Computation of High-Order Sensitivities of Model Responses to Model Parameters—II: Introducing the Second-Order Adjoint Sensitivity Analysis Methodology for Computing Response Sensitivities to Functions/Features of Parameters

Abstract

This work introduces a new methodology, which generalizes the extant second-order adjoint sensitivity analysis methodology for computing sensitivities of model responses to primary model parameters. This new methodology enables the computation, with unparalleled efficiency, of second-order sensitivities of responses to functions of uncertain model parameters, including uncertain boundaries and internal interfaces, for linear and/or nonlinear models. Such functions of primary model parameters customarily describe characteristic “features” of the system under consideration, including correlations modeling material properties, flow regimes, etc. The number of such “feature” functions is considerably smaller than the total number of primary model parameters. By enabling the computations of exact expressions of second-order sensitivities of model responses to model “features”, the number of required large-scale adjoint computations is greatly reduced. As shown in this work, obtaining the first- and second-order sensitivities to the primary model parameters from the corresponding response sensitivities to the feature functions can be performed analytically, thereby involving just the respective function/feature of parameters rather than the entire model. By replacing large-scale computations involving the model with relatively trivial computations involving just the feature functions, this new second-order adjoint sensitivity analysis methodology reaches unsurpassed efficiency. The applicability and unparalleled efficiency of this “2nd-Order Function/Feature Adjoint Sensitivity Analysis Methodology” (2nd-FASAM) is illustrated using a paradigm particle transport model that involves feature functions of many parameters, while admitting closed-form analytic solutions. Ongoing work will generalize the mathematical framework of the 2nd-FASAM to enable the computation of arbitrarily high-order sensitivities of model responses to functions/features of model parameters.

Introducing the n<sup>th</sup>-Order Features Adjoint Sensitivity Analysis Methodology for Nonlinear Systems (n<sup>th</sup>-FASAM-N): I. Mathematical Framework

“…3) For computing the exact expressions of the second-order response sensitivities with respect to the primary model's parameters, the 2 nd -FASAM-N methodology [32] requires as many large-scale "adjoint" computations as there are "feature functions of parameters"…”

“…Comparing the mathematical framework of the 1 st -FASAM-N [32] to the framework of the 1 st -CASAM-N [31] suggests that the component "features"…”

“…2) Prove that the general pattern underlying the nth-FASAM-N is valid for the lowest values of n, i.e., n = 1, which should reduce to the mathematical framework underlying the 1st-FASAM-N, which is reproduced from Cacuci [32] in Appendix B.…”

The Second-Order Features Adjoint Sensitivity Analysis Methodology for Response-Coupled Forward/Adjoint Linear Systems (2nd-FASAM-L): Mathematical Framework and Illustrative Application to an Energy System

“…Implement the boundary conditions represented by Equation (31) into Equation (40) and eliminate the remaining unknown boundary-values of the function v (1) (2; x) from the expression of the bilinear concomitant P (1) v (1) (2; x); a (1) (2; x); f; δf α 0 by selecting appropriate boundary conditions for the function a (1) (2; x) ≜ a (1)…”

“…Recently, Cacuci [31] has introduced the "Second-Order Function/Feature Adjoint Sensitivity Analysis Methodology for Nonlinear Systems" (2nd-FASAM-N), which enables a considerable reduction (by comparison to the 2nd-CASAM-N) in the number of largescale computations needed to compute the second-order sensitivities of a model response with respect to the model parameters, thereby becoming the most efficient methodology known for computing second-order sensitivities exactly. Paralleling the construction of the 2nd-FASAM-N, this work introduces the "First-and Second-Order Function/Feature Adjoint Sensitivity Analysis Methodology for Response-Coupled Adjoint/Forward Linear Systems" (1st and 2nd-FASAM-L).…”