Abstract
This work continues the illustrative application of the “Second Order Comprehensive Adjoint Sensitivity Analysis Methodology” (2nd-CASAM) to a benchmark mathematical model that can simulate the evolution and/or transmission of particles in a heterogeneous medium. The model response considered in this work is a reaction-rate detector response, which provides the average interactions of particles with the respective detector or, alternatively, the time-average of the concentration of a mixture of substances in a medium. The definition of this model response includes both uncertain boundary points of the benchmark, thereby providing both direct and indirect contributions to the response sensitivities stemming from the boundaries. The exact expressions for the 1st- and 2nd-order response sensitivities to the boundary and model parameters obtained in this work can serve as stringent benchmarks for inter-comparing the performances of all (deterministic and statistical) sensitivity analysis methods.
Highlights
This work continues the illustrative application of the “Second Order Comprehensive Adjoint Sensitivity Analysis Methodology” (2nd-CASAM) to a benchmark mathematical model that can simulate the evolution and/or transmission of particles in a heterogeneous medium
The results obtained by applying the general 2nd-CASAM presented in [1] to the paradigm evolution/transmission benchmark analyzed in this work indicate the following major characteristics of this powerful methodology for computing exactly and efficiently the 1st- and 2nd-order sensitivities of model responses with respect to model and boundary parameters: 1) For a model comprising Nα distinct but uncertain model parameters and Nβ distinct but uncertain distinct boundary parameters, a single adjoint computation, to solve the 1st-LASS, is necessary for computing exactly all of the
For each 1st-order sensitivity, the solution of each of the 2nd-LASS is at most a two-component 2nd-level adjoint sensitivity function of the form ψ
Summary
This work continues to illustrate the application of the general second-order adjoint sensitivity analysis methodology (2nd-CASAM) presented in [1] by us-. Adding Equations (21) and (15), and identifying the quantities that multiply the respective parameter variations yields the following expressions for the first-order sensitivities of R2 ( ρ;α , β ) in terms of the first-level adjoint function θ (1) (t ) :. As indicated in Equations (25)-(27), the first-order sensitivities of R2 ( ρ;α , β ) with respect to the model parameters ρin , σi , and ni stem exclusively from the indirect effect term (δ R2 )ind and can be computed after having obtained the first-level adjoint function θ (1) (t ) by solving the 1st-LASS, namely Equations (22) and (23).
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