Abstract
This work presents the “Second-Order Comprehensive Adjoint Sensitivity Analysis Methodology (2nd-CASAM)” for the efficient and exact computation of 1st- and 2nd-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 2nd-CASAM requires a single large-scale computation using the First-Level Adjoint Sensitivity System (1st-LASS) for obtaining all of the first-order response sensitivities, and at most N large-scale computations using the Second-Level Adjoint Sensitivity System (2nd-LASS) for obtaining exactly all of the second-order response sensitivities. In contradistinction, forward other methods would require (N2/2 + 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 2nd-LASS requires very little additional effort beyond the construction of the 1st-LASS needed for computing the first-order sensitivities. Solving the equations underlying the 1st-LASS and 2nd-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 1st-LASS and the 2nd-LASS. Since neither the 1st-LASS nor the 2nd-LASS involves any differentials of the operators underlying the original system, the 1st-LASS is designated as a “first-level” (as opposed to a “first-order”) adjoint sensitivity system, while the 2nd-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 2nd-LASS that involve the imprecisely known boundary parameters. Notably, the 2nd-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.
Highlights
The earliest use of adjoint operators for computing exactly and efficiently the first-order sensitivities of responses of a large-scale linear system comprising many parameters has appeared in the report by Wigner [2] on his work on the “nuclear pile” using the linear neutron transport or diffusion equations
The aim of this work is to present the novel “Second-Order Comprehensive Adjoint Sensitivity Analysis Methodology (2nd-CASAM),” which has the following features that generalize and extend all previously published works on this topic: 2) The system response considered within the 2nd-CASAM framework is an operator-valued response that depends on both the forward and adjoint state-functions
Functional-valued responses are subsumed as particular cases
Summary
The earliest use of adjoint operators for computing exactly and efficiently the first-order sensitivities of responses of a large-scale linear system comprising many parameters has appeared in the report by Wigner [2] on his work on the “nuclear pile” (i.e., nuclear reactor) using the linear neutron transport or diffusion equations. Cacuci [2] [3] conceived the rigorous 1st-order adjoint sensitivity analysis methodology for generic large-scale nonlinear systems involving operator responses, comprising functional-type responses as particular cases. First-order perturbation theory in conjunction with adjoint operators was subsequently used [10]-[15], either formally or in conjunction with variational formulations, to obtain approximate first-order sensitivities to boundary parameters of responses that were linear functionals (or ratios thereof) of the neutron flux in the context of linear neutron diffusion or neutron transport problems. None of the above works considered responses that are simultaneously functions of the forward and adjoint state functions
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