An adaptive RBF-HDMR modeling approach under limited computational budget
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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 processNotes
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
- Acar E, Rais-Rohani M (2009) Ensemble of metamodels with optimized weight factors. Struct Multidiscip Optim 37(3):279–294CrossRefGoogle Scholar
- Andrews D W, Whang Y J (1990) Additive interactive regression models: circumvention of the curse of dimensionality. Econometric Theory 6(4):466–479MathSciNetCrossRefGoogle Scholar
- Breiman L, Friedman J, Stone CJ, Olshen RA (1984) Classification and regression trees. CRC press, Boca RatonMATHGoogle 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):1209–1229CrossRefGoogle 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– 407Google Scholar
- Chowdhury R, Rao B (2009) Hybrid high dimensional model representation for reliability analysis. Comput Methods Appl Mech Eng 198(5):753–765CrossRefMATHGoogle 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):1948–1974Google Scholar
- 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
- Fang H, Horstemeyer M F (2006) Global response approximation with radial basis functions. Eng Optim 38(4):407–424MathSciNetCrossRefGoogle Scholar
- Forrester A, Sobester A, Keane A (2008) Engineering design via surrogate modelling: a practical guide. Wiley, HobokenCrossRefGoogle Scholar
- Friedman J H, Stuetzle W (1981) Projection pursuit regression. J Am Stat Assoc 76(376):817–823MathSciNetCrossRefGoogle Scholar
- Goel T, Haftka R T, Shyy W, Queipo N V (2007) Ensemble of surrogates. Struct Multidiscip Optim 33(3):199–216CrossRefGoogle Scholar
- Hardy R L (1971) Multiquadric equations of topography and other irregular surfaces. J Geophys Res 76 (8):1905–1915CrossRefGoogle 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):643–667CrossRefGoogle Scholar
- Johnson M E, Moore L M, Ylvisaker D (1990) Minimax and maximin distance designs. J Stat Plan Inference 26(2):131–148MathSciNetCrossRefGoogle 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):1947–1955MathSciNetCrossRefGoogle Scholar
- Li G, Rosenthal C, Rabitz H (2001a) High dimensional model representations. J Phys Chem A 105 (33):7765–7777Google Scholar
- 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
- 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
- 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
- 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
- Liu H, Xu S, Wang X (2015) Sequential sampling designs based on space reduction. Eng Optim 47 (7):867–884CrossRefGoogle 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):011–404Google Scholar
- 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
- 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
- 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
- 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
- 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
- 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
- Mueller L, Alsalihi Z, Verstraete T (2013) Multidisciplinary optimization of a turbocharger radial turbine. J Turbomach 135(2):021–022Google Scholar
- Rabitz H, Aliṡ ÖF (1999) General foundations of high-dimensional model representations. J Math Chem 25(2):197–233MathSciNetCrossRefMATHGoogle Scholar
- Rabitz H, Aliṡ ÖF, Shorter J, Shim K (1999) Efficient input-output model representations. Comput Phys Commun 117(1-2):11–20CrossRefMATHGoogle Scholar
- Razavi S, Tolson B A, Burn D H (2012) Review of surrogate modeling in water resources. Water Resour Res 48(7):1–32CrossRefGoogle 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):193–210MathSciNetCrossRefMATHGoogle Scholar
- Shan S, Wang G G (2010a) Metamodeling for high dimensional simulation-based design problems. J Mech Des 132(5):051–009Google Scholar
- 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
- Shan S, Wang G G (2011) Turning black-box functions into white functions. J Mech Des 133(3):031–003CrossRefGoogle Scholar
- Sobol I M (1993) Sensitivity estimates for nonlinear mathematical models. Math Model Comput Exper 1 (4):407–414MathSciNetMATHGoogle Scholar
- Sobol I M (2003) Theorems and examples on high dimensional model representation. Reliab Eng Syst Saf 79 (2):187–193MathSciNetCrossRefGoogle Scholar
- Tang L, Wang H, Li G (2013) Advanced high strength steel springback optimization by projection-based heuristic global search algorithm. Mater Des 43:426–437CrossRefGoogle 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):865–883MathSciNetMATHGoogle Scholar
- Ulaganathan S, Couckuyt I, Dhaene T, Degroote J, Laermans E (2016) High dimensional kriging metamodelling utilising gradient information. Appl Math Model 40(9):5256–5270CrossRefGoogle 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):439–457CrossRefGoogle Scholar
- Wang G G, Shan S (2007) Review of metamodeling techniques in support of engineering design optimization. J Mech Des 129(4):370–380CrossRefGoogle 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):4707–4716CrossRefGoogle 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– 009CrossRefGoogle Scholar
- Yang Q, Xue D (2015) Comparative study on influencing factors in adaptive metamodeling. Eng Comput 31(3):561–577CrossRefGoogle Scholar