The nuclear community relies heavily on computer codes both in research and in the operation of installations. The results of such computations are useful only if they are augmented with sensitivity and uncertainty studies. This technical note presents some theoretical considerations regarding traditional first-order sensitivity analysis and uncertainty quantification involving constrained quantities. The focus is on linear constraints, which are often encountered in reactor physics problems due to energy and angle distributions, or the correlation between the isotopic abundances of elements.
A consistent theory is given for the derivation and interpretation of constrained first-order sensitivity coefficients; covariance matrix normalization procedures; their interrelation; and the treatment of constrained inputs with polynomial chaos expansion, which was the main motivation of this research. It is shown that if the covariance matrix violates the “generic zero column and row sum” condition, normalizing it is equivalent to constraining the sensitivities, but since both can be done in many ways, different sensitivity coefficients and uncertainties can be derived. This makes results ambiguous, underlining the need for proper covariance data. Furthermore, it is highlighted that certain constraining procedures can result in biased or unphysical uncertainty estimates. To confirm our conclusions, we demonstrate the presented theory on three analytical and two numerical examples including fission spectrum, isotopic distribution, and power distribution-related uncertainties, as well as the correlation between mass, volume, and density.