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Experimental Planning and Sequential Kriging EXPERIMENTAL PLANNING AND SEQUENTIAL KRIGING OPTIMIZATION USING VARIABLE FIDELITY DATA DISSERTATION Presented in Partial Fulfillment of the Requirements for the Degree Doctor of Philosophy in the Graduate School of The Ohio State University By Deng Huang, B.S., M.S. ***** The Ohio State University 2005 Dissertation Committee: Approved By Dr. R. Allen Miller, Co-Adviser ________________________ Co-Adviser Dr. Theodore T. Allen , Co-Advisor ________________________ Co-Adviser Dr. William I. Notz Industrial and Systems Engineering Graduate Program Dr. Clark A. Mount-Campbell ABSTRACT Engineers in many industries routinely need to improve the product or process designs using data from the field, lab experiments, and computer experiments. Historically, designers have performed a calibration exercise to “fix” the lab system or computer model and then used an analysis method or optimization procedure that ignores the fact that systematic differences between products in the field and other environments necessarily exist. A new line of research is not based on the assumption that calibration is perfect and seeks to develop experimental planning and optimization schemes using data form multiple experimental sources. We use the term "fidelity" to refer to the extent to which a surrogate experimental system can reproduce results of the system of interest. For experimental planning, we present perhaps the first optimal designs for variable fidelity experimentation, using an extension of the Expected Integrated Mean Squared Error (EIMSE) criterion, where the Generalized Least Squares (GLS) method was used to generate the predictions. Numerical tests are used to compare the method performance with alternatives and to investigate the robustness to incorporated assumptions. The method is applied to automotive engine valve heat treatment process design in which real world data were mixed with data from two types of computer simulations. ii Sequential Kriging Optimization (SKO) is a method developed in recent years for solving expensive black-box problems in areas such as large-scale circuit board design and manufacturing process improvement. We propose an extension of the SKO method, named Multiple Fidelity Sequential Kriging Optimization (MFSKO), where surrogate systems are exploited to reduce the total evaluation cost. As a pre- step to MFSKO, we extended SKO to address stochastic black-box systems. In the empirical studies using numerical test functions, SKO compared favorably with alternatives in terms of consistency in finding global optima and efficiency as measured by number of evaluations. Also, in the presence of noise, the new expected improvement function for infill sample selection appears to achieve the desired balance between the need for global and local searches. In the proposed MFSKO method, data on all experimental systems are integrated to build a kriging meta-model that provides a global prediction of the system of interest and a measure of prediction uncertainty. The location and fidelity level of the next evaluation are selected by maximizing an augmented expected improvement function, which is connected with the evaluation costs. The proposed method was applied to test functions from the literature and metal-forming process design problems via Finite Element simulations. The method manifests sensible search patterns, robust performance, and appreciable reduction in total evaluation cost as compared to the original method. iii For a more peaceful world –– so that hopefully we will not have to invade another country to find out whether it has Weapons of Mass Destruction. iv ACKNOWLEDGMENTS I like to express my sincere gratitude to my advisors, Dr. R. Allen Miller and Dr. Theodore T. Allen, for their inspiration, encouragement and support which made this thesis possible, and for their patience in correcting both my scientific and stylistic errors. I also appreciate Dr. Wei-Tsu Wu of Scientific Forming Technologies Corporation for his role of mentor in the area of industrial practice. My appreciation also goes to committee members Dr. William I. Notz and Dr. Clark A. Mount- Campbell for their valuable comments and suggestions. An incomplete list of those who contributed to my work would include Dr. Thomas Santner, Dr. Emily Patterson, Dr. Mikhail Bernshteyn, Ning Zheng, Dr. Navara Chantarat, Dr. Matthias Schonlau, Dr. Michael Sasena, and Ofelia Marin, who provided ideas, references, source codes, documentation, and discussions. I am also grateful to Dr. Allen’s son, Andrew, for allowing me to take away his invaluable dad during bathing time. And a thousand thanks go to my wife, Shiling Ruan, for feeding me the right food, keeping my spirits high, and sharing knowledge in statistics. This research was partially funded by Scientific Forming Technologies Corporation, Columbus, Ohio. v VITA Oct. 23,1975 ………………………………. Born - Fujian, P. R. China 1997 ………………………………….……. B. E. Materials Science and Engineering, Tsinghua University, Beijing 1999 ……………………………………….. M. S. Materials Science and Engineering The Ohio State University Columbus, Ohio 2000 ……………………………………….. M. S. Industrial and Systems Engineering The Ohio State University Columbus, Ohio 2000 – present …………………………….. Research Scientist, Scientific Forming Technologies Corp. Columbus, Ohio PUBLICATIONS Research Publications Huang, D., Allen, T. T., Notz, W. I., and Miller, R. A., “Sequential Kriging Optimization Using Multiple Fidelity Evaluations”, to be submitted to the Structural and Multidisciplinary Optimization. Huang, D., Allen, T. T., Notz, W. I., and Zheng, N., “Global Optimization of Stochastic Black-Box Systems via Sequential Kriging Meta-Models”, submitted to the Journal of Global Optimization. vi Huang, D., Allen T. T., “Design and Analysis of Variable Fidelity Experimentation Applied to Engine Valve Heat Treatment Process Design”, Journal of Royal Statistics, Series C. 54, Part 2, p. 443-463, (2005). Chantarat, N., N. Zheng, T. T. Allen, and Huang, D., “Optimal Experimental Design for Systems Involving Both Quantitative and Qualitative Factors,” Proceedings of the Winter Simulation Conference, R. D. M. Ferrin and P. Sanchez (www.wintersim.org), (2003). FIELDS OF STUDY Major Field: Industrial and Systems Engineering Minor Field: Quantitative Microeconomics, Computer Graphics vii TABLE OF CONTENTS ABSTRACT ..............................................................................................................ii ACKNOWLEDGMENTS ........................................................................................v VITA.........................................................................................................................vi LIST OF TABLES....................................................................................................x LIST OF FIGURES .................................................................................................xi Chapter 1 Introduction.............................................................................................1 Chapter 2 Design and Analysis of Variable Fidelity Experimentation Applied to Engine Valve Heat Treatment Process Design ........................................................4 Abstract ..................................................................................................................4 2.1 Introduction ......................................................................................................5 2.2 Literature Review..............................................................................................9 2.3 Modeling and Prediction .................................................................................12 2.3.1 Response Model .......................................................................................13 2.3.2 Modeling Covariance ...............................................................................15 2.3.3 Pre-Estimation of Hyper-Parameters.........................................................17 2.4 Experimental Design.......................................................................................19 2.4.1 Design Criterion .......................................................................................20 2.4.2 Assumptions about the Potential Terms ....................................................21 2.4.3 Optimization Methods ..............................................................................22 2.5 Numerical Test Examples................................................................................23 2.5.1 Example 1 ................................................................................................23 2.5.2 Example 2 ................................................................................................26 2.5.3 Example 3 ................................................................................................37 2.6 Application Study: Engine Valve Process Design............................................42 2.7 Conclusions and future work...........................................................................44 2.8 Appendix ........................................................................................................45 Chapter 3 Global Optimization of Stochastic Black-Box Systems via Sequential Kriging Optimization..............................................................................................48 Abstract ................................................................................................................48 viii 3.1 Introduction
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