systems.