×

zbMATH — the first resource for mathematics

High dimensional Kriging metamodelling utilising gradient information. (English) Zbl 07160033
Summary: Kriging-based metamodels are popular for approximating computationally expensive black-box simulations, but suffer from an exponential growth of required training samples as the dimensionality of the problem increases. While a Gradient Enhanced Kriging metamodel with less training samples is able to approximate more accurately than a Kriging-based metamodel, it is prohibitively expensive to build for high dimensional problems. This limits the applicability of Gradient Enhanced Kriging for high dimensional metamodelling. In this work, this limitation is alleviated by coupling Gradient Enhanced Kriging with High Dimensional Model Representation. The approach, known as Gradient Enhanced Kriging based High Dimensional Model Representation, is accompanied by a highly efficient sequential sampling scheme LOLA-Voronoi and is applied to various high dimensional benchmark functions and one real-life simulation problem of varying dimensionality (10D–100D). Test results show that the combination of inexpensive gradient information and the high dimensional model representation can break or at least loosen the limitations associated with high dimensional Kriging metamodelling.

MSC:
62F15 Bayesian inference
65C05 Monte Carlo methods
62M30 Inference from spatial processes
62L05 Sequential statistical design
Software:
SUMO; ooDACE
PDF BibTeX Cite
Full Text: DOI
References:
[1] Forrester, A. I.; Sóbester, A.; Keane, A. J., Engineering Design via Surrogate Modelling: A Practical Guide (2008), Wiley
[2] Kennedy, M. C.; O’Hagan, A., Predicting the output from a complex computer code when fast approximations are available, Biometrika, 87, 1, 1-13 (2000) · Zbl 0974.62024
[3] Yamazaki, W.; Rumpfkeil, M. P.; Mavriplis, D. J., Design optimization utilizing Gradient/Hessian enhanced surrogate model, Proceedings of the 28th AIAA Applied Aerodynamics Conference, AIAA paper 2010-4363, Chicago, Illinois, USA (2010)
[4] Couckuyt, I.; Dhaene, T.; Demeester, P., ooDACE toolbox: A flexible object-oriented Kriging implementation, J. Mach. Learn. Res., 15, 3183-3186 (2014) · Zbl 1319.62001
[5] Zhao, H.; Yue, Z.; Liu, Y.; Gao, Z.; Zhang, Y., An efficient reliability method combining adaptive importance sampling and Kriging metamodel, Appl. Math. Model. (2014)
[6] Simpson, T.; Poplinski, J.; Koch, P. N.; Allen, J., Metamodels for computer-based engineering design: Survey and recommendations, Eng. Comput., 17, 2, 129-150 (2001) · Zbl 0985.68599
[7] Jin, R.; Chen, W.; Simpson, T. W., Comparative studies of metamodeling techniques under multiple modeling criteria, Struct. Multidiscip. Optim., 23, 1-13 (2000)
[8] Wang, G. G.; Shan, S., Review of metamodeling techniques in support of engineering design optimization, J. Mech. Des., 129, 4, 370-380 (2006)
[9] Sacks, J.; Welch, W. J.; Mitchell, T. J.; Wynn, H. P., Design and analysis of computer experiments, Stat. Sci., 4, 4, 409-423 (1989) · Zbl 0955.62619
[10] Morris, M. D.; Mitchell, T. J.; Ylvisaker, D., Bayesian design and analysis of computer experiments: Use of gradients in surface prediction, Technometrics, 35, 3, 243-255 (1993) · Zbl 0785.62025
[11] Chung, H.-S.; Alonso, J. J., Using gradients to construct cokriging approximation models for high-dimensional design optimization problems, Proceedings of the Problems, 40th AIAA Aerospace Sciences Meeting and Exhibit, AIAA, Reno, NV (2002)
[12] Dalbey, K. R., Efficient and Robust Gradient Enhanced Kriging Emulators, Technical Report SAND2013-7022 (2013), Sandia National Laboratories: Sandia National Laboratories Albuquerque, New Mexico 87185 and Livermore, California 94550
[13] Friedman, J.; Stuetzle, W., Projection pursuit regression, J. Am. Stat. Assoc., 76, 372, 817-823 (1981)
[14] Friedman, J., Multivariate adaptive regressive splines, Annals Stat., 19, 1, 1-67 (1991)
[16] Sobol, I., Sensitivity estimates for nonlinear mathematical models, Math. Model. Comput. Exp., 1, 4, 407-414 (1993) · Zbl 1039.65505
[17] Chowdhury, R.; Adhikari, S., High dimensional model representation for stochastic finite element analysis, Appl. Math. Model., 34, 12, 3917-3932 (2010) · Zbl 1201.65009
[18] Shan, S.; Wang, G. G., 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 (2010) · Zbl 1274.74291
[19] Rabitz, H.; Alis, O. F., General foundations of high-dimensional model representations, J. Math. Chem., 25, 197-233 (1999) · Zbl 0957.93004
[20] Rabitz, H.; Alis, O. F.; Shorter, J.; Shim, K., Efficient input-output model representations, Comput. Phys. Commun., 117, 11-20 (1999) · Zbl 1015.68219
[21] Wang, S. W.; Georgopoulos, P. G.; Li, G. Y.; Rabitz, H., 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., 107, 4707-4716 (2003)
[22] Li, G.; Hu, J.; Wang, S.-W.; Georgopoulos, P. G.; Schoendorf, J.; Rabitz, H., Random sampling-high dimensional model representation (RS-HDMR) and orthogonality of its different order component functions, J. Phys. Chem., 110, 7, 2474-2485 (2006)
[23] Shan, S.; Wang, G. G., Metamodeling for high dimensional simulation-based design problems, J. Mech. Des., 132, 5, 051009 (2010)
[24] Li, G.; Rosenthal, C.; Rabitz, H., High dimensional model representations, J. Phys. Chem., 105, 33, 7765-7777 (2001)
[25] Li, G.; Schoendorf, J.; Ho, T.-S.; Rabitz, H. A., Multicut-HDMR with an application to an ionospheric model., J. Comput. Chem., 25, 9, 1149-1156 (2004)
[26] Crombecq, K.; Gorissen, D.; Deschrijver, D.; Dhaene, T., Novel hybrid sequential design strategy for global surrogate modeling of computer experiments, SIAM J. Sci. Comput., 33, 4, 1948-1974 (2011) · Zbl 1227.62059
[27] Crombecq, K.; Laermans, E.; Dhaene, T., Efficient space-filling and non-collapsing sequential design strategies for simulation-based modeling, Eur. J. Oper. Res., 214, 3, 683-696 (2011)
[28] Kleijnen, J., Kriging metamodeling in simulation: a review, Eur. J. Oper. Res., 192, 3, 707-716 (2009) · Zbl 1157.90544
[29] Stein, M. L., Interpolation of Spatial Data: Some Theory for Kriging (1999), Springer: Springer New York · Zbl 0924.62100
[30] Rasmussen, C. E.; Williams, C. K.I., Gaussian Processes for Machine Learning (2006), The MIT Press: The MIT Press Cambridge, MA, USA · Zbl 1177.68165
[31] Ulaganathan, S.; Couckuyt, I.; Ferranti, F.; Laermans, E.; Dhaene, T., Performance study of multi-fidelity gradient enhanced Kriging, Struct. Multidiscip. Optim., 51, 5, 1017-1033 (2015)
[32] Ulaganathan, S.; Couckuyt, I.; Dhaene, T.; Degroote, J.; Laermans, E., Performance study of gradient enhanced Kriging, Eng. Comput., 1-20 (2015)
[33] Zimmermann, R., On the maximum likelihood training of gradient-enhanced spatial gaussian processes, SIAM J. Sci. Comput., 35, 6, A2554-A2574 (2013) · Zbl 1283.62201
[34] Lin, Y., An Efficient Robust Concept Exploration Method and Sequential Exploratory Experimental Design (2004), Mechanical Engineering, Georgia Institute of Technology: Mechanical Engineering, Georgia Institute of Technology Atlanda, Ph.D. thesis
[35] Jin, R.; Chen, W.; Sudjianto, A., An efficient algorithm for constructing optimal design of computer experiments, J. Stat. Plan. Inference, 134, 1, 268-287 (2005) · Zbl 1066.62075
[36] Jin, R.; Chen, W.; Sudjianto, A., On sequential sampling for global metamodeling in engineering design, Proceedings of the ASME 2002 Design Engineering Technical Conferences and Computer and Information in Engineering Conference, Montreal, Canada (2002)
[37] Sasena, M.; Parkinson, M.; Goovaerts, P.; Papalambros, P.; Reed, M., Adaptive experimental design applied to an ergonomics testing procedure, Proceedings of the ASME Design Engineering Technical Conferences, paper DETC2002/DAC-34091, Montreal, Canada (2002)
[38] Gorissen, D.; Couckuyt, I.; Demeester, P.; Dhaene, T.; Crombecq, K., A surrogate modeling and adaptive sampling toolbox for computer based design, J. Mach. Learn. Res., 11, 2051-2055 (2010)
[40] Degroote, J.; Hojjat, M.; Stavropoulou, E.; Wüchner, R.; Bletzinger, K.-U., Partitioned solution of an unsteady adjoint for strongly coupled fluid-structure interactions and application to parameter identification of a one-dimensional problem, Struct. Multidiscip. Optim., 47, 1, 77-94 (2013) · Zbl 1274.74103
[41] Degroote, J.; Bathe, K.-J.; Vierendeels, J., Performance of a new partitioned procedure versus a monolithic procedure in fluid-structure interaction, Comput. Struct., 87, 11-12, 793-801 (2009)
[42] Griewank, A.; Walther, A., Evaluating Derivatives: Principles and Techniques of Algorithmic Differentiation. Evaluating Derivatives: Principles and Techniques of Algorithmic Differentiation, Frontiers in Applied Mathematics, number 19 (2008), Society for Industrial and Applied Mathematics: Society for Industrial and Applied Mathematics Philadelphia, PA, USA · Zbl 1159.65026
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.