Second-order adjoint sensitivity analysis procedure (SO-ASAP) for   computing exactly and efficiently first- and second-order sensitivities in   large-scale linear systems:II. Illustrative application to a paradigm   particle diffusion problem
  This work presents an illustrative application of the second-order adjoint sensitivity analysis procedure (SO-ASAP) to a paradigm neutron diffusion problem, which is sufficiently simple to admit an exact solution, thereby making transparent the mathematical derivations underlying the SO-ASAP. 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 significantly less than 2*N+1 per response. For this illustrative problem, four (4) 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 SASS requires very little additional effort beyond the construction of the adjoint sensitivity system needed for computing the first-order sensitivities. Only the sources on the right-sides of the diffusion operator needed to be modified; the left-side of the differential equations remained unchanged. Most of the second-order relative sensitivities are just as large as or larger than the first-order ones. We show that the second-order sensitivities cause the expected value of the response to differ from the computed nominal value of the response; and they contribute decisively to causing asymmetries in the response distribution. 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 would likely be missed. We expect the SO-ASAP to affect significantly other fields that need efficiently computed second-order response sensitivities, e.g., optimization, data assimilation/adjustment, model calibration, and predictive modeling. 
Dan G. Cacuci
math-ph math.MP