The nth-order features adjoint sensitivity analysis methodology for response-coupled forward/adjoint linear systems (nth-FASAM-L): I. mathematical framework

Abstract

This work presents the mathematical/theoretical framework of the “nth-Order Feature Adjoint Sensitivity Analysis Methodology for Response-Coupled Forward/Adjoint Linear Systems” (abbreviated as “nth-FASAM-L”), which enables the most efficient computation of exactly obtained mathematical expressions of arbitrarily-high-order (nth-order) sensitivities of a generic system response with respect to all of the parameters (including boundary and initial conditions) underlying the respective forward/adjoint systems. Responses of linear models can depend simultaneously on both the forward and the adjoint state functions. This is in contradistinction to responses for nonlinear systems, which can depend only on the forward state functions since nonlinear operators do not admit bona-fide adjoint operators. Among the best-known model responses that depend simultaneously on both the forward and adjoint state functions are Lagrangians used for system optimization, the Schwinger and Roussopoulos functionals for analyzing reaction rates and ratios thereof, and the Rayleigh quotient for analyzing eigenvalues and/or separation constants. The sensitivity analysis of such responses makes it necessary to treat linear models/systems in their own right, rather than treating them just as particular cases of nonlinear systems. The unparalleled efficiency and accuracy of the nth-FASAM-L methodology stems from the maximal reduction of the number of adjoint computations (which are “large-scale” computations) for computing high-order sensitivities, since the number of large-scale computations when applying the nth-FASAM-N methodology is proportional to the number of model features as opposed to the number of model parameters (which are considerably more than the number of features). The mathematical framework underlying the nth-FASAM-L is developed in linearly increasing higher-dimensional Hilbert spaces, as opposed to the exponentially increasing “parameter-dimensional” spaces in which response sensitivities are computed by other methods (statistical, finite differences, etc.), thus providing the basis for overcoming the curse of dimensionality in sensitivity analysis and all other fields (uncertainty quantification, predictive modeling, etc.) which need such sensitivities.