First-Order Comprehensive Adjoint Method for Computing Operator-Valued Response Sensitivities to Imprecisely Known Parameters, Internal Interfaces and Boundaries of Coupled Nonlinear Systems: I. Mathematical Framework

Abstract

This work presents the first-order comprehensive adjoint sensitivity analysis methodology (1st-CASAM) for computing efficiently the first-order sensitivities (i.e., functional derivatives) of operator-valued responses (i.e., model results) of general models of coupled nonlinear physical systems characterized by imprecisely known or and/or uncertain parameters, external boundaries, and internal interfaces between the coupled systems. The explicit mathematical formalism developed within the 1st-CASAM for computing the first-order sensitivities of operator-valued response to uncertain internal interfaces and external boundaries in the models’ phase–space enables this methodology to generalize all of the previously published methodologies for computing first-order response sensitivities. The computational resources needed for using forward versus adjoint operators in conjunction with spectral versus collocation methods for computing the response sensitivities are analyzed in detail. By enabling the exact computations of operator-valued response sensitivities to internal interfaces and external boundary parameters and conditions, the 1st-CASAM presented in this work makes it possible, inter alia, to quantify the effects of manufacturing tolerances on operator-valued responses of physical and engineering systems.