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This work presents an illustrative application of the second-order adjointsensitivity analysis procedure (SO-ASAP) to a paradigm neutron diffusionproblem, which is sufficiently simple to admit an exact solution, therebymaking transparent the mathematical derivations underlying the SO-ASAP. Theillustrative application presented in this work shows that the actual number ofadjoint computations needed for computing all of the first- and second-orderresponse sensitivities may significantly less than 2*N+1 per response. For thisillustrative problem, four (4) large-scale adjoint computations sufficed forthe complete and exact computations of all 4 first- and 10 distinctsecond-order derivatives. Furthermore, the construction and solution of theSASS requires very little additional effort beyond the construction of theadjoint sensitivity system needed for computing the first-order sensitivities.Only the sources on the right-sides of the diffusion operator needed to bemodified; the left-side of the differential equations remained unchanged. Mostof the second-order relative sensitivities are just as large as or larger thanthe first-order ones. We show that the second-order sensitivities cause theexpected value of the response to differ from the computed nominal value of theresponse; and they contribute decisively to causing asymmetries in the responsedistribution. Neglecting the second-order sensitivities would nullify thethird-order response correlations, and hence would nullify the skewness of theresponse; consequently, any events occurring in a response's long and/or shorttails, which are characteristic of rare but decisive events would likely bemissed. We expect the SO-ASAP to affect significantly other fields that needefficiently computed second-order response sensitivities, e.g., optimization,data assimilation/adjustment, model calibration, and predictive modeling.

journal of computational physics

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