Application of the Second-Order Comprehensive Adjoint Sensitivity Analysis Methodology to Compute First- and Second-Order Sensitivities of Flux Functionals in a Multiplying System With Source

This work presents an application of the Second-Order Adjoint Sensitivity Analysis Methodology (2nd-ASAM) to the neutron transport Boltzmann equation that models a multiplying subcritical system comprising a nonfission neutron source to compute efficiently and exactly all of the first- and second-order functional derivatives (sensitivities) of a detector’s response to all of the model’s parameters, including isotopic number densities, microscopic cross sections, fission spectrum, sources, and detector response function. As indicated by the general theoretical considerations underlying the 2nd-ASAM, the number of computations required to obtain the first and second orders increases linearly in augmented Hilbert spaces as opposed to increasing exponentially in the original Hilbert space. The results presented in this work are currently being implemented in several production-oriented three-dimensional neutron transport code systems for analyzing specific subcritical systems

Towards Overcoming the Curse of Dimensionality: The Third-Order Adjoint Method for Sensitivity Analysis of Response-Coupled Linear Forward/Adjoint Systems, Uncertainty Quantification and Predictive Modeling with Applications to Nuclear Energy Systems

This work presents the Third-Order Adjoint Sensitivity Analysis Methodology (3rd-ASAM) for response-coupled forward and adjoint linear systems. The 3rd-ASAM enables the efficient computation of the exact expressions of the 3rd-order functional derivatives (“sensitivities”) of a general system response, which depends on both the forward and adjoint state functions, with respect to all of the parameters underlying the respective forward and adjoint systems. Such responses are often encountered when representing mathematically detector responses and reaction rates in reactor physics problems. The 3rd-ASAM extends the 2nd-ASAM in the quest to overcome the “curse of dimensionality” in sensitivity analysis, uncertainty quantification and predictive modeling. This work also presents new formulas that incorporate the contributions of the 3rd-order sensitivities into the expressions of the first four cumulants of the response distribution in the phase-space of model parameters. Using these newly developed formulas, this work also presents a new mathematical formalism, called the 2nd/3rd-BERRU-PM “Second/Third-Order Best-Estimated Results with Reduced Uncertainties Predictive Modeling”) formalism, which combines experimental and computational information in the joint phase-space of responses and model parameters, including not only the 1st-order response sensitivities, but also the complete hessian matrix of 2nd-order second-sensitivities and also the 3rd-order sensitivities, all computed using the 3rd-ASAM. The 2nd/3rd-BERRU-PM uses the maximum entropy principle to eliminate the need for introducing and “minimizing” a user-chosen “cost functional quantifying the discrepancies between measurements and computations,” thus yielding results that are free of subjective user-interferences while generalizing and significantly extending the 4D-VAR data assimilation procedures. Incorporating correlations, including those between the imprecisely known model parameters and computed model responses, the 2nd/3rd-BERRU-PM also provides a quantitative metric, constructed from sensitivity and covariance matrices, for determining the degree of agreement among the various computational and experimental data while eliminating discrepant information. The mathematical framework of the 2nd/3rd-BERRU-PM formalism requires the inversion of a single matrix of size Nr Nr, where Nr denotes the number of considered responses. In the overwhelming majority of practical situations, the number of responses is much less than the number of model parameters. Thus, the 2nd-BERRU-PM methodology overcomes the curse of dimensionality which affects the inversion of hessian matrices in the parameter space