Skip to main content

Low-dimensional offshore wave input for extreme event quantification

Abstract

In offshore engineering design, nonlinear wave models are often used to propagate stochastic waves from an input boundary to the location of an offshore structure. Each wave realization is typically characterized by a high-dimensional input time-series, and a reliable determination of the extreme events is associated with substantial computational effort. As the sea depth decreases, extreme events become more difficult to evaluate. We here construct a low-dimensional characterization of the candidate input time series to circumvent the search for extreme wave events in a high-dimensional input probability space. Each wave input is represented by a unique low-dimensional set of parameters for which standard surrogate approximations, such as Gaussian processes, can estimate the short-term exceedance probability efficiently and accurately. We demonstrate the advantages of the new approach with a simple shallow-water wave model based on the Korteweg–de Vries equation for which we can provide an accurate reference solution based on the simple Monte Carlo method. We furthermore apply the method to a fully nonlinear wave model for wave propagation over a sloping seabed. The results demonstrate that the Gaussian process can learn accurately the tail of the heavy-tailed distribution of the maximum wave crest elevation based on only \(1.7\%\) of the required Monte Carlo evaluations.

This is a preview of subscription content, access via your institution.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12

References

  1. Bigoni D, Engsig-Karup AP, Eskilsson C (2016) Efficient uncertainty quantification of a fully nonlinear and dispersive water wave model with random inputs. J Eng Math 101(1):87–113

    MathSciNet  Article  Google Scholar 

  2. Bredmose H, Dixen M, Ghadirian A, Larsen TJ, Schløer S, Andersen SJ, Wang S, Bingham HB, Lindberg O, Christensen E, Vested MH, Carstensen S, Engsig-Karup AP, Petersen OS, Hansen HF, Mariegaard JS, Taylor PH, Adcock TAA, Obhrai C, Gudmestad OT, Tarp-Johansen NJ, Meyer CP, Krokstad JR, Suja-Thauvin L, Hanson TD (2016) DeRisk—accurate prediction of ULS wave loads. Outlook and first results. Energy Procedia 90:379–387

    Article  Google Scholar 

  3. Ghadirian A, Bredmose H (2019) Pressure impulse theory for a slamming wave on a vertical circular cylinder. J Fluid Mech 867:R1

    MathSciNet  Article  Google Scholar 

  4. Yildirim B, Karniadakis GE (2015) Stochastic simulations of ocean waves: an uncertainty quantification study. Ocean Model 86:15–35

    Article  Google Scholar 

  5. Rackwitz R (2001) Reliability analysis—a review and some perspectives. Struct Saf 23:365–395

    Article  Google Scholar 

  6. Owen AB (2013) Monte Carlo theory, methods and examples. Open Access

  7. Li J, Xiu D (2010) Evaluation of failure probability via surrogate models. J Comput Phys 229(23):8966–8980

    MathSciNet  Article  Google Scholar 

  8. Schöbi R, Sudret B, Marelli S (2016) Rare event estimation using polynomial-chaos kriging. ASCE ASME J Risk Uncertain Eng Syst A Civ Eng 3(2):D4016002

    Google Scholar 

  9. Constantine PG (2015) Active subspaces: emerging ideas for dimension reduction in parameter studies. SIAM, Philadelphia

    Book  Google Scholar 

  10. Nicodemi M (2012) Extreme value statistics. Springer, New York

    Google Scholar 

  11. Dematteis G, Grafke T, Vanden-Eijnden E (2018) Rogue waves and large deviations in deep sea. Proc Natl Acad Sci USA 115(5):855–860

    MathSciNet  Article  Google Scholar 

  12. Varadhan SRS (1984) Large deviations and applications. SIAM, Philadelphia

    Book  Google Scholar 

  13. Risken H (1989) The Fokker–Planck equation methods of solution and applications, 2nd edn. Springer, Berlin, Heidelberg, New York

    Book  Google Scholar 

  14. Mohamad MA, Sapsis TP (2018) Sequential sampling strategy for extreme event statistics in nonlinear dynamical systems. Proc Natl Acad Sci USA 115(44):11138–11143

    MathSciNet  Article  Google Scholar 

  15. Anastopoulos PA, Spyrou KJ, Bassler CC, Belenky V (2016) Towards an improved critical wave groups method for the probabilistic assessment of large ship motions in irregular seas. Prob Eng Mech 44(2016):18–27

    Article  Google Scholar 

  16. Boccotti P (1983) Some new results on statistical properties of wind waves. Appl Ocean Res 5(3):134–140

    Article  Google Scholar 

  17. Cousins W, Sapsis TP (2016) Reduced-order precursors of rare events in unidirectional nonlinear water waves. J Fluid Mech 790:368–388

    MathSciNet  Article  Google Scholar 

  18. Lindgren G (1970) Some properties of a normal process near a local maximum. Ann Math Stat 41(6):1870–1883

    MathSciNet  Article  Google Scholar 

  19. Tromans PS, Anaturk AR, Hagemeijer A (1991) A new model for the kinematics of large ocean waves-application as a design wave. In: Proceedings of the First International Offshore and Polar Engineering Conference, pp 64–71

  20. Šehić K, Bredmose H, Sørensen JD, Karamehmedović M (2021) Activesubspace analysis of exceedance probability for shallow-water waves. J Eng Math 126: 1

  21. Jolliffe I (2002) Principal component analysis, 2nd edn. Springer, New York

    MATH  Google Scholar 

  22. Sclavounos PD (2012) Karhunen–Loeve representation of stochastic ocean waves. Proc R Soc 468:2574–2594

    MathSciNet  Article  Google Scholar 

  23. Bouhlel M, Bartoli N, Otsmane A, Joseph Morlier J (2016) Improving kriging surrogates of high-dimensional design models by partial least squares dimension reduction. Struct Multidiscip Opt 53:935–952

    MathSciNet  Article  Google Scholar 

  24. Papaioannou I, Ehre M, Straub D (2019) PLS-based adaptation for efficient PCE representation in high dimensions. J Comput Phys 387:186–204

    MathSciNet  Article  Google Scholar 

  25. Gonzalez FJ, Balajewicz M (2018) Deep convolutional recurrent autoencoders for learning low-dimensional feature dynamics of fluid systems. CoRR abs/180801346

  26. Esmael B, Arnaout A, Fruhwirth RK, Thonhauser G (2013) A statistical feature-based approach for operations recognition in drilling time series. IJCISIM 5(2150–7988):454–461

    Google Scholar 

  27. Engsig-Karup AP, Bingham H, Lindberg O (2009) An efficient flexible-order model for 3D nonlinear water waves. J Comput Phys 228:2100–2118

    MathSciNet  Article  Google Scholar 

  28. Rasmussen CE, Williams CKI (2006) Gaussian processes for machine learning. MIT Press, Cambridge

    MATH  Google Scholar 

  29. Shahriari B, Swersky K, Wang Z, Adams R, de Freitas N (2016) Taking the human out of the loop: a review of bayesian optimization. Proc IEEE 104(1):148–175

    Article  Google Scholar 

  30. Gramacy RB, Apley DW (2015) Local gaussian process approximation for large computer experiments. J Comput Graph Stat 24:561–578

    MathSciNet  Article  Google Scholar 

  31. Naess A, Moan T (2012) Stochastic dynamics of marine structures. Cambridge University Press, Cambridge

    Book  Google Scholar 

  32. Paulsen BT (2019) OceanWave3D-Fortran90. GitHub repos. https://github.com/boTerpPaulsen/OceanWave3D-Fortran90

  33. Bredmose H (1999) Evolution equations for wave–wave interaction. Master’s thesis, Technical University of Denmark, Lyngby, Denmark

  34. Conrad PR, Marzouk YM, Pillai NS, Smith A (2016) Accelerating asymptotically exact MCMC for computationally intensive models via local approximations. J Am Stat Assoc 111(516):1591–1607

    MathSciNet  Article  Google Scholar 

  35. Engsig-Karup AP, Glimberg LS, Nielsen AS, Lindberg O (2013) Fast hydrodynamics on heterogenous many-core hardware. In: Couturier R (ed) Designing Scientific Applications on GPUs. Chap. 11. Taylor Francis, Abingdon, pp 251–294

    Google Scholar 

  36. Schløer S, Bredmose H, Ghadirian A (2017) Experimental and numerical statistics of storm wave forces on a monopile in uni- and multidirectional seas. In: Proceedings of the ASME 2017 36th Int Conf Omae 10:OMAE2017–61676

  37. Ullmann E, Papaioannou I (2015) Multilevel estimation of rare events. SIAM/ASA J UQ 3:922–953

    MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

This research was funded by the DeRisk project of Innovation Fund Denmark, Grant Number 4106-00038B. The authors very much appreciate the support by the Innovation Fund Denmark.

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Kenan Šehić.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Šehić, K., Bredmose, H., Sørensen, J.D. et al. Low-dimensional offshore wave input for extreme event quantification. J Eng Math 126, 13 (2021). https://doi.org/10.1007/s10665-021-10091-w

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s10665-021-10091-w

Keywords

  • Dimensionality reduction
  • Extreme events
  • Gaussian process
  • Offshore applications
  • Sequential design