Structural and Multidisciplinary Optimization

, Volume 57, Issue 3, pp 1233–1250 | Cite as

An adaptive RBF-HDMR modeling approach under limited computational budget

  • Haitao Liu
  • Jaime-Rubio Hervas
  • Yew-Soon Ong
  • Jianfei Cai
  • Yi Wang
RESEARCH PAPER
First Online:
Received:
Revised:
Accepted:
  • 137 Downloads

Abstract

The metamodel-based high-dimensional model representation (e.g., RBF-HDMR) has recently been proven to be very promising for modeling high dimensional functions. A frequently encountered scenario in practical engineering problems is the need of building accurate models under limited computational budget. In this context, the original RBF-HDMR approach may be intractable due to the independent and successive treatment of the component functions, which translates in a lack of knowledge on when the modeling process will stop and how many points (simulations) it will cost. This article proposes an adaptive and tractable RBF-HDMR (ARBF-HDMR) modeling framework. Given a total of N m a x points, it first uses N i n i points to build an initial RBF-HDMR model for capturing the characteristics of the target function f, and then keeps adaptively identifying, sampling and modeling the potential cuts with the remaining N m a x − N i n i points. For the second-order ARBF-HDMR, N i n i ∈ [2n + 2,2n 2 + 2] not only depends on the dimensionality n but also on the characteristics of f. Numerical results on nine cases with up to 30 dimensions reveal that the proposed approach provides more accurate predictions than the original RBF-HDMR with the same computational budget, and the version that uses the maximin sampling criterion and the best-model strategy is a recommended choice. Moreover, the second-order ARBF-HDMR model significantly outperforms the first-order model; however, if the computational budget is strictly limited (e.g., 2n + 1 < N m a x ≪ 2n 2 + 2), the first-order model becomes a better choice. Finally, it is noteworthy that the proposed modeling framework can work with other metamodeling techniques.

Keywords

Metamodeling Adaptive high dimensional model representation Limited computational budget Tractable process 
This is a preview of subscription content, log in to check access

Notes

Acknowledgements

The majority of this work was finished before joining the Lab. We appreciate the support from the National Research Foundation (NRF) Singapore under the Corp Lab@University Scheme for completing the research. It is also partially supported by the Data Science and Artificial Intelligence Research Center (DSAIR) and the School of Computer Science and Engineering at Nanyang Technological University.

References

  1. Acar E, Rais-Rohani M (2009) Ensemble of metamodels with optimized weight factors. Struct Multidiscip Optim 37(3):279–294CrossRefGoogle Scholar
  2. Andrews D W, Whang Y J (1990) Additive interactive regression models: circumvention of the curse of dimensionality. Econometric Theory 6(4):466–479MathSciNetCrossRefGoogle Scholar
  3. Breiman L, Friedman J, Stone CJ, Olshen RA (1984) Classification and regression trees. CRC press, Boca RatonMATHGoogle Scholar
  4. Cai X, Qiu H, Gao L, Yang P, Shao X (2016) An enhanced RBF-HDMR integrated with an adaptive sampling method for approximating high dimensional problems in engineering design. Struct Multidiscip Optim 53 (6):1209–1229CrossRefGoogle Scholar
  5. Cheng G H, Younis A, Hajikolaei K H, Wang G G (2015) Trust region based mode pursuing sampling method for global optimization of high dimensional design problems. J Mech Des 137(2):021– 407Google Scholar
  6. Chowdhury R, Rao B (2009) Hybrid high dimensional model representation for reliability analysis. Comput Methods Appl Mech Eng 198(5):753–765CrossRefMATHGoogle Scholar
  7. Crombecq K, Gorissen D, Deschrijver D, Dhaene T (2011a) A novel hybrid sequential design strategy for global surrogate modeling of computer experiments. SIAM J Sci Comput 33(4):1948–1974Google Scholar
  8. Crombecq K, Laermans E, Dhaene T (2011b) Efficient space-filling and non-collapsing sequential design strategies for simulation-based modeling. Eur J Oper Res 214(3):683–696Google Scholar
  9. Fang H, Horstemeyer M F (2006) Global response approximation with radial basis functions. Eng Optim 38(4):407–424MathSciNetCrossRefGoogle Scholar
  10. Forrester A, Sobester A, Keane A (2008) Engineering design via surrogate modelling: a practical guide. Wiley, HobokenCrossRefGoogle Scholar
  11. Friedman J H, Stuetzle W (1981) Projection pursuit regression. J Am Stat Assoc 76(376):817–823MathSciNetCrossRefGoogle Scholar
  12. Goel T, Haftka R T, Shyy W, Queipo N V (2007) Ensemble of surrogates. Struct Multidiscip Optim 33(3):199–216CrossRefGoogle Scholar
  13. Hardy R L (1971) Multiquadric equations of topography and other irregular surfaces. J Geophys Res 76 (8):1905–1915CrossRefGoogle Scholar
  14. Huang Z, Qiu H, Zhao M, Cai X, Gao L (2015) An adaptive SVR-HDMR model for approximating high dimensional problems. Eng Comput 32(3):643–667CrossRefGoogle Scholar
  15. Johnson M E, Moore L M, Ylvisaker D (1990) Minimax and maximin distance designs. J Stat Plan Inference 26(2):131–148MathSciNetCrossRefGoogle Scholar
  16. Li E, Wang H, Li G (2012) High dimensional model representation (HDMR) coupled intelligent sampling strategy for nonlinear problems. Comput Phys Commun 183(9):1947–1955MathSciNetCrossRefGoogle Scholar
  17. Li G, Rosenthal C, Rabitz H (2001a) High dimensional model representations. J Phys Chem A 105 (33):7765–7777Google Scholar
  18. Li G, Wang S W, Rosenthal C, Rabitz H (2001b) High dimensional model representations generated from low dimensional data samples. i. mp-Cut-HDMR. J Math Chem 30(1):1–30Google Scholar
  19. Li G, Wang S W, Rabitz H (2002) Practical approaches to construct RS-HDMR component functions. J Phys Chem A 106(37):8721–8733CrossRefGoogle Scholar
  20. Li G, Hu J, Wang S W, Georgopoulos P G, Schoendorf J, Rabitz H (2006) Random sampling-high dimensional model representation (RS-HDMR) and orthogonality of its different order component functions. J Phys Chem A 110(7):2474–2485CrossRefGoogle Scholar
  21. Li G, Rabitz H, Hu J, Chen Z, Ju Y (2008) Regularized random-sampling high dimensional model representation (RS-HDMR). J Math Chem 43(3):1207–1232MathSciNetCrossRefMATHGoogle Scholar
  22. Liu H, Xu S, Wang X (2015) Sequential sampling designs based on space reduction. Eng Optim 47 (7):867–884CrossRefGoogle Scholar
  23. Liu H, Xu S, Ma Y, Chen X, Wang X (2016a) An adaptive bayesian sequential sampling approach for global metamodeling. J Mech Des 138(1):011–404Google Scholar
  24. Liu H, Xu S, Wang X (2016b) Sampling strategies and metamodeling techniques for engineering design: comparison and application. In: ASME Turbo Expo 2016: Turbomachinery Technical Conference and Exposition, ASME, pp V02CT45A019–V02CT45A019Google Scholar
  25. Liu H, Xu S, Wang X, Meng J, Yang S (2016c) Optimal weighted pointwise ensemble of radial basis functions with different basis functions. AIAA J 54(10):3117–3133Google Scholar
  26. Liu H, Ong Y S, Cai J (2017a) An adaptive sampling approach for kriging metamodeling by maximizing expected prediction error. Comput Chem Eng 106:171–182Google Scholar
  27. Liu H, Wang X, Xu S (2017b) Generalized radial basis function-based high-dimensional model representation handling existing random data. J Mech Des 139(1):011–404Google Scholar
  28. Liu Y, Hussaini M Y, Ökten G (2016d) Accurate construction of high dimensional model representation with applications to uncertainty quantification. Reliab Eng Syst Saf 152:281–295Google Scholar
  29. Morris M D, Mitchell T J, Ylvisaker D (1993) Bayesian design and analysis of computer experiments: use of derivatives in surface prediction. Technometrics 35(3):243–255MathSciNetCrossRefMATHGoogle Scholar
  30. Mueller L, Alsalihi Z, Verstraete T (2013) Multidisciplinary optimization of a turbocharger radial turbine. J Turbomach 135(2):021–022Google Scholar
  31. Rabitz H, Aliṡ ÖF (1999) General foundations of high-dimensional model representations. J Math Chem 25(2):197–233MathSciNetCrossRefMATHGoogle Scholar
  32. Rabitz H, Aliṡ ÖF, Shorter J, Shim K (1999) Efficient input-output model representations. Comput Phys Commun 117(1-2):11–20CrossRefMATHGoogle Scholar
  33. Razavi S, Tolson B A, Burn D H (2012) Review of surrogate modeling in water resources. Water Resour Res 48(7):1–32CrossRefGoogle Scholar
  34. Rippa S (1999) An algorithm for selecting a good value for the parameter c in radial basis function interpolation. Adv Comput Math 11(2):193–210MathSciNetCrossRefMATHGoogle Scholar
  35. Shan S, Wang G G (2010a) Metamodeling for high dimensional simulation-based design problems. J Mech Des 132(5):051–009Google Scholar
  36. Shan S, Wang G G (2010b) Survey of modeling and optimization strategies to solve high-dimensional design problems with computationally-expensive black-box functions. Struct Multidiscip Optim 41(2):219–241Google Scholar
  37. Shan S, Wang G G (2011) Turning black-box functions into white functions. J Mech Des 133(3):031–003CrossRefGoogle Scholar
  38. Sobol I M (1993) Sensitivity estimates for nonlinear mathematical models. Math Model Comput Exper 1 (4):407–414MathSciNetMATHGoogle Scholar
  39. Sobol I M (2003) Theorems and examples on high dimensional model representation. Reliab Eng Syst Saf 79 (2):187–193MathSciNetCrossRefGoogle Scholar
  40. Tang L, Wang H, Li G (2013) Advanced high strength steel springback optimization by projection-based heuristic global search algorithm. Mater Des 43:426–437CrossRefGoogle Scholar
  41. Tunga M A, Demiralp M (2005) A factorized high dimensional model representation on the nodes of a finite hyperprismatic regular grid. Appl Math Comput 164(3):865–883MathSciNetMATHGoogle Scholar
  42. Ulaganathan S, Couckuyt I, Dhaene T, Degroote J, Laermans E (2016) High dimensional kriging metamodelling utilising gradient information. Appl Math Model 40(9):5256–5270CrossRefGoogle Scholar
  43. Viana F A, Haftka R T, Steffen V (2009) Multiple surrogates: how cross-validation errors can help us to obtain the best predictor. Struct Multidiscip Optim 39(4):439–457CrossRefGoogle Scholar
  44. Wang G G, Shan S (2007) Review of metamodeling techniques in support of engineering design optimization. J Mech Des 129(4):370–380CrossRefGoogle Scholar
  45. Wang S W, Georgopoulos P G, Li G, Rabitz H (2003) Random sampling- high dimensional model representation (RS-HDMR) with nonuniformly distributed variables: Application to an integrated multimedia/multipathway exposure and dose model for trichloroethylene. J Phys Chem A 107(23):4707–4716CrossRefGoogle Scholar
  46. Xu S, Liu H, Wang X, Jiang X (2014) A robust error-pursuing sequential sampling approach for global metamodeling based on voronoi diagram and cross validation. J Mech Des 136(7):071– 009CrossRefGoogle Scholar
  47. Yang Q, Xue D (2015) Comparative study on influencing factors in adaptive metamodeling. Eng Comput 31(3):561–577CrossRefGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany 2017

Authors and Affiliations

  • Haitao Liu
    • 1
  • Jaime-Rubio Hervas
    • 2
  • Yew-Soon Ong
    • 2
    • 3
  • Jianfei Cai
    • 2
  • Yi Wang
    • 4
  1. 1.Rolls-Royce@NTU Corporate LaboratoryNanyang Technological UniversitySingaporeSingapore
  2. 2.School of Computer Science and EngineeringNanyang Technological UniversitySingaporeSingapore
  3. 3.Data Science and Artificial Intelligence Research CenterNanyang Technological UniversitySingaporeSingapore
  4. 4.Applied Technology GroupRolls-Royce SingaporeSingaporeSingapore

Personalised recommendations