SECOND-ORDER ADJOINT SENSITIVITY ANALYSIS METHODOLOGY (2nd-ASAM) FOR COMPUTING EXACTLY AND EFFICIENTLY FIRST-AND SECOND-ORDER SENSITIVITIES IN LARGE-SCALE SYSTEMS

摘要

This work presents the second-order adjoint sensitivity analysis methodology (2nd -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_α parameters and N_r responses, forward methods require a total of at least (N_α~2 + 3N_α /2) large-scale computations for obtaining all of the first- and second-order sensitivities, for all N_r system responses. On the other hand, for one functional-type system response, the 2nd-ASAM requires one large-scale computation using the first-level adjoint sensitivity system (1st-LASS) for obtaining all of the first-order sensitivities, followed by at most (2N_α +1) large-scale computations using the second-level adjoint sensitivity systems (2nd -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 2nd-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 2nd-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 2nd-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.