Elements of High-Order Predictive Model Calibration Algorithms with Applications to Large-Scale Reactor Physics Systems

Abstract

The inevitable discrepancies between experimental and computational results provide the basic motivation for performing quantitative model verification, validation, and predictive estimation. Loosely speaking, “code verification” addresses the question “are you solving the mathematical model correctly?”, while model validation addresses the question “does the model represent reality?” Ultimately, one aims at obtaining a probabilistic description of possible future outcomes based on all recognized errors and uncertainties, from all steps in the sequence of modeling and simulation processes that leads to a prediction using a computational model. Achieving this goal requires the combination (“assimilation”) of computational and experimental results in order to adjust (“calibrate”) the model parameters for predicting results (“responses”) more accurately—the so-called “best-estimate” results, with smaller uncertainties. The mathematical frameworks for combining experimental and computational quantities are customarily called “data adjustment” (for time-independent reactor physics applications) or “data assimilation” (for time-dependent geophysical applications). Notably, the current state-of-the-art procedures for data adjustment and/or assimilation are restricted to the use of second-order uncertainties (i.e., covariance matrices), and do not have provisions for incorporating response derivatives higher than first-order (in data adjustment procedures) or second-order (in some limited-scope research-versions of data assimilation procedures). Furthermore, neither the data adjustment nor the data assimilation procedures are currently able of computing higher-order moments (e.g., skewness and kurtosis) of the response distribution. In the absence of these higher-order moments of the response distribution, the predicted response distribution must implicitly be assumed as being Gaussian, since it is not possible to quantify the departures, if any, of the predicted responses from the assumed Gaussian distribution.

An important aspect of the novel contributions presented in this dissertation is the development of highly parallel and scalable algorithms for application of data adjustment and assimilation to large (peta)-scale systems, thereby significantly extending the practical feasibility and applicability of predictive model calibration activities. These new algorithms also include mathematical verification procedures for identifying non-physical covariance matrices, as well as quantifying the consistency of computational and experimental information. Furthermore, the dissertation presents expressions for computing the skewness and kurtosis of response distributions, to be used for quantifying non-Gaussian features of computed response distributions. A novel method, using adjoint functions, for computing very efficiently second-order mixed derivatives of responses to parameters, is also presented in this work.

The significant impact of the above algorithmic advances is demonstrated by using the neutron transport code Denovo, a highly parallel (one the order of tens of thousands of processors) code that runs on ORNL's leadership-class computer Jaguar, in conjunction with experimental results from the Lady Godiva and Jezebel benchmarks, as well as the “LEU-COMP-THERM-008” (shorthand: LCT) assembly. We recall here that the Lady Godiva benchmark is a bare sphere containing 94 wt% ²³⁵U , Jezebel is a critical assembly containing ²³⁹Pu , and the LCT assembly models a 3 × 3 array of Pressurized Water Reactor fuel assemblies comprising 4808 fuel rods and 153 water holes. Noteworthy new results in this dissertation are also obtained by using the remarkable efficiency of the “adjoint sensitivity analysis procedure for operator-type responses”, originally developed by Cacuci in 1981, to compute the sensitivities (derivatives) of the spatially dependent (as opposed to point-values of) neutron fluxes to cross sections.

The results obtained in this work represent first-of-a-kind computations of response skewness and kurtosis, thus enabling a quantitative assessment of non-Gaussian features of predicted responses (results). In particular, the illustrative results presented for the Godiva, Jezebel, and LCT benchmarks show that the response skewness and kurtosis are relatively small, thus quantitatively confirming the intuitive feeling (based on the presumed applicability of the central limit theorem) that simple reactor physics problems involving small cross section uncertainties tend to produce reaction rate responses that are nearly normally distributed. Finally, yet importantly, the algorithmic advances and results presented in this dissertation represent a fundamental first step towards developing a high-order predictive model calibration procedure capable of Bayesian combination of non-Gaussian model parameter features with non-Gaussian experimental distributions. Such developments are currently underway, and their successful completion is expected to enable more accurate predictions of “best-estimate results” including corresponding predicted non-Gaussian features, for large (peta- and exa-) scale systems.