1Blurb:: 2(Experimental) Post-calibration calculation of model discrepancy correction 3 4Description:: 5The goal of parameter calibration is to minimize the difference between 6experimental data, \f$d(x)\f$, and model observations, \f$M(\theta, x)\f$, 7where \f$\theta\f$ are the model parameters and \f$x\f$ is a configuration 8variable, such as temperature or pressure. However, it is not uncommon that, at 9the conclusion of parameter calibration, the agreement between experimental 10data and the calibrated model is not "close enough." This is often due to 11model form or structural error. In this case, the canonical equation 12 13\f[ 14d (x) = M (\theta, x) + \varepsilon 15\f] 16 17is replaced by one that also includes model discrepancy \f$\delta(x)\f$, 18 19\f[ 20d (x) = M (\theta, x) + \delta(x) + \varepsilon. 21\f] 22 23In the Dakota implementation, the calculation of \f$\delta(x)\f$ is performed 24after the model parameters \f$\theta\f$ are calibrated. For each observable 25\f$d_i\f$, the discrepancy 26 27\f[ 28\delta_i(x_j) = d_i(x_j) - M_i(\theta^*, x_j) 29\f] 30 31is calculated for each value \f$x_j\f$ of the configuration variable, where 32\f$\theta^*\f$ is the optimal parameter value obtained during the 33calibration. For scalar responses, the model discrepancy is only a function of 34the configuration variables, and there is one discrepancy regression model for 35each observable \f$d_{i}\f$. 36This set of discrepancy 37models may be specified to be either Gaussian process or polynomial regression 38models of constant, linear, or quadratic order, and each model is fit to the 39calculated discrepancy values. See the 40\ref method-bayes_calibration-model_discrepancy-discrepancy_type 41command for more details regarding these options. 42For field responses, the model discrepancy is 43a function of the configuration variables as well as the independent field 44coordinates (such as time or space), and there is one discrepancy regression 45model for each field. 46In this case, the discrepancy models are Gaussian process models. The 47calculation of model discrepancy has not been tested for cases in which 48responses are mixed scalar and field responses. 49 50 51 52The user 53may then specify new or "prediction" configurations at which the corrected 54model \f$M(\theta^*, x_{new}) + \delta(x_{new})\f$ should be calculated, using 55one of the \c num_prediction_configs, \c prediction_configs, or 56\c import_prediction_configs keywords. If none of these keywords is specified, 57the number of prediction configurations is set to 10 for scalar responses. 58The corresponding prediction variances are also calculated, according to 59 60\f[ 61\Sigma_{total}(x) = \Sigma_{\delta}(x) + \sigma^2_{exp}I. 62\f] 63 64Here, \f$\Sigma_{\delta}(x)\f$ is the (co)variance calculated from the 65Gaussian process or polynomial regression model, and \f$\sigma^2_{exp}\f$ is 66the maximum variance, taken over all configuration variables for that 67observation. In the case of field responses, the default prediction 68configurations are set equal to the input configurations, and the variance 69information contains only the variance calculated from the Gaussian process 70correction model. Further details can be found in the Dakota User's and Theory 71Manuals. 72 73<b> Usage Tips </b> 74 75For field responses, the keyword \ref 76responses-calibration_terms-field_calibration_terms-read_field_coordinates 77<i>must</i> be specified when computing the model discrepancy. See \ref 78responses-calibration_terms-field_calibration_terms for more information 79regarding options for calibration with field responses. 80 81<b> Expected Output </b> 82 83The resulting values of 84\f$\delta(x_{new})\f$ will be exported to the file specified using 85\c export_discrepancy_file or to the default file 86\c dakota_discrepancy_tabular.dat. The values of the corrected model at the 87specified prediction configurations will be exported to the file specified 88using \c export_corrected_model_file or to the default file 89\c dakota_corrected_model_tabular.dat, and the corresponding prediction 90variances will be exported to \c dakota_discrepancy_variance_tabular.dat 91or the file specified with \c export_corrected_variance_file. 92 93Examples:: 94Extensive examples can be found in the Dakota User's and Theory Manuals. 95The input files below illustrate the options available to \c model_discrepancy 96 97\verbatim 98model_discrepancy 99 discrepancy_type gaussian_process 100 num_prediction_configs 11 101 export_discrepancy_file "discrepancy_values.txt" 102 export_corrected_model_file "corrected_model.txt" 103 export_corrected_variance_file "prediction_variance.txt" 104\endverbatim 105 106\verbatim 107model_discrepancy 108 discrepancy_type polynomial 109 correction_order constant 110 prediction_configs 1 1.5 2 2.5 3 111\endverbatim 112 113\verbatim 114model_discrepancy 115 discrepancy_type polynomial 116 correction_order linear 117 import_prediction_configs "prediction_configs.txt" 118\endverbatim 119 120 121