The Second-Order Adjoint Sensitivity Analysis Methodology for Nonlinear Systems—I: Theory

The use of adjoint methods for computing first-order sensitivities (i.e., functional derivatives) of results (responses) produced by a computational model to the model’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 system’s effective multiplication factor to cross sections were computationally intensive to obtain, requiring O(N_α²) large-scale computations per response, for a system comprising N_α 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 (2^nd-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 system’s model parameters. For a physical system comprising N_α parameters, forward methods require a total of (N_α²/2+3N_α/2) large-scale computations for obtaining all of the first- and second-order sensitivities, for all system responses. In contradistinction, the 2^nd-ASAM requires one large-scale computation using the first-level adjoint sensitivity system (1^st-LASS) for obtaining all of the first-order sensitivities, followed by at most N_α 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 2^nd-ASAM require very little additional effort beyond the construction of the 1^st-LASS needed for computing the first-order sensitivities. Furthermore, because of the symmetry properties of the second-order sensitivities, the 2^nd-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 2^nd-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.