Efficient computation of operator‐type response sensitivities for uncertainty quantification and predictive modeling: illustrative application to a spent nuclear fuel dissolver model

Abstract

SUMMARYThis work honors the 75th birthday of Professor Ionel Michael Navon by presenting original results highlighting the computational efficiency of the adjoint sensitivity analysis methodology for functionvalued operator responses by means of an illustrative paradigm dissolver model. The dissolver model analyzed in this work has been selected because of its applicability to material separations and its potential role in diversion activities associated with proliferation and international safeguards. This dissolver model comprises eight active compartments in which the 16 time-dependent nonlinear differential equations modeling the physical and chemical processes comprise 619 scalar and time-dependent model parameters, related to the model's equation of state and inflow conditions. The most important response for the dissolver model is the time-dependent nitric acid in the compartment furthest away from the inlet, where measurements are available at 307 time instances over the transient's duration of 10.5 h. The sensitivities to all model parameters of the acid concentrations at each of these instances in time are computed efficiently by applying the adjoint sensitivity analysis methodology for operator-valued responses.The uncertainties in the model parameters are propagated using the above-mentioned sensitivities to compute the uncertainties in the computed responses. A predictive modeling formalism is subsequently used to combine the computational results with the experimental information measured in the compartment furthest from the inlet and then predict optimal values and uncertainties throughout the dissolver. This predictive modeling methodology uses the maximum entropy principle to construct an optimal approximation of the unknown a priori distribution for the a priori known mean values and uncertainties characterizing the model parameters and the computed and experimentally measured model responses. This approximate a priori distribution is subsequently combined using Bayes' theorem with the "likelihood" provided by the multiphysics computational models. Finally, the posterior distribution is evaluated using the saddle-point method to obtain analytical expressions for the optimally predicted values for the parameters and responses of both multi-physics models, along with corresponding reduced uncertainties. This work shows that even though the experimental data pertains solely to the compartment furthest from the inlet (where the data were measured), the predictive modeling procedure used herein actually improves the predictions and reduces the predicted uncertainties for the entire dissolver, including the compartment furthest from the measurements, because this predictive modeling methodology combines and transmits information simultaneously over the entire phase-space, comprising all time steps and spatial locations.

Illustrative Application of the 2<sup>nd</sup>-Order Adjoint Sensitivity Analysis Methodology to a Paradigm Linear Evolution/Transmission Model: Reaction-Rate Detector Response

“…5) As has been generally shown by Cacuci [1] [2] [3], the mixed 2 nd -order sensitivities are obtained twice, stemming from distinct 2 nd -LASS. This fact enables the 2 nd -CASAM to provide an inherent independent verification of the correctness and accuracy of the 2 nd -level adjoint sensitivity functions that are used to compute the respective mixed 2 nd -order sensitivities.…”

“…This work continues to illustrate the application of the general second-order adjoint sensitivity analysis methodology (2 nd -CASAM) presented in [1] by us-ing the evolution/transmission mathematical benchmark model introduced in [2], but considering a "reaction-rate" detector response, as opposed to the "point-detector" response considered in [2]. As in [2], the mathematical model considered in this work could represent [3] [4] the time-evolution of the concentration of a substance in a homogeneous mixture of materials or, alternatively, it could represent [4] [5] [6] the transmission/attenuation of the flux of uncollided particles (e.g., photons) travelling through a one-dimensional homogenized multi-material slab of imprecisely known thickness.…”

Illustrative Application of the 2<sup>nd</sup>-Order Adjoint Sensitivity Analysis Methodology to a Paradigm Linear Evolution/Transmission Model: Point-Detector Response

“…The simple evolution system represented by Equations (1) and (2) occurs in the mathematical modeling of many physical systems. δ − denotes the well-known Dirac-delta (impulse) functional.…”

“…The application of the general second-order adjoint sensitivity analysis metho-dology presented in [1] is illustrated in this work by means of a simple mathematical model which expresses a conservation law of the model's state function. This paradigm model is representative of transmission of particles and/or radiation through materials [2] [3], chemical kinetics processes [4] [5], radioactive decay modeled by the Bateman equation, etc.…”

Model reduction and inverse problems and data assimilation with geophysical applications. A special issue in honor of I. Michael Navon's 75th birthday

“…, implementation of computationally efficient operator‐type response sensitivities for uncertainty quantification and predictive modeling – Cacuci et al . , and the development of a second‐order adjoint model to better understand the impact of the changes in pollutant emission onto a target region – Le Dimet et al . .…”