Skip to main content
SpringerLink
Log in

An adaptive RBF-HDMR modeling approach under limited computational budget

  • RESEARCH PAPER
  • Published:
Structural and Multidisciplinary Optimization Aims and scope Submit manuscript

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.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8

Similar content being viewed by others

References

  • Acar E, Rais-Rohani M (2009) Ensemble of metamodels with optimized weight factors. Struct Multidiscip Optim 37(3):279–294

    Article  Google Scholar 

  • Andrews D W, Whang Y J (1990) Additive interactive regression models: circumvention of the curse of dimensionality. Econometric Theory 6(4):466–479

    Article  MathSciNet  Google Scholar 

  • Breiman L, Friedman J, Stone CJ, Olshen RA (1984) Classification and regression trees. CRC press, Boca Raton

    MATH  Google Scholar 

  • 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):1209–1229

    Article  Google Scholar 

  • 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– 407

    Google Scholar 

  • Chowdhury R, Rao B (2009) Hybrid high dimensional model representation for reliability analysis. Comput Methods Appl Mech Eng 198(5):753–765

    Article  MATH  Google Scholar 

  • 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):1948–1974

  • 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):683–696

  • Fang H, Horstemeyer M F (2006) Global response approximation with radial basis functions. Eng Optim 38(4):407–424

    Article  MathSciNet  Google Scholar 

  • Forrester A, Sobester A, Keane A (2008) Engineering design via surrogate modelling: a practical guide. Wiley, Hoboken

    Book  Google Scholar 

  • Friedman J H, Stuetzle W (1981) Projection pursuit regression. J Am Stat Assoc 76(376):817–823

    Article  MathSciNet  Google Scholar 

  • Goel T, Haftka R T, Shyy W, Queipo N V (2007) Ensemble of surrogates. Struct Multidiscip Optim 33(3):199–216

    Article  Google Scholar 

  • Hardy R L (1971) Multiquadric equations of topography and other irregular surfaces. J Geophys Res 76 (8):1905–1915

    Article  Google Scholar 

  • 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):643–667

    Article  Google Scholar 

  • Johnson M E, Moore L M, Ylvisaker D (1990) Minimax and maximin distance designs. J Stat Plan Inference 26(2):131–148

    Article  MathSciNet  Google Scholar 

  • Li E, Wang H, Li G (2012) High dimensional model representation (HDMR) coupled intelligent sampling strategy for nonlinear problems. Comput Phys Commun 183(9):1947–1955

    Article  MathSciNet  Google Scholar 

  • Li G, Rosenthal C, Rabitz H (2001a) High dimensional model representations. J Phys Chem A 105 (33):7765–7777

  • 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):1–30

  • Li G, Wang S W, Rabitz H (2002) Practical approaches to construct RS-HDMR component functions. J Phys Chem A 106(37):8721–8733

    Article  Google Scholar 

  • 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):2474–2485

    Article  Google Scholar 

  • 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):1207–1232

    Article  MathSciNet  MATH  Google Scholar 

  • Liu H, Xu S, Wang X (2015) Sequential sampling designs based on space reduction. Eng Optim 47 (7):867–884

    Article  Google Scholar 

  • 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):011–404

  • 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–V02CT45A019

  • 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):3117–3133

  • Liu H, Ong Y S, Cai J (2017a) An adaptive sampling approach for kriging metamodeling by maximizing expected prediction error. Comput Chem Eng 106:171–182

  • 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):011–404

  • Liu Y, Hussaini M Y, Ökten G (2016d) Accurate construction of high dimensional model representation with applications to uncertainty quantification. Reliab Eng Syst Saf 152:281–295

  • 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):243–255

    Article  MathSciNet  MATH  Google Scholar 

  • Mueller L, Alsalihi Z, Verstraete T (2013) Multidisciplinary optimization of a turbocharger radial turbine. J Turbomach 135(2):021–022

    Google Scholar 

  • Rabitz H, Aliṡ ÖF (1999) General foundations of high-dimensional model representations. J Math Chem 25(2):197–233

    Article  MathSciNet  MATH  Google Scholar 

  • Rabitz H, Aliṡ ÖF, Shorter J, Shim K (1999) Efficient input-output model representations. Comput Phys Commun 117(1-2):11–20

    Article  MATH  Google Scholar 

  • Razavi S, Tolson B A, Burn D H (2012) Review of surrogate modeling in water resources. Water Resour Res 48(7):1–32

    Article  Google Scholar 

  • Rippa S (1999) An algorithm for selecting a good value for the parameter c in radial basis function interpolation. Adv Comput Math 11(2):193–210

    Article  MathSciNet  MATH  Google Scholar 

  • Shan S, Wang G G (2010a) Metamodeling for high dimensional simulation-based design problems. J Mech Des 132(5):051–009

  • 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):219–241

  • Shan S, Wang G G (2011) Turning black-box functions into white functions. J Mech Des 133(3):031–003

    Article  Google Scholar 

  • Sobol I M (1993) Sensitivity estimates for nonlinear mathematical models. Math Model Comput Exper 1 (4):407–414

    MathSciNet  MATH  Google Scholar 

  • Sobol I M (2003) Theorems and examples on high dimensional model representation. Reliab Eng Syst Saf 79 (2):187–193

    Article  MathSciNet  Google Scholar 

  • Tang L, Wang H, Li G (2013) Advanced high strength steel springback optimization by projection-based heuristic global search algorithm. Mater Des 43:426–437

    Article  Google Scholar 

  • 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):865–883

    MathSciNet  MATH  Google Scholar 

  • Ulaganathan S, Couckuyt I, Dhaene T, Degroote J, Laermans E (2016) High dimensional kriging metamodelling utilising gradient information. Appl Math Model 40(9):5256–5270

    Article  Google Scholar 

  • 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):439–457

    Article  Google Scholar 

  • Wang G G, Shan S (2007) Review of metamodeling techniques in support of engineering design optimization. J Mech Des 129(4):370–380

    Article  Google Scholar 

  • 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):4707–4716

    Article  Google Scholar 

  • 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– 009

    Article  Google Scholar 

  • Yang Q, Xue D (2015) Comparative study on influencing factors in adaptive metamodeling. Eng Comput 31(3):561–577

    Article  Google Scholar 

Download references

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.

Author information

Authors and Affiliations

  1. Rolls-Royce@NTU Corporate Laboratory, Nanyang Technological University, Singapore, 637460, Singapore

    Haitao Liu

  2. School of Computer Science and Engineering, Nanyang Technological University, Singapore, 639798, Singapore

    Jaime-Rubio Hervas, Yew-Soon Ong & Jianfei Cai

  3. Data Science and Artificial Intelligence Research Center, Nanyang Technological University, Singapore, 639798, Singapore

    Yew-Soon Ong

  4. Applied Technology Group, Rolls-Royce Singapore, 6 Seletar Aerospace Rise, Singapore, 797575, Singapore

    Yi Wang

Authors
  1. Haitao Liu
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Jaime-Rubio Hervas
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Yew-Soon Ong
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Jianfei Cai
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. Yi Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Haitao Liu.

Appendix

Appendix

Tables 5 and 6 offer the expressions of the employed five benchmark functions and four engineering examples, respectively.

Table 5 Five benchmark functions
Full size table
Table 6 Four engineering examples
Full size table

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Liu, H., Hervas, JR., Ong, YS. et al. An adaptive RBF-HDMR modeling approach under limited computational budget. Struct Multidisc Optim 57, 1233–1250 (2018). https://doi.org/10.1007/s00158-017-1807-0

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00158-017-1807-0

Keywords

Search

Navigation