This work presents the second-order forward and adjoint sensitivity analysis methodologies (2nd-FSAM and 2nd-ASAM) for computing exactly and efficiently the second-order functional derivatives of physical (engineering, biological, etc.) system responses (i.e., "system performance parameters") to the system's model parameters. The definition of "system parameters" used in this work includes all computational input data, correlations, initial and/or boundary conditions, etc. For a physical system comprising Nα parameters and Nr responses, we note that the 2nd-FSAM requires a total of (Nα2/2+3Nα/2) large-scale computations for obtaining all of the first- and second-order sensitivities, for all Nr system responses. On the other hand, for one functional-type system response, the 2nd-ASAM requires one large-scale computation using the first-level adjoint sensitivity system 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. Therefore, the 2nd-FSAM should be used when Nr≫Nα, while the 2nd-ASAM should be used when Nα≫Nr. The original 2nd-ASAM presented in this work should enable the hitherto very difficult, if not intractable, exact computation of all of the second-order response sensitivities (i.e., functional Gateaux-derivatives) for large-systems involving many parameters, as usually encountered in practice. Very importantly, the implementation of the 2nd-ASAM requires very little additional effort beyond the construction of the adjoint sensitivity system needed for computing the first-order sensitivities.
Cacuci, D. G. (2015). Second-order adjoint sensitivity analysis methodology (2nd-ASAM) for computing exactly and efficiently first- and second-order sensitivities in large-scale linear systems: I. Computational methodology. Journal of Computational Physics, 284, 687–699. https://doi.org/10.1016/j.jcp.2014.12.042