Dan G. Cacuci | Searcharxiv

arXiv2014

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.

math-ph↗

arXiv2014

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.

math-ph↗