Abstract
Surrogate models or metamodels are commonly used to exploit expensive computational simulations within a design optimization framework. The application of multi-fidelity surrogate modeling approaches has recently been gaining ground due to the potential for further reductions in simulation effort over single fidelity approaches. However, given a black box problem when exactly should a designer select a multi-fidelity approach over a single fidelity approach and vice versa? Using a series of analytical test functions and engineering design examples from the literature, the following paper illustrates the potential pitfalls of choosing one technique over the other without a careful consideration of the optimization problem at hand. These examples are then used to define and validate a set of guidelines for the creation of a multi-fidelity Kriging model. The resulting guidelines state that the different fidelity functions should be well correlated, that the amount of low fidelity data in the model should be greater than the amount of high fidelity data and that more than 10 % and less than 80 % of the total simulation budget should be spent on low fidelity simulations in order for the resulting multi-fidelity model to perform better than the equivalent costing high fidelity model.
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Notes
The sampling plan data for this calculation has been kindly provided by Brooks, Forrester, Keane and Shahpar
References
Bettebghor D, Blondeau C, Toal DJ, Eres H (2013) Bi-objective optimization of pylon-engine-nacelle assembly: Weight vs. tip clearnce criterion. Struct Multidiscip Optim 48(3):637–652
Brooks C, Forrester A, Keane A, Shahpar S (2011) Multi-fidelity design optimisation of a transonic compressor rotor. In: 9th European Turbomachinery Conference, Istanbul, Turkey, 21st-25th March
Forrester A (2010) Black-box calibration for complex systems simulation. Phil Trans R Soc A 368(1924). doi:10.1098/rsta.2010.0051
Forrester A, Keane A (2009) Recent advances in surrogate-based optimization. Prog Aerosp Sci 45(1-3):50–79. doi:10.1016/j.paerosci.2008.11.001
Forrester A, Sóbester A, Keane A (2007) Multi-fidelity optimization via surrogate modelling. Proc R Soc A 463(2088):3251–3269. doi:10.1098/rspa.2007.1900
Forrester A, Sóbester A, Keane A (2008) Engineering Design via Surrogate Modelling. Wiley-Blackwell
Ghoreyshi M, Badcock K, Woodgate M (2009) Accelerating the numerical generation of aerodynamic models for flight simulation. J Aircr 46(3):972–980. doi:10.2514/1.39626
Goldberg D (1989) Genetic algorithms in search, optimization & machine learning. Addison-Wesley
Han Z, Görtz S (2012) Hierarchical kriging model for variable-fidelity surrogate modeling. AIAA J 50(9):1885–1896
Han Z, Görtz S, Zimmermann R (2013) Improving variable-fidelity surrogate modeling via gradient-enhanced kriging and a generalised hybrid bridge function. Aerosp Sci Technol 25(1):177–189
Han Z, Zimmermann R, Görtz S (2012) Alternative cokriging model for variable-fidelity surrogate modeling. AIAA J 50(5):1205–1210
Jones D (2001) A taxonomy of global optimization methods based on response surfaces. J Glob Optim 21(4):345–383. doi:10.1023/A:1012771025575
Jones D, Schonlau M, Welch W (1998) Efficient global optimization of expensive black-box functions. J Glob Optim 13(4):455–492. doi:10.1023/A:1008306431147
Kennedy M, O’Hagan A (2000) Predicting the output from a complex computer code when fast approximations are available. Biometrika 87(1):1–13. doi:10.1093/biomet/87.1.1
Kirkpatrick S, Gelatt C, Vecchi M (1983) Optimization by simulated annealing. Science 220:671–680
Krige D (1951) A statistical approach to some basic mine valuation problems on the witwatersrand. J Chem Metallurigical Min Eng Soc South Affrica 52(6):119–139. doi:10.2307/3006914
Kuya Y, Takeda K, Zhang X, Forrester A (2011) Multifidelity surrogate modeling of experimental and computational aerodynamic data sets. AIAA J 49(2):289–298. doi:10.2514/1.53410
Lapworth B, Shahpar S (2004) Design of gas turbine engines using cfd. In: ECCOMAS
Queipo N, Haftka R, Shyy W, Goel T, Vaidyanathan R, Tucker P (2005) Surrogate-based analysis and optimization. Prog Aerosp Sci 41:1–28. doi:10.1016/j.paerosci.2005.02.001
Reid L, Moore R (1978) Design and Overall Performance of Four Highly-Loaded, High-Speed Inlet Stages for an Advanced, High-Pressure-Ratio Core Compressor. NASA TP-1337
Sacks J, Welch W, Mitchell T, Wynn H (1989) Design and analysis of computer experiments. Stat Sci 4(4):409–435. doi:10.2307/2245858
Simpson T, Peplinski J, Kock P, Allen J (2001) Metamodels for computer-based engineering design: Survey and recommendations. Eng Comput 17(2):129–150. doi:10.1007/PL00007198
Toal D, Bressloff N, Keane A, Holden C (2011) The development of a hybridized particle swarm for kriging hyperparameter tuning. Eng Optim 43(6):675–699. doi:10.1080/0305215X.2010.508524
Toal D, Keane A (2011) Efficient multi-point aerodynamic design optimization via co-kriging. J Aircr 48(5):1685–1695. doi:10.2514/1.C031342
Wankhede M, Bressloff N, Keane A (2011) Combustor design optimisation using co-kriging of steady and unsteady turbulent combustion. In: Proceedings of ASME Turbo Expo 2011
Yamazaki W, Mavriplis D (2013) derivative-enhanced variable fidelity surrogate modeling for aerodynamic functions. AIAA J 51:126–137
Acknowledgments
The research leading to these results has received funding from the European Community’s Seventh Framework Programme (FP7/2007-2013) under grant agreement no. 234344 (www.crescendo-fp7.eu).
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Toal, D.J.J. Some considerations regarding the use of multi-fidelity Kriging in the construction of surrogate models. Struct Multidisc Optim 51, 1223–1245 (2015). https://doi.org/10.1007/s00158-014-1209-5
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DOI: https://doi.org/10.1007/s00158-014-1209-5