The nth-Order Comprehensive Adjoint Sensitivity Analysis Methodology for Nonlinear Systems (nth-CASAM-N): Mathematical Framework

Abstract

This work presents the nth-Order Comprehensive Adjoint Sensitivity Analysis Methodology for Nonlinear Systems (nth-CASAM-N), which enables the most efficient computation of exactly determined expressions of arbitrarily high-order sensitivities of generic nonlinear system responses with respect to model parameters, uncertain boundaries, and internal interfaces in the model’s phase space. The mathematical framework underlying the nth-CASAM-N is proven to be correct by using mathematical induction. The nth-CASAM-N is formulated in linearly increasing higher-dimensional Hilbert spaces—as opposed to exponentially increasing parameter-dimensional spaces—thus overcoming the curse of dimensionality in sensitivity analysis of nonlinear systems.

Illustrative Application of the nth-Order Comprehensive Adjoint Sensitivity Analysis Methodology for Nonlinear Systems to the Nordheim–Fuchs Reactor Dynamics/Safety Model

“…[δϕ(t), δT(t), δW(t)] † , δϕ(t) ϕ(t) − ϕ 0 (t), δT(t) T(t) − T 0 (t), δW(t) W(t) − W 0 (t), in the state functions. The first-order G-differential δR u 0 ; α 0 δu; δα is defined as follows: δR u 0 ; α 0 ; δv (1) ; δα d dε t f 0 F u 0 + εv (1) ; α 0 + εδα dt ε=0 δR u 0 ; α 0 ; δα dir + δR u 0 ; α 0 ; δv (1) ind…”

“…The direct-effect term can be evaluated immediately, but the indirect-effect term can be evaluated only after having obtained the variations v (1) (t)…”

“…[δϕ(t), δT(t), δW(t)] † . For given variations δα α − α 0 , the variations v (1) (t) [δϕ(t), δT(t), δW(t)] † are the solutions of the 1st-Level Variational Sensitivity System (1st-LVSS), which is obtained by determining the first-order G-differentials of the equations underlying the model, in which it is convenient to use Equation (9) instead of Equation (1). Applying the definition of the G-differential to these equations yields the following relations:…”

“…The nth-Order Comprehensive Adjoint Sensitivity Analysis Methodology for Nonlinear Systems (nth-CASAM-N) has been presented in [1]. The nth-CASAM-N methodology enables the most efficient computation of exactly-determined expressions of arbitrarily high-order sensitivities of results of interest (called "responses") produced by models that are nonlinear in their underlying state functions, with respect to the model's parameters, uncertain boundaries, and internal interfaces in the model's phase-space.…”

“…Furthermore, since this reducedorder model was shown in [6] to possess large sensitivities of its state functions to its model parameters, it would be prudent to quantify the effects of higher-order sensitivities on the responses of such dynamical models to establish the actual importance of higherorder sensitivities quantitatively. The development of the nth-CASAM-N [1] makes it now possible to quantify the effects of arbitrarily-high order of response sensitivities for nonlinear models. The application of the nth-CASAM-N to a nonlinear dynamical model will be illustrated in this work by considering a well-known paradigm model that describes a short-time self-limiting power excursion in a nuclear reactor system having a negative temperature coefficient in which a large amount of reactivity is suddenly inserted, either intentionally or by accident.…”