model_discrepancy

(Experimental) Post-calibration calculation of model discrepancy correction

Specification

• Alias: None

• Arguments: None

Child Keywords:

Required/Optional	Description of Group	Dakota Keyword	Dakota Keyword Description
Optional		discrepancy_type	Specify the type of model discrepancy
Optional		num_prediction_configs	Specify number of prediction locations for model discrepancy
Optional		prediction_configs	List prediction locations for model discrepancy
Optional		import_prediction_configs	Specify text file containing prediction configurations for model discrepancy
Optional		export_discrepancy_file	Output file for prediction discrepancy calculations
Optional		export_corrected_model_file	Output file for corrected model prediction calculations
Optional		export_corrected_variance_file	Output file for prediction variance calculations

Description

The goal of parameter calibration is to minimize the difference between experimental data, \(d(x)\) , and model observations, \(M(\theta, x)\) , where \(\theta\) are the model parameters and \(x\) is a configuration variable, such as temperature or pressure. However, it is not uncommon that, at the conclusion of parameter calibration, the agreement between experimental data and the calibrated model is not “close enough.” This is often due to model form or structural error. In this case, the canonical equation

\[d (x) = M (\theta, x) + \varepsilon\]

is replaced by one that also includes model discrepancy \(\delta(x)\) ,

\[d (x) = M (\theta, x) + \delta(x) + \varepsilon.\]

In the Dakota implementation, the calculation of \(\delta(x)\) is performed after the model parameters \(\theta\) are calibrated. For each observable \(d_i\) , the discrepancy

\[\delta_i(x_j) = d_i(x_j) - M_i(\theta^*, x_j)\]

is calculated for each value \(x_j\) of the configuration variable, where \(\theta^*\) is the optimal parameter value obtained during the calibration. For scalar responses, the model discrepancy is only a function of the configuration variables, and there is one discrepancy regression model for each observable \(d_{i}\) . This set of discrepancy models may be specified to be either Gaussian process or polynomial regression models of constant, linear, or quadratic order, and each model is fit to the calculated discrepancy values. See the method-bayes_calibration-model_discrepancy-discrepancy_type command for more details regarding these options. For field responses, the model discrepancy is a function of the configuration variables as well as the independent field coordinates (such as time or space), and there is one discrepancy regression model for each field. In this case, the discrepancy models are Gaussian process models. The calculation of model discrepancy has not been tested for cases in which responses are mixed scalar and field responses.

The user may then specify new or “prediction” configurations at which the corrected model \(M(\theta^*, x_{new}) + \delta(x_{new})\) should be calculated, using one of the num_prediction_configs, prediction_configs, or import_prediction_configs keywords. If none of these keywords is specified, the number of prediction configurations is set to 10 for scalar responses. The corresponding prediction variances are also calculated, according to

\[\Sigma_{total}(x) = \Sigma_{\delta}(x) + \sigma^2_{exp}I.\]

Here, \(\Sigma_{\delta}(x)\) is the (co)variance calculated from the Gaussian process or polynomial regression model, and \(\sigma^2_{exp}\) is the maximum variance, taken over all configuration variables for that observation. In the case of field responses, the default prediction configurations are set equal to the input configurations, and the variance information contains only the variance calculated from the Gaussian process correction model. Further details can be found in the Dakota User’s and Theory Manuals.

Usage Tips

For field responses, the keyword responses-calibration_terms-field_calibration_terms-read_field_coordinates <i>must</i> be specified when computing the model discrepancy. See responses-calibration_terms-field_calibration_terms for more information regarding options for calibration with field responses.

Expected Output

The resulting values of \(\delta(x_{new})\) will be exported to the file specified using export_discrepancy_file or to the default file dakota_discrepancy_tabular.dat. The values of the corrected model at the specified prediction configurations will be exported to the file specified using export_corrected_model_file or to the default file dakota_corrected_model_tabular.dat, and the corresponding prediction variances will be exported to dakota_discrepancy_variance_tabular.dat or the file specified with export_corrected_variance_file.