Research Papers

Adaptive Designs of Experiments for Accurate Approximation of a Target Region

[+] Author and Article Information
Victor Picheny

Department of Applied Mathematics and Systems, Ecole Centrale Paris, Chatenay-Malabry 92295, Francevictor.picheny@ecp.fr

David Ginsbourger

Institute of Mathematical Statistics and Actuarial Science, University of Bern, Bern, CH-3012 Bern, Switzerlanddavid.ginsbourger@stat.unibe.ch

Olivier Roustant

 Ecole des Mines de St Etienne, Saint Etienne 42023, Franceroustant@emse.fr

Raphael T. Haftka

Department of Mechanical and Aerospace Engineering, University of Florida, Gainesville, FL 32611haftka@ufl.edu

Nam-Ho Kim

Department of Mechanical and Aerospace Engineering, University of Florida, Gainesville, FL 32611nkim@ufl.edu

J. Mech. Des 132(7), 071008 (Jun 29, 2010) (9 pages) doi:10.1115/1.4001873 History: Received February 15, 2009; Revised May 12, 2010; Published June 29, 2010; Online June 29, 2010

This paper addresses the issue of designing experiments for a metamodel that needs to be accurate for a certain level of the response value. Such a situation is common in constrained optimization and reliability analysis. Here, we propose an adaptive strategy to build designs of experiments that is based on an explicit trade-off between reduction in global uncertainty and exploration of regions of interest. A modified version of the classical integrated mean square error criterion is used that weights the prediction variance with the expected proximity to the target level of response. The method is illustrated by two simple examples. It is shown that a substantial reduction in error can be achieved in the target regions with reasonable loss of global accuracy. The method is finally applied to a reliability analysis problem; it is found that the adaptive designs significantly outperform classical space-filling designs.

Copyright © 2010 by American Society of Mechanical Engineers
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Figure 1

One-dimensional illustration of the target region. Here, T=1 and ε=0.2. The target region consists of two distinct intervals.

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Figure 2

Illustration of the weights functions. Upper graph: true function, observations, kriging mean, and confidence intervals; the target region is represented by the horizontal lines at T−ε, T, and T+ε. Lower graph: weight functions. Both weights are large where the true function is not only inside the target region but also signaling regions of high uncertainties (around x=0.65 and 0.85).

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Figure 3

Optimal design after 11 iterations. The contour lines correspond to the true function at levels T (bold line) and [T−σε,T+σε], which delimit the actual target regions. Most of the training points are chosen close to the target region. The kriging variance is very small in these regions and large in noncritical regions.

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Figure 4

Evolution of kriging target contour line (thin line) compared with actual (bold line) during the sequential process: (a) Initial, (b) after four iterations, (c) after eight iterations, and (d) final

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Figure 5

Comparison of error distribution for two 90 points DoEs: optimal DoE (top) and classical LHS (bottom). The x-axis is the difference between the true function and the threshold, the y-axis is the error. Three vertical bars are drawn at −2σε, 0, and +2σε for the target region. The error is on average smaller for the LHS design but the optimal DoE reduces substantially the error in the target region.

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Figure 6

Boxplots of errors for the 90 points LHS and optimal designs for the test points where responses are inside the domain [T−2σε,T+2σε]. Error at these points is about 2.5 smaller for the optimal designs.

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Figure 7

Optimal design with (a) uniform integration measure and (b) input distribution integration measure; (c) full-factorial designs with 16 points. Plain green line shows the limit of the failure region; input distribution is shown in (d).



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