Demonstration of prospective application of the dual number automatic differentiation for uncertainty propagation in neutronic calculations

Abstract

Automatic differentiation (AD) is a set of techniques which allows the numeric evaluation of derivatives of functions calculated by a computer program. In recent years, interest in AD has grown significantly in many disciplines, especially in the context of gradient-based optimization algorithms. Sensitivity analysis is another natural application area for AD methods. However, despite the large body of sensitivity and uncertainty (S/U) analysis publications produced in the field of nuclear reactor science and engineering in the last decade, the use of AD by the community has been very limited. The purpose of the present paper is to fill this gap and to demonstrate how AD can be employed in conjunction with some traditionally used sensitivity analysis and uncertainty propagation techniques. Specifically, the forward mode of AD based on dual number arithmetic was considered in the study. We provide a short overview of dual number algebra and dual number automatic differentiation (DNAD) methods, as well as of the tools available for the practical implementation of DNAD, followed by a discussion of its application to S/U analysis. As illustration, we solve a simplistic example of an infinite, homogeneous diffusion problem using parameters that correspond to a plate-type, Material Testing Reactor fuel assembly. Homogenized cross-sections and uncertainty (covariance) data for the test problem are generated with the SCALE code in six energy groups. The diffusion problem is solved through the power iteration algorithm with the algebra of dual matrices, which yields sensitivity information for use in the sandwich formula. DNAD is also used to calculate partial derivatives of the production and loss operators in the perturbation formula in the context of the adjoint-weighted technique. Both of these methods yield uncertainty values for the multiplication factor that are within three pcm of the reference value. Automatic differentiation can, therefore, be useful for uncertainty propagation in the framework of local sensitivity analysis in addition to traditionally employed sampling methods or in conjunction with the perturbation method.