Second-order adjoint sensitivity analysis methodology (2nd-ASAM) for computing exactly and efficiently first- and second-order sensitivities in large-scale linear systems

This work presents an illustrative application of the second-order adjoint sensitivity analysis methodology (2nd-ASAM) to a paradigm neutron diffusion problem, which is sufficiently simple to admit an exact solution, thereby making transparent the underlying mathematical derivations. The general theory underlying 2nd-ASAM indicates that, for a physical system comprising N α parameters, the computation of all of the first- and second-order response sensitivities requires (per response) at most ( 2 N α + 1 ) "large-scale" computations using the first-level and, respectively, second-level adjoint sensitivity systems (1st-LASS and 2nd-LASS). Very importantly, however, the illustrative application presented in this work shows that the actual number of adjoint computations needed for computing all of the first- and second-order response sensitivities may be significantly less than ( 2 N α + 1 ) per response. For this illustrative problem, four "large-scale" adjoint computations sufficed for the complete and exact computations of all 4 first- and 10 distinct second-order derivatives. Furthermore, the construction and solution of the 2nd-LASS requires very little additional effort beyond the construction of the adjoint sensitivity system needed for computing the first-order sensitivities. Very significantly, only the sources on the right-sides of the diffusion (differential) operator needed to be modified; the left-side of the differential equations (and hence the "solver" in large-scale practical applications) remained unchanged.All of the first-order relative response sensitivities to the model parameters have significantly large values, of order unity. Also importantly, most of the second-order relative sensitivities are just as large, and some even up to twice as large as the first-order sensitivities. In the illustrative example presented in this work, the second-order sensitivities contribute little to the response variances and covariances. However, they have the following major impacts on the computed moments of the response distribution: (a) they cause the "expected value of the response" to differ from the "computed nominal value of the response"; and (b) they contribute decisively to causing asymmetries in the response distribution. Indeed, neglecting the second-order sensitivities would nullify the third-order response correlations, and hence would nullify the skewness of the response. Consequently, any events occurring in a response's long and/or short tails, which are characteristic of rare but decisive events (e.g., major accidents, catastrophes), would likely be missed. The 2nd-ASAM is expected to affect significantly other fields that need efficiently computed second-order response sensitivities, e.g., optimization, data assimilation/adjustment, model calibration, and predictive modeling. We apply the second-order adjoint sensitivity analysis methodology (2nd-ASAM) to a paradigm diffusion problem.Four 2nd-ASAM adjoint computations exactly determine all fourteen 1st- and 2nd-order sensitivities.Non-zero second-order response sensitivities decisively skew response distributions.