On the Need to Determine Accurately the Impact of Higher-Order Sensitivities on Model Sensitivity Analysis, Uncertainty Quantification and Best-Estimate Predictions

Dan Gabriel Cacuci

doi:10.3390/en14196318

Abstract

This work aims at underscoring the need for the accurate quantification of the sensitivities (i.e., functional derivatives) of the results (a.k.a. “responses”) produced by large-scale computational models with respect to the models’ parameters, which are seldom known perfectly in practice. The large impact that can arise from sensitivities of order higher than first has been highlighted by the results of a third-order sensitivity and uncertainty analysis of an OECD/NEA reactor physics benchmark, which will be briefly reviewed in this work to underscore that neglecting the higher-order sensitivities causes substantial errors in predicting the expectation and variance of model responses. The importance of accurately computing the higher-order sensitivities is further highlighted in this work by presenting a text-book analytical example from the field of neutron transport, which impresses the need for the accurate quantification of higher-order response sensitivities by demonstrating that their neglect would lead to substantial errors in predicting the moments (expectation, variance, skewness, kurtosis) of the model response’s distribution in the phase space of model parameters. The incorporation of response sensitivities in methodologies for uncertainty quantification, data adjustment and predictive modeling currently available for nuclear engineering systems is also reviewed. The fundamental conclusion highlighted by this work is that confidence intervals and tolerance limits on results predicted by models that only employ first-order sensitivities are likely to provide a false sense of confidence, unless such models also demonstrate quantitatively that the second- and higher-order sensitivities provide negligibly small contributions to the respective tolerance limits and confidence intervals. The high-order response sensitivities to parameters underlying large-scale models can be computed most accurately and most efficiently by employing the high-order comprehensive adjoint sensitivity analysis methodology, which overcomes the curse of dimensionality that hampers other methods when applied to large-scale models involving many parameters.

Highlights

The state-of-the-art computational tool for sensitivity analysis, uncertainty quantification and “data adjustment” of model responses in reactor physics and reactor criticality safety is embodied in the moduleTSURFER [1] of the code system SCALE, developed at the Oak Ridge National Laboratory.The methodology underlying TSURFER is the so-called “generalized linear least squares adjustment (GLLSA)” procedure [2], which is limited to incorporating just the first-order sensitivities of the response with respect to the model parameters
The first-order response sensitivities to the many thousands of model parameters involved in reactor physics computations are computed efficiently and accurately using the first-order adjoint sensitivity analysis method
At least second-order sensitivities must be used in order to estimate the third-order moment of the response distribution

Summary

Introduction

The state-of-the-art computational tool for sensitivity analysis, uncertainty quantification and “data adjustment” of model responses (effective multiplication factors, reaction rates) in reactor physics and reactor criticality safety is embodied in the moduleTSURFER [1] of the code system SCALE, developed at the Oak Ridge National Laboratory.The methodology underlying TSURFER is the so-called “generalized linear least squares adjustment (GLLSA)” procedure [2], which is limited to incorporating just the first-order sensitivities of the response with respect to the model parameters. Since a general-purpose adjoint sensitivity analysis methodology capable of computing second-order sensitivities efficiently, reliably and accurately for large-scale systems (such as encountered in reactor physics) was unavailable in the past, the potential effects of higher-order sensitivities were not considered. A general-purpose, efficient and accurate methodology was recently developed by Cacuci [3,4,5] and was applied, in a sequence of pioneering works [6,7,8,9,10,11], to perform a comprehensive second-order sensitivity and uncertainty analysis of an OECD/NEA polyethylene-reflected plutonium (acronym: PERP) reactor physics benchmark [12], which comprised 21,976 uncertain models parameters (including 180 group-averaged total microscopic cross-sections, 21,600 group-averaged scattering microscopic cross-sections, 120 parameters describing the fission process, 60 parameters describing the fission spectrum, 10 parameters describing the system’s sources and six isotopic number densities). The PERP benchmark comprises 21,976 firstorder sensitivities of the leakage response with respect to the model parameters and

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Conclusion