Nuclear Science and Engineering / Volume 184 / Number 1 / September 2016 / Pages 16-30
Technical Paper / dx.doi.org/10.13182/NSE16-16
The use of adjoint methods for computing first-order sensitivities (i.e., functional derivatives) of results (responses) produced by a computational model to the modelâ€™s parameters was initiated in the nuclear engineering sciences in the 1940s. The field of nuclear science and engineering also provided pioneering works, during the 1970s, for computing second-order response sensitivities of responses associated with the adjoint neutron and radiation transport and/or diffusion equations. These works generally indicated that the second-order sensitivities of responses such as reaction rates and the systemâ€™s effective multiplication factor to cross sections were computationally intensive to obtain, requiring O(Nα2) large-scale computations per response, for a system comprising Nα model parameters, and were considerably smaller than the corresponding first-order sensitivities. These results likely gave rise to the generally held opinion that second-order sensitivities are generally insignificant in reactor physics, which may, in turn, have led to diminishing interest in developing efficient methods for computing second-order sensitivities for nuclear engineering systems.
This work presents the second-order adjoint sensitivity analysis methodology (2nd-ASAM) for nonlinear systems, which yields exactly and efficiently the second-order functional derivatives of physical system responses (i.e., system performance parameters) to the systemâ€™s model parameters. For a physical system comprising Nα parameters, forward methods require a total of (Nα2/2+3Nα/2) large-scale computations for obtaining all of the first- and second-order sensitivities, for all system responses. In contradistinction, the 2nd-ASAM requires one large-scale computation using the first-level adjoint sensitivity system (1st-LASS) for obtaining all of the first-order sensitivities, followed by at most Nα large-scale computations using the second-level adjoint sensitivity systems, for obtaining exactly all of the second-order sensitivities of a functional-type response. The construction, implementation, and solution of the 2nd-ASAM require very little additional effort beyond the construction of the 1st-LASS needed for computing the first-order sensitivities. Furthermore, because of the symmetry properties of the second-order sensitivities, the 2nd-ASAM comprises the inherent automatic solution verification of the correctness and accuracy of the second-level adjoint functions used for the efficient and exact computation of the second-order sensitivities. The use of the 2nd-ASAM to compute exactly all of the second-order response sensitivities to model input parameters is expected to enable significant advances in related scientific disciplines, particularly the areas of uncertainty quantification and predictive modeling, including model validation, reduced-order modeling, data assimilation, model calibration, and extrapolation.