Engineering with Computers

, Volume 32, Issue 1, pp 15–34 | Cite as

Performance study of gradient-enhanced Kriging

  • Selvakumar UlaganathanEmail author
  • Ivo Couckuyt
  • Tom Dhaene
  • Joris Degroote
  • Eric Laermans
Original Article


The use of surrogate models for approximating computationally expensive simulations has been on the rise for the last two decades. Kriging-based surrogate models are popular for approximating deterministic computer models. In this work, the performance of Kriging is investigated when gradient information is introduced for the approximation of computationally expensive black-box simulations. This approach, known as gradient-enhanced Kriging, is applied to various benchmark functions of varying dimensionality (2D-20D). As expected, results from the benchmark problems show that additional gradient information can significantly enhance the accuracy of Kriging. Gradient-enhanced Kriging provides a better approximation even when gradient information is only partially available. Further comparison between gradient-enhanced Kriging and an alternative formulation of gradient-enhanced Kriging, called indirect gradient-enhanced Kriging, highlights various advantages of directly employing gradient information, such as improved surrogate model accuracy, better conditioning of the correlation matrix, etc. Finally, gradient-enhanced Kriging is used to model 6- and 10-variable fluid–structure interaction problems from bio-mechanics to identify the arterial wall’s stiffness.


Kriging Surrogate modelling Gradient enhancement Fluid structure interaction 



This research has been funded by the Interuniversity Attraction Poles Programme BESTCOM initiated by the Belgian Science Policy Office. Additionally, this research has been supported by the Fund for Scientific Research in Flanders (FWO-Vlaanderen). Ivo Couckuyt and Joris Degroote are post-doctoral research fellows of the Research Foundation Flanders (FWO).


  1. 1.
    Brezillon J, Dwight R (2005) Discrete adjoint of the Navier–Stokes equations for aerodynamic shape optimization. In: Evolutionary and deterministic methods for design, optimisation and control with applications to industrial and societal problems (EUROGEN 2005). Munich, GermanyGoogle Scholar
  2. 2.
    Chung HS, Alonso JJ (2002) Using gradients to construct cokriging approximation models for high-dimensional design optimization problems. Problems., G 40th AIAA Aerospace Sciences Meeting and Exhibit, AIAAReno, NV, pp. 2002–0317Google Scholar
  3. 3.
    Couckuyt I, Dhaene T, Demeester P (2014) ooDACE toolbox: a flexible object-oriented kriging implementation. J Mach Learn Res 15:3183–3186zbMATHGoogle Scholar
  4. 4.
    Degroote J, Bathe KJ, Vierendeels J (2009) Performance of a new partitioned procedure versus a monolithic procedure in fluid–structure interaction. Comput Struct 87(11–12):793–801CrossRefGoogle Scholar
  5. 5.
    Degroote J, Hojjat M, Stavropoulou E, Wüchner R, Bletzinger KU (2013) Partitioned solution of an unsteady adjoint for strongly coupled fluid–structure interactions and application to parameter identification of a one-dimensional problem. Struct Multidiscip Optim 47(1):77–94zbMATHMathSciNetCrossRefGoogle Scholar
  6. 6.
    Dwight RP, Han ZH (2009) Efficient uncertainty quantification using gradient-enhanced kriging. In: 11th AIAA Non-Deterministic Approaches Conference. Palm Springs, California, USAGoogle Scholar
  7. 7.
    Forrester AI, Sóbester A, Keane AJ (2007) Multi-fidelity optimization via surrogate modelling. Proc R Soc 463:3251–3269zbMATHCrossRefGoogle Scholar
  8. 8.
    Forrester AI, Sóbester A, Keane AJ (2008) Engineering design via surrogate modelling: a practical guide, 1 edn. Wiley, New York (2008)Google Scholar
  9. 9.
    Ginsbourger D (2009) Multiples métamodéles pour l’approximation et l’optimisation de fonctions numériques multivariables. Ph.D. thesis, École des Mines de Saint-ÉtienneGoogle Scholar
  10. 10.
    Ginsbourger D, Dupuis D, Badea A, Carraro L, Roustant O (2009) A note on the choice and the estimation of kriging models for the analysis of deterministic computer experiments. Appl Stoch Models Bus Ind 25(2):115–131. Special issue: Computer experiments versus physical experimentsGoogle Scholar
  11. 11.
    Ginsbourger D, Helbert C, Carraro L (2008) Discrete mixtures of kernels for kriging-based optimization. Qual Reliabil Eng Int 24(6):681–691CrossRefGoogle Scholar
  12. 12.
    Griewank A (2000) Evaluating derivatives: principles and techniques of algorithmic differentiation. No. 19 in Frontiers in Appl Math SIAM, Philadelphia, PA (2000)Google Scholar
  13. 13.
    Husslage B, Rennen G, van Dam E, den Hertog D (2011) Space-filling latin hypercube designs for computer experiments. Optim Eng 12(4):611–630zbMATHCrossRefGoogle Scholar
  14. 14.
    Kennedy MC, O’Hagan A (2000) Predicting the output from a complex computer code when fast approximations are available. Biometrika 87(1):1–13zbMATHMathSciNetCrossRefGoogle Scholar
  15. 15.
    Killeya MRH (2004) Thinking inside the box: using derivatives to improve Bayesian black box emulation of computer simulators with application to compart mental models. Ph.D. thesis, Durham theses, Durham University, EnglandGoogle Scholar
  16. 16.
    Kleijnen J (2009) Kriging metamodeling in simulation: a review. Eur J Oper Res 192(3):707–716zbMATHMathSciNetCrossRefGoogle Scholar
  17. 17.
    Laurenceau J, Meaux M, Montagnac M, Sagaut P (2010) Comparison of gradient-based and gradient-enhanced response-surface-based optimizers. Am Inst Aeronaut Astronaut J 48(5):981–994CrossRefGoogle Scholar
  18. 18.
    Laurenceau J, Sagaut P (2008) Building efficient response surfaces of aerodynamic functions with kriging and cokriging. AIAA 46(2):498–507CrossRefGoogle Scholar
  19. 19.
    Laurent S, Cockcroft J, Van Bortel L, Boutouyrie P, Giannattasio C, Hayoz D, Pannier B, Vlachopoulos C, Wilkinson I, Struijker-Boudier H (2006) Expert consensus document on arterial stiffness: methodological issues and clinical applications. Eur Heart J 27(21):2588–2605CrossRefGoogle Scholar
  20. 20.
    Liu W (2003) Development of gradient-enhanced kriging approximations for multidisciplinary design optimisation. Ph.D. thesis, University of Notre Dame, Notre Dame, IndianaGoogle Scholar
  21. 21.
    Lockwood BA, Anitescu M (2010) Gradient-enhanced universal kriging for uncertainty propagation. Preprint ANL/MCS-P1808-1110Google Scholar
  22. 22.
    Meckesheimer M, Booker AJ, Barton RR, Simpson TW (2002) Computationally inexpensive metamodel assessment strategies. AIAA 40(10):2053–2060CrossRefGoogle Scholar
  23. 23.
    Morris MD, Mitchell TJ, Ylvisaker D (1993) Bayesian design and analysis of computer experiments: use of gradients in surface prediction. Technometrics 35(3):243–255zbMATHMathSciNetCrossRefGoogle Scholar
  24. 24.
    Näther W, Šimák J (2003) Effective observation of random processes using derivatives. Metrika Springer 58:71–84zbMATHGoogle Scholar
  25. 25.
    Rasmussen CE, Williams CKI (2006) Gaussian processes for machine learning. The MIT Press, CambridgezbMATHGoogle Scholar
  26. 26.
    Sacks J, Schiller SB, Welch WJ (1989) Designs for computer experiments. Technometrics 31(1):41–47MathSciNetCrossRefGoogle Scholar
  27. 27.
    Sacks J, Welch WJ, Mitchell TJ, Wynn HP (1989) Design and analysis of computer experiments. Stat Sci 4(4):409–423zbMATHMathSciNetCrossRefGoogle Scholar
  28. 28.
    Schneider R (2012) Feins: Finite element solver for shape optimization with adjoint equations. In: Progress in industrial mathematics at ECMI 2010 Conference, pp. 573–580Google Scholar
  29. 29.
    Shao W, Deng H, Ma Y, Wei Z (2012) Extended Gaussian kriging for computer experiments in engineering design. Eng Comput 28(2):161–178CrossRefGoogle Scholar
  30. 36.
    Šimák J (2002) On experimental designs for derivative random fields. Ph.D. thesis, TU Bergakademie Freiberg, Freiberg, GermanyGoogle Scholar
  31. 30.
    Simpson T, Poplinski J, Koch PN, Allen J (2001) Metamodels for computer-based engineering design: survey and recommendations. Eng Comput 17(2):129–150zbMATHCrossRefGoogle Scholar
  32. 31.
    Stein ML (1999) Interpolation of spatial data: some theory for Kriging. Springer, New YorkzbMATHCrossRefGoogle Scholar
  33. 32.
    Stephenson G (2010) Using derivative information in the statistical analysis of computer models. Ph.D. thesis, University of Southampton, Southampton, UKGoogle Scholar
  34. 33.
    Toal DJ, Forrester AI, Bressloff NW, Keane AJ, Holden C (2009) An adjoint for likelihood maximization. Proc R Soc A 8 465(2111):3267–3287 (2009)Google Scholar
  35. 34.
    Ulaganathan S, Couckuyt I, Ferranti F, Laermans E, Dhaene T (2014) Performance study of multi-fidelity gradient enhanced Kriging. Struct Multidiscip Optim. doi: 10.1007/s00158-014-1192-x Google Scholar
  36. 35.
    Vignon-Clementel I, Figueroa C, Jansen K, Taylor C (2010) Outflow boundary conditions for 3D simulations of non-periodic blood flow and pressure fields in deformable arteries. Comput Methods Biomech Biomed Eng 13(5):625–640CrossRefGoogle Scholar
  37. 37.
    Wang GG, Shan S (2006) Review of metamodeling techniques in support of engineering design optimization. J Mech Design 129(4):370–380CrossRefGoogle Scholar
  38. 38.
    Yamazaki W, Rumpfkeil MP, Mavriplis DJ (2010) Design optimization utilizing Gradient/Hessian enhanced surrogate model. In: 28th AIAA Applied Aerodynamics Conference, AIAA paper 2010–4363. Chicago, Illinois, USAGoogle Scholar
  39. 39.
    Ying X, JunHua X, WeiHua Z, YuLin Z (2009) Gradient-based kriging approximate model and its application research to optimization design. Sci China Technol Sci 52(4):1117–1124zbMATHCrossRefGoogle Scholar
  40. 40.
    Zhao D, Xue D (2011) A multi-surrogate approximation method for metamodeling. Eng Comput 27(2):139–153CrossRefGoogle Scholar

Copyright information

© Springer-Verlag London 2015

Authors and Affiliations

  • Selvakumar Ulaganathan
    • 1
    Email author
  • Ivo Couckuyt
    • 1
  • Tom Dhaene
    • 1
  • Joris Degroote
    • 2
  • Eric Laermans
    • 1
  1. 1.Department of Information Technology (INTEC)Ghent University-iMINDSGhentBelgium
  2. 2.Department of Flow, Heat and Combustion MechanicsGhent UniversityGhentBelgium

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