1 Calibration of System Parameters Under Model Uncertainty 3 f(X) ymax PDF 0 y X Fig. 1.1 Simple implementation of slice sampling 1.2.2 Uncertainty Quantification The experimental output Yobs is expressed in terms of the experimental input X, the errors and the model output as follows: Yobs C"obs DG.XC"in; / C"surr C"mf (1.3) where "in: input experimental error "mf : model form error (calculated automatically within the code) "surr: surrogate model error "obs: output experimental error In order to get accurate calibration results, these errors need to be included in the calibration [21]. The experimental measurement errors "in and "obs are represented as random variables, with known (or assumed) distributions. The surrogate model error "surr reflects the uncertainty we have regarding the replacement of the original model with a response surface model, as is shown in Eq. 1.2. As for the model form error "mf , it is calibrated along with model parameters using experimental data, following Eq. 1.3. 1.2.3 Bayesian Calibration Three approaches are available for calibration: least squares, maximum likelihood, and Bayesian calibration. This paper uses Bayesian calibration since it is the most comprehensive approach, allowing uncertainty quantification of the calibration result. The Bayesian calibration is based on Bayes’ theorem: fX.xj D/ /fX.x/:L.x/ (1.4) wherefX(xj D) is the posterior distribution of the variableXafter calibration using the data D, fX(x) is the prior distribution of X(assumed by the user), andL(x) is the likelihood function (probability of observing the data D, given a calibration parameter value). Sampling from the posterior is done using a Markov Chain Monte Carlo (MCMC) algorithm. Several algorithms are available for MCMC sampling: Metropolis-Hastings [22], Gibbs sampling [23], slice sampling [24], etc. Slice sampling is used in the numerical example to evaluate Eq. 1.4. It is based on the observation that to sample a random variable, one can sample uniformly from the region under the PDF. The simplest implementation (for a uni-variate distribution without the need to reject any points) consists of first sampling a random value y between 0 and the maximum PDF value ymax. The next step is to sample x from the slice under the PDF, as shown in Fig. 1.1. Slice sampling requires many simulations of the model used. Low-fidelity models are fast, but inaccurate. High-fidelity models are more reliable, but time-consuming. Therefore we propose correcting the low-fidelity model first with high-fidelity simulations, and using the corrected model for calibration with experimental data.
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