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Abstract
Design of oﬀshore structures is severely inﬂuenced by considerations of the eﬀects of extreme events. It is wellknown 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 shallowwater wave models, in order to achieve accelerated and improved predictions – compared to the stateoftheart – 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 limitstate 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 limitstate 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 highdimensional models of environmental conditions. To address this, we identify and use a lowdimensional but suﬃcient subspace of the input parameters to improve standard regression methods. To this end, we employ the activesubspace 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 gradientfree 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 autoencoderbased 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 lowdimensional 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.
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 shallowwater wave models, in order to achieve accelerated and improved predictions – compared to the stateoftheart – 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 limitstate 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 limitstate 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 highdimensional models of environmental conditions. To address this, we identify and use a lowdimensional but suﬃcient subspace of the input parameters to improve standard regression methods. To this end, we employ the activesubspace 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 gradientfree 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 autoencoderbased 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 lowdimensional 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.
Original language  English 

Publisher  Technical University of Denmark 

Number of pages  171 
Publication status  Published  2020 
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 1 Finished

Numerical Uncertainty Quantification for Stochastic Wave Loads
Sehic, K., Karamehmedovic, M., Bredmose, H., Sørensen, J. D., EngsigKarup, A. P., Hesthaven, J. & Le Maitre, O. P.
15/01/2017 → 13/05/2020
Project: PhD