Multilevel Uncertainty Integration

This chapter discusses a Bayesian methodology to integrate model verification, validation, and calibration activities for the purpose of overall uncertainty quantification in model-based prediction. The methodology is first developed for single-level models and then extended to systems that are studied using multilevel models that interact with each other. Two types of interactions among multilevel models are considered: (1) Type-I, where the output of a lower-level model (component and/or subsystem) becomes an input to a higher-level system model, and (2) Type-II, where parameters of the system model are inferred using lower-level models and tests (that describe simplified components and/or isolated physics). The various models; their inputs, parameters, and outputs; experimental data; and various sources of model error are connected through a Bayesian network. The results of calibration, verification, and validation with respect to each individual model are integrated using the principles of conditional probability and total probability and propagated through the Bayesian network in order to quantify the overall system-level prediction uncertainty. For Type-II model, the relevance of each lower-level output to the system-level quantity of interest is quantified by comparing Sobol indices, thus measuring the extent to which a lower-level test represents the characteristics of the system so that the calibration results can be reliably used in the system level. The proposed methodology is illustrated with numerical examples that deal with heat conduction and structural dynamics.