Second-order adjoint sensitivity analysis methodology (2ND-ASAM) for computing exactly and efficiently first- and second-order sensitivities in large-scale systems - 14356

Abstract

This work presents the second-order adjoint sensitivity analysis methodology (2.-ASAM) for computing exactly and efficiently the second-order functional derivatives ('sensitivities') of system responses to the system's model parameters. For a physical system comprising N{sub α} parameters and N{sub r} responses, forward methods require a total of at least (N{sup 2}{sub α}/2+3N{sub α}/2) large-scale computations for obtaining all of the first- and second-order sensitivities, for all N{sub r} system responses. On the other hand, for one functional-type system response, the 2.-ASAM requires one large-scale computation using the first-level adjoint sensitivity system (1.-LASS) for obtaining all of the first-order sensitivities, followed by at most (2N{sub α}+1) large-scale computations using the second-level adjoint sensitivity systems (2.-LASS) for obtaining all of the second-order sensitivities. The second-order sensitivities contribute decisively to causing asymmetries in the response distribution, since they are the leading contributors to the third-order response correlations (skewness). They also cause the 'expected value of the response' to differ from the 'computed nominal value of the response'. The implementation of the 2.-ASAM 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 (differential) operator needed to be modified; the left-side of the differential equations (and hence the 'solver' in large-scale practical applications) remain unchanged. The application of the 2.-ASAM is illustrated on a benchmark heat conduction/convection problem, which makes transparent the underlying mathematical derivations. For this illustrative problem, 4 'large-scale' adjoint computations suffice for computing exactly all of the 6 first- and 21 distinct second-order derivatives. The 2.-ASAM presented in this work should enable the hitherto very difficult, if not intractable, exact computation of all of the second-order response sensitivities for large-systems involving many parameters, which is expected to affect significantly other fields that need efficiently computed second-order response sensitivities, such as optimization, data assimilation/adjustment, model calibration, and predictive modeling. (author)