Illustrative Application of the 2<sup>nd</sup>-Order Adjoint Sensitivity Analysis Methodology to a Paradigm Linear Evolution/Transmission Model: Point-Detector Response

Abstract

This work illustrates the application of the "Second Order Comprehensive Adjoint Sensitivity Analysis Methodology" (2 nd-CASAM) to a mathematical model that can simulate the evolution and/or transmission of particles in a heterogeneous medium. The model response is the value of the model's state function (particle concentration or particle flux) at a point in phase-space, which would simulate a pointwise measurement of the respective state function. This paradigm model admits exact closed-form expressions for all of the 1 stand 2 nd-order response sensitivities to the model's uncertain parameters and domain boundaries. These closed-form expressions can be used to verify the numerical results of production and/or commercial software, e.g., particle transport codes. Furthermore, this paradigm model comprises many uncertain parameters which have relative sensitivities of identical magnitudes. Therefore, this paradigm model could serve as a stringent benchmark for inter-comparing the performances of all deterministic and statistical sensitivity analysis methods, including the 2 nd-CASAM.

Fourth-Order Comprehensive Adjoint Sensitivity Analysis (4th-CASAM) of Response-Coupled Linear Forward/Adjoint Systems: I. Theoretical Framework

“…where S[ϕ(x), ψ(x); α] denotes a suitably Gateaux-(G-) differentiable function of the indicated arguments. Specific examples that illustrate the effects of the first-and second-order sensitivities of responses of the form shown in Equations ( 14)-( 20) to imprecisely known model and domain-boundary parameters are provided in [25][26][27][28][29][30][31][32][33][34].…”

Illustrative Application of the 2<sup>nd</sup>-Order Adjoint Sensitivity Analysis Methodology to a Paradigm Linear Evolution/Transmission Model: Reaction-Rate Detector Response

“…As has been discussed in [2], Equations (1) and (2) An important typical response of interest for the physical problem modeled by Equations (1) and (2) is a "reaction rate" detector response, which will be denoted as…”

“…; , R ρ α β can be derived following the same procedure as outlined above, and as also outlined in [2] since these derivations would not illustrate any new concepts, they will not be presented in this work.…”

“…This work continues to illustrate the application of the general second-order adjoint sensitivity analysis methodology (2 nd -CASAM) presented in [1] by us-ing the evolution/transmission mathematical benchmark model introduced in [2], but considering a "reaction-rate" detector response, as opposed to the "point-detector" response considered in [2]. As in [2], the mathematical model considered in this work could represent [3] [4] the time-evolution of the concentration of a substance in a homogeneous mixture of materials or, alternatively, it could represent [4] [5] [6] the transmission/attenuation of the flux of uncollided particles (e.g., photons) travelling through a one-dimensional homogenized multi-material slab of imprecisely known thickness.…”

Second-Order Adjoint Sensitivity Analysis Methodology for Computing Exactly Response Sensitivities to Uncertain Parameters and Boundaries of Linear Systems: Mathematical Framework

“…Section 4 offers concluding remarks. An accompanying work [18] presents an illustrative application of the general 2 nd -CASAM methodology to a paradigm model of generic particle/radiation transmission and/or evolution which has fundamental applications in many fields, including nuclear engineering (e.g., radiation detection, chemical reprocessing of spent reactor fuel, radioactive decay, etc. ).…”