Second-Order Adjoint Sensitivity Analysis Methodology for Computing Exactly Response Sensitivities to Uncertain Parameters and Boundaries of Linear Systems: Mathematical Framework

Abstract

This work presents the “Second-Order Comprehensive Adjoint Sensitivity Analysis Methodology (2 nd -CASAM)” for the efficient and exact computation of 1 st - and 2 nd -order response sensitivities to uncertain parameters and domain boundaries of linear systems. The model’s response ( i.e. , model result of interest) is a generic nonlinear function of the model’s forward and adjoint state functions, and also depends on the imprecisely known boundaries and model parameters. In the practically important particular case when the response is a scalar-valued functional of the forward and adjoint state functions characterizing a model comprising N parameters, the 2 nd -CASAM requires a single large-scale computation using the First-Level Adjoint Sensitivity System (1 st -LASS) for obtaining all of the first-order response sensitivities, and at most N large-scale computations using the Second-Level Adjoint Sensitivity System (2 nd -LASS) for obtaining exactly all of the second-order response sensitivities. In contradistinction, forward other methods would require ( N 2/2 + 3 N /2) large-scale computations for obtaining all of the first- and second-order sensitivities. This work also shows that constructing and solving the 2 nd -LASS requires very little additional effort beyond the construction of the 1 st -LASS needed for computing the first-order sensitivities. Solving the equations underlying the 1 st -LASS and 2 nd -LASS requires the same computational solvers as needed for solving ( i.e. , “inverting”) either the forward or the adjoint linear operators underlying the initial model. Therefore, the same computer software and “solvers” used for solving the original system of equations can also be used for solving the 1 st -LASS and the 2 nd -LASS. Since neither the 1 st -LASS nor the 2 nd -LASS involves any differentials of the operators underlying the original system, the 1 st -LASS is designated as a “ first-level ” (as opposed to a “first-order”) adjoint sensitivity system, while the 2 nd -LASS is designated as a “ second-level ” (rather than a “second-order”) adjoint sensitivity system. Mixed second-order response sensitivities involving boundary parameters may arise from all source terms of the 2 nd -LASS that involve the imprecisely known boundary parameters. Notably, the 2 nd -LASS encompasses an automatic, inherent, and independent “solution verification” mechanism of the correctness and accuracy of the 2nd-level adjoint functions needed for the efficient and exact computation of the second-order sensitivities.