First order comprehensive adjoint sensitivity analysis methodology (1st-casam) for computing efficiently the exact response sensitivities for physical systems with imprecisely known boundaries and parameters: General theory and illustrative paradigm applications

Abstract

This work presents the novel first-order comprehensive adjoint sensitivity analysis methodology (1st-CASAM) for computing efficiently, exactly, and exhaustively, the first-order response sensitivities for nonlinear physical systems characterized by imprecisely known (uncertain) parameters model and domain boundary parameters. This novel methodology is designed for generic operator-valued model responses, treating scalar-valued responses as particular cases. The 1st-CASAM highlights the conclusion that response sensitivities to the imprecisely known domain boundaries can arise both from the definition of the system’s response as well as from the equations and boundary conditions defining the model on its imprecisely known domain. The application of this new methodology is illustrated on two paradigm models, each admitting exact closed-form expressions for the sensitivities of typical model responses to the model’s imprecisely known (uncertain) model and boundary parameters, as follows: (a) a nonlinear heat conduction equation; and (b) a neutron diffusion equation modeling a subcritical reactor with an external source. Very importantly, the diffusion equation is subject to vacuum boundary conditions on the extrapolated system boundary, thus illustrating the computation of response sensitivities to boundary parameters involving not only imprecisely known geometrical dimensions but also imprecisely known nuclear data. For the nonlinear heat conduction equation, the value of the temperature that would be measured at a point in phase-space is chosen as a representative scalar-valued response, while the temperature distribution in the entire phase-space, having uncertain boundaries, of the model’s independent variable has been chosen as a representative function-valued response. For the neutron diffusion equation, a reaction-rate within the imprecisely known domain boundaries has been chosen as a representative scalar-valued response, and the neutron flux distribution within the imprecisely known domain boundaries has been chosen as a representative function-valued response. The application of the 1st-CASAM for the operator-valued responses is illustrated using: (a) a pseudo-spectral collocation representation for the sensitivities of the temperature distribution response; and (b) a spectral Fourier representation for the sensitivities of the neutron flux distribution response. These paradigm examples highlight the details of applying the general 1st-CASAM methodology, underscoring the effects of imprecisely known domain boundaries on both the system’s state functions and responses. By enabling, in premiere, the exact computations of function-valued response sensitivities to boundary parameters and conditions, the novel adjoint sensitivity analysis methodology presented in this work enables the quantification of the effects of manufacturing tolerances on the responses of physical and engineering systems.