Special section: Methods for Uncertainty Characterizations in Existing Models Through Uncertainly Quantification or Calibration

Quantification of Model Uncertainty: Calibration, Model Discrepancy, and Identifiability

[+] Author and Article Information
Paul D. Arendt

 Department of Mechanical Engineering, Northwestern University, 2145 Sheridan Road Room B214, Evanston, IL, 60208paularendt2012@u.northwestern.edu

Daniel W. Apley

 Department of Industrial Engineering and Management Sciences, Northwestern University, 2145 Sheridan Road Room C150, Evanston, IL, 60208apley@northwestern.edu

Wei Chen1

 Department of Mechanical Engineering, Northwestern University, 2145 Sheridan Road Room A216, Evanston, IL, 60208weichen@northwestern.edu


Corresponding author Wilson-Cook Professor in Engineering Design.

J. Mech. Des 134(10), 100908 (Sep 28, 2012) (12 pages) doi:10.1115/1.4007390 History: Received August 27, 2011; Revised July 03, 2012; Published September 21, 2012; Online September 28, 2012

To use predictive models in engineering design of physical systems, one should first quantify the model uncertainty via model updating techniques employing both simulation and experimental data. While calibration is often used to tune unknown calibration parameters of a computer model, the addition of a discrepancy function has been used to capture model discrepancy due to underlying missing physics, numerical approximations, and other inaccuracies of the computer model that would exist even if all calibration parameters are known. One of the main challenges in model updating is the difficulty in distinguishing between the effects of calibration parameters versus model discrepancy. We illustrate this identifiability problem with several examples, explain the mechanisms behind it, and attempt to shed light on when a system may or may not be identifiable. In some instances, identifiability is achievable under mild assumptions, whereas in other instances, it is virtually impossible. In a companion paper, we demonstrate that using multiple responses, each of which depends on a common set of calibration parameters, can substantially enhance identifiability.

Copyright © 2012 by American Society of Mechanical Engineers
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Figure 1

Flowchart of model updating, model validation, and model refinement

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Figure 2

Example of a response considering a calibration parameter (a) without and (b) with the need for a discrepancy function

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Figure 3

Sources of uncertainty with calibration parameters and a discrepancy function

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Figure 4

Depiction of (a) prior distribution for a GP model with constant mean 0 and constant variance; and (b) and (c) posterior distributions for the same GP model after collecting response observations (bullets). The solid black lines are the predicted mean from Eq. 5, and the shaded regions are 95% prediction intervals.

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Figure 5

Flowchart of the modular Bayesian approach

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Figure 6

Schematic of the simply supported beam

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Figure 7

Material stress–strain curves for (a) the computer model and (b) the “physical experiments.” E is Young’s modulus and the calibration parameter. σy is the yield stress (225 MPa).

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Figure 8

The posterior distributions for (a) the experimental response, (b) the discrepancy function, and (c) the calibration parameter showing a lack of identifiability

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Figure 9

The computer response ym (x,θ), the experimental response ye (x) (bullets indicate experimental data), and the estimated discrepancy function δ̂(x,θ)=ye(x)-ym(x,θ) for (a) θ = 150 GPa and (b) θ = 250 GPa

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Figure 10

Posterior distributions for the discrepancy function and the calibration parameter for the three prior distributions in Table 2

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Figure 11

Computer model and experimental response for Eqs. 11,12 with (a) θ = 1.0 and (c) θ = 3.0. Corresponding estimated discrepancy function δ̂(x,θ) for (b) θ = 1.0 and (d) θ = 3.0.

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Figure 12

Fitted GP model of the computer model of Eq. 11 based on 168 simulation runs, ym (black dots)

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Figure 13

Posterior distributions for (a) the experimental response (black dots indicate experimentally observed response values), (b) the discrepancy function, and (c) the calibration parameter θ




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