4 G.N. Absi and S. Mahadevan Higher Fidelity Lower Fidelity... esu emf Fig. 1.2 Variation of "sur and "mf with model fidelity 1.2.4 Validation Model validation is used to determine the degree of agreement of a model prediction with observed experimental data, and compare multiple models to determine which one is better supported by the data. Many methods have been proposed that include parametric uncertainty [25]. For two models (denoted byMi andMj) with prior probabilities of acceptance P(Mi) and P(Mj), the relative posterior probabilities can be computed using Bayes’ rule [26, 27]: P Mi ˇ ˇ ˇ Observation P Mjˇ ˇ ˇ Observation D P Observationˇ ˇ ˇ Mi P Observationˇ ˇ ˇ Mj P .Mi / P Mj (1.5) The likelihood ratio P Observationˇ ˇ ˇ Mi ! P Observationˇ ˇ ˇ Mj ! is referred to as “Bayes factor”, and is used as the metric to assess the data support toModel Mi relative to model Mj. If the Bayes factor is greater than 1.0, then it can be concluded that the model Mi ismore supported by the data. 1.2.5 Multi-fidelity Implementation Let N1 and N2 denote the number of simulations available for each original model G1 and G2, uniformly distributed over the problem domain. These realizations are used to build the corresponding surrogate models. Because of time and budget constraints, we assume that higher the fidelity of the model, the lower the number of simulations available, i.e.: N2 . N1. This results in a surrogate model error for G2 larger than that for G1. However, since G2 is of a higher fidelity than G1, the model form error inG1 is larger than that for G2, i.e., "mf(1) "mf(2). Figure 1.2 shows a notional diagram of how the surrogate model error and model form error might vary with the fidelity of the model. The proposed approach avoids the need to build a surrogate model for the high-fidelity simulations, and thus avoids the high surrogate model error that comes with it. It uses the available simulations of the high-fidelity model to correct the low-fidelity surrogate model, and uses the latter in the calibration process. The multi-fidelity calibration algorithm is as follows: (i) Run the Low(G1) andHigh(G2) fidelity models to obtain N1 andN2 sets of outputs, respectively. (ii) Build S1, the surrogate model replacing G1. In this step, the variance of S1 is calculated to account for the surrogate model error. (iii) Define the priors of the calibration parameters, and the discrepency between the models D2,1. (iv) Calibrate the parameters of low-fidelity model as well as the discrepency with the high-fidelity simulations.
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