Hierarchical Kriging Model for Variable-Fidelity Surrogate Modeling

The efficiency of building a surrogate model for the output of a computer code can be dramatically improved via variable-fidelity surrogate modeling techniques. In this article, a hierarchical kriging model is proposed and used for variable-fidelity surrogate modeling problems. Here, hierarchical kriging refers to a surrogate model of a highfidelity function that uses a kriging model of a sampled lower-fidelity function as a model trend. As a consequence, the variation in the lower-fidelity data is mapped to the high-fidelity data, and a more accurate surrogate model for the high-fidelity function is obtained. A self-contained derivation of the hierarchical kriging model is presented. The proposed method is demonstrated with an analytical example and used for modeling the aerodynamic data of an RAE 2822 airfoil and an industrial transport aircraft configuration. The numerical examples show that it is efficient, accurate, and robust. It is also observed that hierarchical kriging provides a more reasonable mean-squared-error estimation than traditional cokriging. It can be applied to the efficient aerodynamic analysis and shape optimization of aircraft or any other research areas where computer codes of varying fidelity are in use.

[1]  D. Ginsbourger,et al.  Kriging is well-suited to parallelize optimization , 2010 .

[2]  Kenji Takeda,et al.  Multifidelity surrogate modeling of experimental and computational aerodynamic data sets , 2011 .

[3]  Timothy W. Simpson,et al.  Design and Analysis of Computer Experiments in Multidisciplinary Design Optimization: A Review of How Far We Have Come - Or Not , 2008 .

[4]  Stefan Görtz,et al.  Alternative Cokriging Method for Variable-Fidelity Surrogate Modeling , 2012 .

[5]  A. Jameson.  Optimum aerodynamic design using CFD and control theory , 1995 .

[6]  Wolfgang Ponweiser,et al.  Clustered multiple generalized expected improvement: A novel infill sampling criterion for surrogate models , 2008, 2008 IEEE Congress on Evolutionary Computation (IEEE World Congress on Computational Intelligence).

[7]  P. A. Newman,et al.  Optimization with variable-fidelity models applied to wing design , 1999 .

[8]  D. Krige.  A statistical approach to some basic mine valuation problems on the Witwatersrand, by D.G. Krige, published in the Journal, December 1951 : introduction by the author , 1951 .

[9]  Alexander I. J. Forrester,et al.  Multi-fidelity optimization via surrogate modelling , 2007, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences.

[10]  Raphael T. Haftka,et al.  Surrogate-based Analysis and Optimization , 2005 .

[11]  Stefan Görtz,et al.  Improving variable-fidelity surrogate modeling via gradient-enhanced kriging and a generalized hybrid bridge function , 2013 .

[12]  Scott L. Lawrence,et al.  Generation of Aerodynamic Data using a Design of Experiment and Data Fusion Approach , 2005 .

[13]  John E. Renaud,et al.  Variable Fidelity Optimization Using a Kriging Based Scaling Function , 2004 .

[14]  G. Gary Wang,et al.  Survey of Modeling and Optimization Strategies for High-Dimensional Design Problems , 2008 .

[15]  Søren Nymand Lophaven,et al.  DACE - A Matlab Kriging Toolbox , 2002 .

[16]  N. M. Alexandrov,et al.  A trust-region framework for managing the use of approximation models in optimization , 1997 .

[17]  Ilan Kroo,et al.  Multi-fidelity Design Optimization of Low-boom Supersonic Business Jets , 2004 .

[18]  Jacques Rivoirard.  On the Structural Link Between Variables in Kriging with External Drift , 2002 .

[19]  Andy J. Keane,et al.  Recent advances in surrogate-based optimization , 2009 .

[20]  Raphael T. Haftka,et al.  Sensitivity-based scaling for approximating. Structural response , 1993 .

[21]  C. Lemieux.  Monte Carlo and Quasi-Monte Carlo Sampling , 2009 .

[22]  G. Gary Wang,et al.  Review of Metamodeling Techniques in Support of Engineering Design Optimization , 2007 .

[23]  Ilan Kroo,et al.  Two-Level Multifidelity Design Optimization Studies for Supersonic Jets , 2005 .

[24]  T. Gerhold,et al.  On the Validation of the DLR-TAU Code , 1999 .

[25]  Søren Nymand Lophaven,et al.  DACE - A Matlab Kriging Toolbox, Version 2.0 , 2002 .

[26]  A. O'Hagan,et al.  Predicting the output from a complex computer code when fast approximations are available , 2000 .

[27]  Ralf Zimmermann,et al.  Simplified Cross-Correlation Estimation For Multi-Fidelity Surrogate Cokriging Models , 2010 .

[28]  L. Watson,et al.  Why Not Run the Efficient Global Optimization Algorithm with Multiple Surrogates , 2010 .

[29]  K. Deb.  An Efficient Constraint Handling Method for Genetic Algorithms , 2000 .

[30]  Volker Schulz,et al.  Efficient Response Surface Methods Based on Generic Surrogate Models , 2012, SIAM J. Sci. Comput..

[31]  Donald R. Jones,et al.  Efficient Global Optimization of Expensive Black-Box Functions , 1998, J. Glob. Optim..

[32]  G. Matheron.  Principles of geostatistics , 1963 .

[33]  W. K. Anderson,et al.  First-Order Model Management With Variable-Fidelity Physics Applied to Multi-Element Airfoil Optimization , 2000 .

[34]  Farrokh Mistree,et al.  Kriging Models for Global Approximation in Simulation-Based Multidisciplinary Design Optimization , 2001 .

[35]  David J. J. Toal,et al.  Kriging Hyperparameter Tuning Strategies , 2008 .

[36]  Mehdi Ghoreyshi,et al.  Accelerating the Numerical Generation of Aerodynamic Models for Flight Simulation , 2009 .

[37]  T. Simpson,et al.  Use of Kriging Models to Approximate Deterministic Computer Models , 2005 .

[38]  Farrokh Mistree,et al.  Statistical Approximations for Multidisciplinary Design Optimization: The Problem of Size , 1999 .

[39]  John E. Renaud,et al.  Update Strategies for Kriging Models for Use in Variable Fidelity Optimization , 2005 .

[40]  Thomas J. Santner,et al.  The Design and Analysis of Computer Experiments , 2003, Springer Series in Statistics.

[41]  Jens K. Fassbender,et al.  MEGAFLOW - Numerical Flow Simulation for Aircraft Design , 2005 .

[42]  Stefan Görtz,et al.  A Variable-Fidelity Modeling Method for Aero-Loads Prediction , 2010 .

[43]  Leifur Þ. Leifsson,et al.  Aerodynamic Design Optimization: Physics-based Surrogate Approaches for Airfoil and Wing Design , 2014 .