Second-order adjoint sensitivity analysis methodology (2nd-ASAM) for computing exactly and efficiently first- and second-order sensitivities in large-scale linear systems: I. Computational methodology

Abstract

This work presents the second-order forward and adjoint sensitivity analysis procedures (SO-FSAP and SO-ASAP) for computing exactly and efficiently the second-order functional derivatives of physical (engineering, biological, etc.) system responses (i.e., "system performance parameters") to the system's model parameters. The definition of "system parameters" used in this work includes all computational input data, correlations, initial and/or boundary conditions, etc. For a physical system comprising N α parameters and r N responses, we note that the SO-FSAP requires a total of ( )computations for obtaining all of the first-and second-order sensitivities, for all r N system responses. On the other hand, the SO-ASAP requires a total of ( )large-scale computations for obtaining all of the first-and second-order sensitivities, for one functionaltype system responses. Therefore, the SO-FSAP should be used when r N N α  , while the SO-ASAP should be used when r N N α  . The original SO-ASAP presented in this work should enable the hitherto very difficult, if not intractable, exact computation of all of the second-order response sensitivities (i.e., functional Gateaux-derivatives) for large-systems involving many parameters, as usually encountered in practice. Very importantly, the implementation of the SO-ASAP requires very little additional effort beyond the construction of the adjoint sensitivity system needed for computing the first-order sensitivities.KEYWORDS: second-order adjoint sensitivity analysis procedure (SO-ASAP); exact and efficient computation of first-and second-order functional derivatives; large-scale systems.2

Predictive Modeling of a Paradigm Mechanical Cooling Tower Model: II. Optimal Best-Estimate Results with Reduced Predicted Uncertainties

“…The adjoint sensitivity analysis methodology used in PART I [1] for computing exactly and efficiently the 1 st -order response sensitivities to model parameters has been recently extended to computing efficiently and exactly the 2nd-order response sensitivities to parameters for linear [13] and nonlinear [14] large-scale systems. As has been shown in [15][16][17][18], the 2nd-order response sensitivities have the following major impacts on the computed moments of the response distribution: (a) they cause the "expected value of the response" to differ from the "computed nominal value of the response"; and (b) they contribute decisively to causing asymmetries in the response distribution.…”

Sensitivity analysis applied to a variational data assimilation of a simulated pollution transport problem

“…onˆand ‰, the solution of the system (37)-(42) can be obtained by a forward integration of Equations (39)-(40) followed by a backward integration of Equations (37)- (38). It is shown in [51] that this process defines 2 .0/ and ƒ 2 .0/ as…”

“…Additional applications of the adjoint approach of the sensitivity can be found in [29][30][31][32][33][34][35]. The general theory and application of the sensitivity analysis using adjoint and non-adjoint methods can be found in [36][37][38][39] and references therein.…”

On the use of the continuous adjoint method to compute nongeometric sensitivities

“…The first one discusses the necessary and sufficient conditions for the existence and uniqueness of adjoint operators, whereas the second extends the scope of the adjoint formalism to a larger variety of responses, which includes general operators. It is worth adding that, along with his collaborators, the author went on to publish a number of relevant references on a wide range of applications of the method …”

“…It is worth adding that, along with his collaborators, the author went on to publish a number of relevant references on a wide range of applications of the method. 5,[17][18][19][20][21][22][23][24][25] Two of the above works by Cacuci are especially relevant to our purposes. Cacuci et al 14 devise a sequence of formal steps to construct the adjoint problem.…”