Abstract
This work illustrates the application of the “Second Order Comprehensive Adjoint Sensitivity Analysis Methodology” (2nd-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 1st- and 2nd-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 2nd-CASAM.
Highlights
This work illustrates the application of the “Second Order Comprehensive Adjoint Sensitivity Analysis Methodology” (2nd-CASAM) to a mathematical model that can simulate the evolution and/or transmission of particles in a heterogeneous medium
These closed-form expressions can be used to verify the numerical results of production and/or commercial software, e.g., particle transport codes. This paradigm model comprises many uncertain parameters which have relative sensitivities of identical magnitudes. This paradigm model could serve as a stringent benchmark for inter-comparing the performances of all deterministic and statistical sensitivity analysis methods, including the 2nd-CASAM
Dology presented in [1] is illustrated in this work by means of a simple mathematical model which expresses a conservation law of the model’s state function. This paradigm model is representative of transmission of particles and/or radiation through materials [2] [3], chemical kinetics processes [4] [5], radioactive decay modeled by the Bateman equation, etc
Summary
The variation δρ (t ) , of the state function ρ (t ) , which appears in Equation (14) is the solution of the following First-Level Forward Sensitivity System (1st-LFSS) obtained by G-differentiating Equations (1) and (2) around the nominal parameter values: d δρ Since neither the direct-effect nor the indirect-effect terms depend on the variation ∂βu , it follows that It is evident from Equations (23) through (27) that the sensitivities of the response R1 ( ρ;α , β ) can be computed by fast quadrature methods applied to the integrals appearing in these expressions, after the 1st-level adjoint function ψ (1) (t ) has been obtained by solving once the 1st-LASS, which comprises Equations (21) and (22).
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