Second-order adjoint sensitivity analysis procedure (SO-ASAP) for   computing exactly and efficiently first- and second-order sensitivities in   large-scale linear systems: I. Computational methodology
  This work presents the second-order forward and adjoint sensitivity analysis procedures (SO-FSAP and SO-ASAP) for computing exactly and efficiently the second-order functional derivatives of physical (engineering, biological, etc.) system responses 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 N parameters and M responses, we note that the SO-FSAP requires a total of 0.5*N**2+1.5*N large-scale computations for obtaining all of the first- and second-order sensitivities, for all M system responses. On the other hand, the SO-ASAP requires a total of 2*N+1 large-scale computations for obtaining all of the first- and second-order sensitivities, for one functional-type system responses. Therefore, the SO-ASAP should be used when M is much larger than N, while the SO-ASAP should be used when N is much larger than M. The original SO-ASAP 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 SO-ASAP requires very little additional effort beyond the construction of the adjoint sensitivity system needed for computing the first-order sensitivities. 
Dan G. Cacuci
math-ph math.MP