Numerical Uncertainty Quantification for Extreme Wave Events

Design of offshore structures is severely influenced by considerations of the effects of extreme events. It is well-known that extreme wave events appear more frequently than predicted by the traditional linear wave theory. In 1995, just off the coast of Norway, an unexpected wave of 18.5m crest height hit the Draupner platform and generated substantial loads.

Measurements failed to provide any evidence of the coming freakwave. In Denmark, wave impacts that lead to severe damages were recorded at the Horns Rev Offshore Wind Farm. Therefore, to design a safe and robust offshore structure, it is crucial to appropriately quantify uncertainties that may generate extreme events.

The current level of mathematical modeling of extreme events is unsatisfactory. This is partly because the methods such as FORM/SORM idealize the failure surface, and partly due to impractical convergence rates. Alternative approaches, such as extreme value or large deviation theories, use only the statistical characteristics of extreme events.

When the probability density for the events features an unusually heavy tail, these methods can be critically limited. The simple Monte Carlo method seems to be the only reliable computational approach to quantify extreme events accurately. The purpose of this Ph.D. project is to develop, implement, validate, and apply numerical uncertainty quantification methods to shallow-water wave models, in order to achieve accelerated and improved predictions – compared to the state-of-the-art – of extreme wave events at offshore structures.

To quantify the occurrence of extreme events sufficiently, we sequentially design numerical experiments and actively explore the wave model input probability space. The idea is to gradually approach the failure region in the input space using a decreasing sequence of nested intermediate failure regions.

Each intermediate step employs a Markov chain Monte Carlo (MCMC) algorithm, and hence requires numerical evaluation of the limit-state function. When this evaluation is expensive, the MCMC algorithm becomes impractical. We therefore propose to approximate each Markov chain locally using standard regression.

To this end, we exploit the local regularity of the limit-state function and of its numerical realization to get a sufficient approximation using only a few evaluations. However, the computational requirements of standard regression usually increase with input dimension, and the computations become infeasible.

As it turns out, offshore applications typically require high-dimensional models of environmental conditions. To address this, we identify and use a low-dimensional but sufficient subspace of the input parameters to improve standard regression methods. To this end, we employ the active-subspace analysis (ASA).

Our numerical experiments show that Gaussian process regression with ASA can sufficiently quantify extreme events. However, ASA requires efficient evaluation of the output gradient, and when this is infeasible, we replace ASA with partial least square (PLS) regression, which is a gradient-free approach.

Being a linear approach, PLS may critically underpredict extreme events. But we show that when it is applied in the local approximation for the subset simulation method, the PLS indeed provides adequate dimension reduction of the input space. Other approaches, such as the principal component analysis (PCA) or the use of autoencoders, fail to include the correlation between the input parameters and the value of the limitstate function.

However, these methods can still prove advantageous in classificationproblems. For example, we have shown that suspensions of microparticles can be classified according to concentration using PCA and autoencoder-based analysis of the optical speckle transmitted through these suspensions. Back in the context of offshore engineering, when classical dimension reduction methods fail, we propose to use standard statistical measures of the wave input as the low-dimensional design parameters.

Our numerical results show that extreme events can in this way be adequately quantified using significantly fewer model evaluations than simple Monte Carlo.