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Coupling multi-fidelity kriging and model-order reduction for the construction of virtual charts

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Abstract

This article presents the coupling between multi-fidelity kriging and a database generated on-the-fly by model reduction to accelerate the generation of a surrogate model. The two-level multi-fidelity kriging method Evofusion is used for data fusion. The remarkable point is the generation of low-fidelity and high-fidelity observations from the same solver using the Proper Generalized Decomposition, a model-order reduction method. A 17 \(\times \) speedup is obtained here on an elasto-viscoplastic test case.

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References

  1. Aversano G, Parra-Alvarez JC, Isaac BJ, Smith ST, Coussement A, Gicquel O, Parente A (2018) PCA and Kriging for the efficient exploration of consistency regions in Uncertainty Quantification. In: Proceedings of the combustion institute. https://doi.org/10.1016/j.proci.2018.07.040

    Article  Google Scholar 

  2. Bhattacharyya M, Fau A, Nackenhorst U, Néron D, Ladevèze P (2017) A LATIN-based model reduction approach for the simulation of cycling damage. Comput Mech 62(4):725–743. https://doi.org/10.1007/s00466-017-1523-z

    Article  MathSciNet  MATH  Google Scholar 

  3. Boucard PA, Buytet S, Guidault PA (2009) A multiscale strategy for structural optimization. Int J Numer Methods Eng 78(1):101–126. https://doi.org/10.1002/nme.2484

    Article  MATH  Google Scholar 

  4. Chatterjee A (2000) An introduction to the proper orthogonal decomposition. Curr Sci 78(7):808–817

    Google Scholar 

  5. Chinesta F, Keunings R, Leygue A (2014) The proper generalized decomposition for advanced numerical simulations. Springer briefs in applied sciences and technology. Springer, Cham

    Book  Google Scholar 

  6. Courrier N, Boucard PA, Soulier B (2016) Variable-fidelity modeling of structural analysis of assemblies. J Glob Optim 64(3):577–613. https://doi.org/10.1007/s10898-015-0345-9

    Article  MathSciNet  MATH  Google Scholar 

  7. Cressie N (2015) Statistics for spatial data. Wiley, New York

    MATH  Google Scholar 

  8. De Lozzo M (2015) Substitution de modèle et approche multifidélité en expérimentation numérique. Journal de la Société Française de Statistique 156(4):21–55

    MATH  Google Scholar 

  9. Forrester AI, Bressloff NW, Keane AJ (2006) Optimization using surrogate models and partially converged computational fluid dynamics simulations. Proc R Soc A Math Phys Eng Sci 462(2071):2177–2204. https://doi.org/10.1098/rspa.2006.1679

    Article  MATH  Google Scholar 

  10. Forrester AIJ, Keane AJ, Bressloff NW (2006) Design and analysis of “Noisy” computer experiments. AIAA J 44(10):2331–2339

    Article  Google Scholar 

  11. Forrester AIJ, Sóbester A, Keane AJ (2007) Multi-fidelity optimization via surrogate modelling. Proc R Soc Lond A Math Phys Eng Sci 463(2088):3251–3269. https://doi.org/10.1098/rspa.2007.1900

    Article  MathSciNet  MATH  Google Scholar 

  12. Han Z, Zimmerman R, Görtz S (2012) Alternative Cokriging method for variable-fidelity surrogate modeling. AIAA J 50(5):1205–1210. https://doi.org/10.2514/1.J051243

    Article  Google Scholar 

  13. Han ZH, Görtz S (2012) A hierarchical kriging model for variable-fidelity surrogate modeling of aerodynamic functions. AIAA J 50(9):1885–1896

    Article  Google Scholar 

  14. Han ZH, Zimmermann R, Görtz S (2010) A new Cokriging method for variable-fidelity surrogate modeling of aerodynamic data. In: 48th AIAA aerospace sciences meeting including the new horizons forum and aerospace exposition

  15. Heyberger C, Boucard PA, Néron D (2013) A rational strategy for the resolution of parametrized problems in the PGD framework. Comput Methods Appl Mech Eng 259:40–49. https://doi.org/10.1016/j.cma.2013.03.002

    Article  MathSciNet  MATH  Google Scholar 

  16. Jones DR (2001) A taxonomy of global optimization methods based on response surface. J Glob Optim 21:345–383

    Article  MathSciNet  Google Scholar 

  17. Jones DR, Schonlau M, Welch WJ (1998) Efficient global optimization of expensive black-box functions. J Glob Optim 13(4):455–492

    Article  MathSciNet  Google Scholar 

  18. Kennedy MC, O’Hagan A (2000) Predicting the output from a complex computer code when fast approximations are available. Biometrika 87(1):1–13

    Article  MathSciNet  Google Scholar 

  19. Kleijnen JP (1998) Experimental design for sensitivity analysis, optimization, and validation of simulation models. Handbook of simulation. Wiley, New York, pp 173–223

    Chapter  Google Scholar 

  20. Kleijnen JPC, van Beers WCM (2004) Application-driven sequential designs for simulation experiments: Kriging metamodelling. J Oper Res Soc 55(8):876–883. https://doi.org/10.1057/palgrave.jors.2601747

    Article  MATH  Google Scholar 

  21. Kramer B, Marques AN, Peherstorfer B, Villa U, Willcox K (2017) Multifidelity probability estimation via fusion of estimators. J Comput Phys 392:385–402

    Article  MathSciNet  Google Scholar 

  22. Ladevèze P (1985) Sur une famille d’algorithmes en mécanique des structures. Comptes-rendus des séances de l’Académie des sciences. Série 2, Mécanique-physique, chimie, sciences de l’univers, sciences de la terre 300(2):41–44

    MathSciNet  MATH  Google Scholar 

  23. Ladevèze P (1999) Nonlinear computational structural mechanics: new approaches and non-incremental methods of calculation. Mechanical engineering series. Springer, New York

    Book  Google Scholar 

  24. Laurent L, Riche RL, Soulier B, Boucard PA (2017) An overview of gradient-enhanced metamodels with applications. Arch. Comput. Methods Eng. 26:61–106. https://doi.org/10.1007/s11831-017-9226-3

    Article  MathSciNet  Google Scholar 

  25. Le Gratiet L (2013) Recursive co-kriging model for design of computer experiments with multiple levels of fidelity with an application to hydrodynamic. Int J Uncertain Quantif 4(5):365–386

    Article  Google Scholar 

  26. Lemaitre J, Chaboche JL (1994) Mechanics of solid materials. Cambridge University Press, Cambridge

    MATH  Google Scholar 

  27. Maday Y, Ronquist E (2004) The reduced basis element method: application to a thermal fin problem. SIAM J Sci Comput 26(1):240–258. https://doi.org/10.1137/S1064827502419932

    Article  MathSciNet  MATH  Google Scholar 

  28. McKay MD, Beckman RJ, Conover WJ (2000) A comparison of three methods for selecting values of input variables in the analysis of output from a computer code. Technometrics 42(1):55–61

    Article  Google Scholar 

  29. Nachar S (2018) Einstein summation for MATLAB. Zenodo. https://doi.org/10.5281/zenodo.1297570

    Article  Google Scholar 

  30. Néron D, Boucard PA, Relun N (2015) Time-space PGD for the rapid solution of 3D nonlinear parametrized problems in the many-query context. Int J Numer Methods Eng 103(4):275–292

    Article  MathSciNet  Google Scholar 

  31. Nouy A (2010) A priori model reduction through proper generalized decomposition for solving time-dependent partial differential equations. Comput Methods Appl Mech Eng 199(23–24):1603–1626. https://doi.org/10.1016/j.cma.2010.01.009

    Article  MathSciNet  MATH  Google Scholar 

  32. Picheny V, Ginsbourger D, Roustant O, Haftka RT, Kim NH (2010) Adaptive designs of experiments for accurate approximation of target regions. J Mech Des 132(7):1–9

    Article  Google Scholar 

  33. Quarteroni A, Manzoni A, Negri F (2016) Reduced basis methods for partial differential equations, UNITEXT, vol 92. Springer, Cham

    MATH  Google Scholar 

  34. Rasmussen CE, Williams CKI (2006) Gaussian processes for machine learning. Adaptive computation and machine learning. MIT Press, Cambridge

    MATH  Google Scholar 

  35. Relun N, Néron D, Boucard PA (2013) A model reduction technique based on the PGD for elastic-viscoplastic computational analysis. Comput Mech 51(1):83–92. https://doi.org/10.1007/s00466-012-0706-x

    Article  MathSciNet  MATH  Google Scholar 

  36. Robinson GK (1991) That BLUP is a good thing: the estimation of random effects. Stat Sci 6(1):15–32. https://doi.org/10.1214/ss/1177011926

    Article  MathSciNet  MATH  Google Scholar 

  37. Vitse M, Néron D, Boucard PA (2014) Virtual charts of solutions for parametrized nonlinear equations. Comput Mech 54(6):1529–1539. https://doi.org/10.1007/s00466-014-1073-6

    Article  MathSciNet  MATH  Google Scholar 

  38. Willcox K (2006) Unsteady flow sensing and estimation via the gappy proper orthogonal decomposition. Comput Fluids 35(2):208–226

    Article  Google Scholar 

  39. Zimmerman DL, Holland DM (2005) Complementary co-kriging: spatial prediction using data combined from several environmental monitoring networks. Environmetrics 16:219–234

    Article  MathSciNet  Google Scholar 

  40. Zimmermann R, Han ZH (2010) Simplified cross-correlation estimation for multi-fidelity surrogate Cokriging models. Adv Appl Math Sci 7(2):181–201

    MathSciNet  MATH  Google Scholar 

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Acknowledgements

This work was supported by Ministry of Higher Education, Research and Innovation (France) and SAFRAN Tech. This work was also performed using HPC resources from the “Mesocentre” computing center of CentraleSupélec and École normale supérieure Paris-Saclay supported by CNRS and Région Île-de-France (http://mesocentre.centralesupelec.fr/).

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Authors and Affiliations

  1. LMT/ENS Paris-Saclay/CNRS/Université Paris-Saclay, 61 Avenue du Président Wilson, 94235, Cachan, France

    Stéphane Nachar, Pierre-Alain Boucard & David Néron

  2. SAFRAN Tech, CS 80112, 78772, Magny-les-Hameaux, France

    Felipe Bordeu

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  1. Stéphane Nachar
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  2. Pierre-Alain Boucard
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  4. Felipe Bordeu
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Correspondence to David Néron.

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Nachar, S., Boucard, PA., Néron, D. et al. Coupling multi-fidelity kriging and model-order reduction for the construction of virtual charts. Comput Mech 64, 1685–1697 (2019). https://doi.org/10.1007/s00466-019-01745-9

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