The results of measurements and computations are never perfectly accurate. On the one hand, results of measurements inevitably reflect the influence of experimental errors, imperfect instruments, or imperfectly known calibration standards. Around any reported experimental value, therefore, there always exists a range of values that may also be plausibly representative of the true but unknown value of the measured quantity. On the other hand, computations are afflicted by errors stemming from numerical procedures, uncertain model parameters, boundary and initial conditions, and/or imperfectly known physical processes or problem geometry.

It is therefore paramount to quantify the impact of the imprecisely known model parameters on the results (called “model responses”) produced by computational models. The functional derivatives of the model responses with respect to the imprecisely known parameters are called “sensitivities.” Computing the sensitivities efficiently and exactly is paramount for validating and optimizing computational models in the presence of uncertainties, model calibration through data assimilation, reduced-order modeling, forward and inverse predictive modeling, etc. Since a model comprising N imprecisely known model parameters has N first-order response sensitivities (functional derivatives), N(N+1)/2 distinct second-order response sensitivities, and so on, it follows that the number of computations required by conventional methods to compute response sensitivities increases exponentially (the “curse of dimensionality”) with the order of the respective sensitivities,. Therefore, conventional methods for sensitivity analysis require O(N^k) computations for obtaining the k-th order response sensitivities, which becomes prohibitively expensive or even impractical for computational models involving large numbers (>10²) of imprecisely known parameters.

Cacuci has conceived [1, 2] the adjoint sensitivity analysis methodology for nonlinear systems, which computes most efficiently and exactly the 1^st-order functional derivatives (“sensitivities”) of a system responses to system parameters, requiring 1 adjoint computation (as opposed to N computations, as required by other methods) for computing exactly all of the 1^st-order sensitivities. Cacuci has extended this original work to computing the higher-order sensitivities of response to parameters [3, 4] and is currently developing the Comprehensive Adjoint Sensitivity Analysis Methodology (CASAM) for computing exactly and efficiently the higher-order sensitivities of model responses to imprecisely known model parameters, internal interfaces and boundaries in phase-space. The CASAM [5-7] mathematical framework is set in linearly increasing higher-dimensional Hilbert spaces, as opposed to exponentially increasing parameter-dimensional spaces. Thus, for a scalar-valued valued response associated with a nonlinear model comprising N parameters, the CASAM requires 1 adjoint computation (as opposed to N, as required by other methods) for computing exactly all of the 1^st-order response sensitivities. All of the (mixed) 2^nd-order sensitivities are computed exactly by the C-ASAM in at most N computations, as opposed to (N+1)N/2 computations required by all other methods, and so on. For every lower-order sensitivity of interest, the CASAM computes the “N next-higher-order” sensitivities in one adjoint computation performed in a linearly increasing higher-dimensional Hilbert space. Very importantly, the CASAM is also applicable to the computation of sensitivities of operator-valued responses, which cannot be handled by current methods when the underlying problem comprises many parameters. The CASAM is expected to revolutionize the fields mentioned above (uncertainty quantification, model validation, optimization, data assimilation, model calibration, sensor fusion, reduced-order modeling, inverse problems, predictive modeling, etc.). The CASAM applies to any computational model (deterministic, statistical, etc.). The larger the number of model parameters, the more efficient the CASAM becomes for computing arbitrarily high-order response sensitivities to imprecisely known parameters.

Knowing just the nominal values of experimentally measured or computed quantities is insufficient for applications. The quantitative uncertainties accompanying measurements and computations are also needed, along with the respective nominal values. Extracting “best estimate” values for model parameters and predicted results, together with “best estimate” uncertainties for these parameters and results requires the combination of experimental and computational data, including their accompanying uncertainties (standard deviations and correlations). The goal of “predictive modeling” is to perform such a combination, which requires reasoning from incomplete, error-afflicted, and occasionally discrepant information, to predict future outcomes based on all recognized errors and uncertainties.

The BERRU predictive modeling methodology currently under development in the Center uses the maximum entropy principle to avoid the need for minimizing a user-chosen “cost functional” (usually a quadratic functional that represents the weighted errors between measured and computed responses) as intrinsically required by the extant “data adjustment” and/or “4D-VAR data assimilation” procedures, thus generalizing and significantly extending these methodologies. The acronym BERRU stands for “Best-Estimate Results with Reduced Uncertainties,” because the application of the BERRU predictive modeling methodology reduces the predicted standard deviations of both the best-estimate predicted responses and parameters. The BERRU predictive modeling methodology also provides a quantitative indicator, constructed from response sensitivities and response and parameter covariance matrices, for determining the consistency (agreement or disagreement) among the a priori computational and experimental information available for parameters and responses. Furthermore, the maximum entropy principles ensures that the more information is assimilated, the more the predicted standard deviations of the predicted responses and parameters are reduced, since the introduction of additional knowledge reduces the state of ignorance (as long as the additional information is consistent with the physical underlying system), as would also be expected based on principles of information theory.

The Center addresses applications to many of the areas of nuclear science and engineering covered in the Handbook of Nuclear Engineering (Dan G. Cacuci, Ed., Five Volumes, ca. 3600 pages; ISBN: 978-0-387-98150-5, Springer New York / Berlin, 2010). Illustrative applications, ranging from reactor physics experimental benchmarks to reprocessing facilities for spent nuclear fuel and cooling towers for nuclear reactors, are presented in the references below.

The BERRU Predictive Modeling methodology has also been applied to improve the modeling and performance of many other physical and engineering systems. Notable applications to fuel cells and solar collector facilities are presented in Refs. 15 and 16, respectively. Ongoing research aims at applying the BERRU Predictive Modeling methodology to improve the modeling and performance of lithium ion and other innovative batteries.