Theoretical investigations of the new Cokriging method for variable-fidelity surrogate modeling
- 88 Downloads
A mandatory requirement for the well-posedness of the Cokriging emulator is the positive definiteness of the associated Cokriging correlation matrix. Spatial correlations are usually modeled by positive definite correlation kernels, which are guaranteed to yield positive definite correlation matrices for mutually distinct sample points. However, in applications, low-fidelity information is often available at high-fidelity sample points and the Cokriging predictor may benefit from the additional information provided by such an inclusive sampling. We investigate the positive definiteness of the Cokriging covariance matrix in both of the aforementioned cases and derive sufficient conditions for the well-posedness of the Cokriging predictor.
The approximation quality of the Cokriging predictor is highly dependent on a number of model- and hyper-parameters. These parameters are determined by the method of maximum likelihood estimation. For standard Kriging, closed-form optima of the model parameters along hyper-parameter profile lines are known. Yet, these do not readily transfer to the setting of Cokriging, since additional parameters arise, which exhibit a mutual dependence. In previous work, this obstacle was tackled via a numerical optimization. Here, we derive closed-form optima for all Cokriging model parameters along hyper-parameter profile lines. The findings are illustrated by numerical experiments.
KeywordsCokriging Surrogate modeling Variable-fidelity methods Multifidelity methods Response surface Maximum likelihood estimation Covariance matrix
Mathematics Subject Classification 201060G15 62M20 62K20 65C20 62P30 65D15
Unable to display preview. Download preview PDF.
- 3.Fernández-Godino, M.G., Park, C., Kim, N.H., Haftka, R.T.: Review of multi-fidelity models. arXiv:1609.07196v3, 1–46 (2017)
- 5.Forrester, A.I., Bressloff, N.W., Keane, A.J.: Optimization using surrogate models and partially converged computational fluid dynamics simulations. Proceedings of the Royal Society A: Mathematical. Phys. Eng. Sci. 462(2071), 2177–2204 (2006). https://doi.org/10.1098/rspa.2006.1679CrossRefMATHGoogle Scholar
- 10.Han, Z.H., Zimmermann, R., Goretz, S.: A New Cokriging Method for Variable-Fidelity Surrogate Modeling of Aerodynamic Data. In: 48Th AIAA Aerospace Sciences Meeting including the New Horizons Forum and Aerospace Exposition, Aerospace Sciences Meetings. American Institute of Aeronautics and Astronautics (2010). https://doi.org/10.2514/6.2010-1225
- 11.Journel, A.G., Huijbregts, C.J.: Mining Geostatistics. Academic Press, London (1978)Google Scholar
- 13.Koehler, J.R., Owen, A.B.: Computer experiments. In: Ghosh, S., Rao, C.R. (eds.) Design and Analysis of Experiments, Handbook of Statistics, vol. 13, pp. 261–308. Elsevier, Amsterdam (1996)Google Scholar
- 14.Krige, D.: A statistical approach to some basic mine valuation problems on the witwatersrand. J Chemical, Metallurgical Mining Eng Soc South Africa 52(6), 119–139 (1951)Google Scholar
- 15.Lophaven, S., Nielsen, H.B., Søndergaard, J.: Dace - a MATLAB Kriging Toolbox, Version 2.0S. Technical Report IMM-TR-2002-12, Technical University of Denmark, Lyngby, Denmark (2002)Google Scholar
- 18.OpenCFD Ltd (ESI Group): OpenFOAM—the open source cfd toolbox. http://www.openfoam.com. Accessed December 5 2016 (2016)
- 25.Zimmermann, R., Han, Z.-H.: Simplified cross-correlation estimation for Multi-Fidelity surrogate cokriging models. Adv. Appl. Math. Sci. 7(2), 181–202 (2010)Google Scholar