Dual number automatic differentiation as applied to two-group cross-section uncertainty propagation

Abstract

This work addresses the problem of propagating uncertainty from group-wise neutron cross-sections to the results of neutronics diffusion calculations. Automatic differentiation based on dual number arithmetic was applied to uncertainty propagation in the framework of local sensitivity analysis. As an illustration, we consider a two-group diffusion problem in an infinite medium, which has a solution in a closed form. We employ automatic differentiation in conjunction with the sandwich formula for uncertainty propagation in three ways. Firstly, by evaluating the analytical expression for the multiplication factor using dual number arithmetic. Then, by solving the diffusion problem with the power iteration algorithm and the algebra of dual matrices. Finally, automatic differentiation is used to calculate the partial derivatives of the production and loss operators in the perturbation formula from the adjoint-weighted technique. The numerical solution of the diffusion problem is verified against the analytical formulas and the results of the uncertainty calculations are compared with those from the global sensitivity analysis approach. The uncertainty values obtained in this work differ from values given in the literature by less than 1•10–5.

Keywords

• automatic differentiation
• dual numbers
• sandwich formula
• sensitivity analysis
• uncertainty propagation
• power iteration