Adaptive design of experiments for global Kriging metamodeling through cross-validation information
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RESEARCH PAPER
Adaptive design of experiments for global Kriging metamodeling through cross-validation information Aikaterini P. Kyprioti 1 & Jize Zhang 1 & Alexandros A. Taflanidis 1,2 Received: 29 August 2019 / Revised: 9 January 2020 / Accepted: 11 February 2020 # Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract This paper discusses a new sequential adaptive design of experiments (DoE) approach for global Kriging metamodeling applications. The sequential implementation is established by using the current metamodel, formulated based on the existing experiments, to guide the selection of the optimal new experiment(s). The score function, defining the DoE objective, combines two components: (1) the metamodel prediction variability, expressed through the predictive variance, and (2) the metamodel bias, approximated through the leave-one-out cross validation (LOOCV) error. The latter is used as a weighting factor to extend traditional DoE approaches that focus solely on the metamodel prediction variability. Two such approaches are considered here, adopting either the integrated mean squared error or the maximum mean squared error as the basic component of the score function. The incorporation of bias information as weighting within these well-established approaches facilitates a direct extension of their respective computational workflows, making the proposed implementation attractive from computational perspective. An efficient optimization scheme for identification of the next experiment, as well as the balancing of exploration and exploitation between the two components of the score function, are also discussed. The incorporation of LOOCV weightings is shown to be highly beneficial in a total of six analytical and engineering examples. Furthermore, these examples demonstrate that for DoE approaches which use LOOCV information as weights, it is preferable to update the predictive variance to explicitly consider the impact of the new experiment, rather than relying strictly on the current metamodel variance. Keywords Adaptive design of experiments (DoE) . Sequential DoE . Kriging . Weighted integrated mean squared error . Weighted maximum mean squared error . Leave-one-out cross-validation weights
1 Introduction Metamodels, also referenced as surrogate models, have emerged as a valuable tool for emulating the response of computationally expensive high-fidelity simulation models for a variety of engineering applications (Kennedy and O'Hagan 2001; Queipo et al. 2005; Forrester and Keane 2009). They correspond to a data driven approximation of the input/output relationship of the high-fidelity model, formulated based on an
Responsible Editor: Shapour Azarm * Alexandros A. Taflanidis [email protected] 1
Department of Civil & Environmental Engineering & Earth Sciences, University of Notre Dame, Notre Dame, IN 46556, USA
2
Department of Aerospace and Mechanical Engineering, University of Notre Dame, Notre Dame, IN 46556, USA
observation set, frequently referenced as experiments, obtained from
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