Sensitivity and Uncertainty Analysis, Data Assimilation, and Predictive Best-Estimate Model Calibration

Abstract

In practice, the results of experiments seldom coincide with the computational results obtained from the mathematical models of the respective experiments. Discrepancies between experimental and computational results stemfrom both experimental and computa- tional uncertainties. Such discrepanciesmotivate the activities ofmodel verification, validation, and predictive estimation. Following a brief review of the classification and origins of experi- mental uncertainties, this chapter presents widely used statistical and deterministic methods for computing response sensitivities tomodel parameters, highlighting, in particular, the novel adjoint sensitivity analysis procedure (ASAP) for augmented nonlinear large-scale systems with feedback. Thepractical useof ASAP is illustrated by a large-scale application for analyzing the dynamic reliability of an accelerator system design for the International Fusion Materials Irradiation Facility (IFMIF). Response sensitivities to parameters and the corresponding uncertainties are the fundamen- tal ingredients for predictive estimation (PE),which aims at providing a probabilistic description of possible future outcomes based on all recognized errors and uncertainties. The key PE- activity is model calibration,which uses data adjustment and data assimilation procedures for addressing the integration of experimental data for updating (calibrating or adjusting) parameters in the simulation model. This chapter also presents a state-of-the-art mathemati- cal framework for time-dependent data assimilation andmodel calibration, using sensitivities and covariance matrices. The basic premise underlying this mathematical framework is that only means and covariance matrices are a priori available, which is the usual situation when analyzing large-scale systems. Under this premise, the maximum entropy principle of statis- tical mechanics is employed in conjunction with information theory to construct a Gaussian prior distribution that takes all of the available information into account while minimizing (in the sense of quadratic loss) the introduction of spurious information. This prior distribu- tion also comprises correlations amongmodel parameters and responses, thus generalizing the state-of-the-art data assimilation algorithms used in geosciences. The posterior distribution for the best-estimate calibrated model parameters and responses is constructed by using Bayes’ theorem. The best-estimate predicted mean values and reduced covariances, which are customarily needed when employing decision theory under “quadratic loss,” are computed by extracting the bulk contributions via the saddle-point method. The minimum value of the quadratic form appearing in the exponent of the Gaussian poste- rior distribution can be used as an indicator of the agreement between the computed and experimentally measured responses. When all information is consistent, the posterior prob- ability density function yields reduced best-estimate uncertainties for the best-estimatemodel parameters and responses. This fact is illustrated in this chapter for a time-dependent thermal- hydraulic systemthat can serve as a benchmark for validating and calibrating thermal-hydraulic codes.The novel features of the data assimilationandmodel calibrationmethodologypresented in this chapter include: (1) treatment of systems involving correlated parameters and responses; (2) simultaneous calibration of all parameters and responses; and (3) simultaneous calibration over all time intervals; this includes the usual two-step time advancement procedures used in geophysical sciences. Open issues (e.g., explicit treatment ofmodeling errors, reducing the computational burden, removing the current restriction to Gaussian distributions) are addressed in the concluding section of this chapter. Since predictive “best-estimate”numerical simulation models are essen- tial for designing newtechnologies and facilities, particularly when the newsystems cannot be readily tested experimentally, the only path to progress is to reduce drastically the uncertainties associated with such simulation tools while enlarging the respective validation domains.