Abstract
This work reviews the state-of-the-art methodologies for the deterministic sensitivity analysis of nonlinear systems and deterministic quantification of uncertainties induced in model responses by uncertainties in the model parameters. The need for computing high-order sensitivities is underscored by presenting an analytically solvable model of neutron scattering in a hydrogenous medium, for which all of the response’s relative sensitivities have the same absolute value of unity. It is shown that the wider the distribution of model parameters, the higher the order of sensitivities needed to achieve a desired level of accuracy in representing the response and in computing the response’s expectation, variance, skewness and kurtosis. This work also presents new mathematical expressions that extend to the sixth-order of the current state-of-the-art fourth-order formulas for computing fourth-order correlations among computed model response and model parameters. Another novelty presented in this work is the mathematical framework of the 3rd-Order Comprehensive Adjoint Sensitivity Analysis Methodology for Nonlinear Systems (3rd-CASAM-N), which enables the most efficient computation of the exact expressions of the 1st-, 2nd- and 3rd-order functional derivatives (“sensitivities”) of a model’s response to the underlying model parameters, including imprecisely known initial, boundary and/or interface conditions. The 2nd- and 3rd-level adjoint functions are computed using the same forward and adjoint computer solvers as used for solving the original forward and adjoint systems. Comparisons between the CPU times are also presented for an OECD/NEA reactor physics benchmark, highlighting the fact that finite-difference schemes would not only provide approximate values for the respective sensitivities (in contradistinction to the 3rd-CASAM-N, which provides exact expressions for the sensitivities) but would simply be unfeasible for computing sensitivities of an order higher than first-order. Ongoing work will generalize the 3rd-CASAM-N to a higher order while aiming to overcome the curse of dimensionality.
Highlights
The functional derivatives of the results produced by models with respect to the underlying model parameters are customarily called the model “response sensitivities”
This work reviewed the state-of-the-art methodologies for the deterministic sensitivity analysis of nonlinear systems and deterministic quantification of uncertainties induced in model responses by uncertainties in the model parameters
It was shown that the wider the distribution of model parameters, the higher the order of sensitivities needed to achieve a desired level of accuracy in representing the response and in computing the response’s expectation, variance, skewness and kurtosis
Summary
The functional derivatives of the results produced by models with respect to the underlying model parameters are customarily called the model “response sensitivities”. Cacuci [23,24,25] conceived the generally applicable “Second-Order Adjoint Sensitivity Analysis Methodology” for both linear [23] and nonlinear [24,25] systems, which enables the most efficient computation of the exact expressions of all of the second-order functional derivatives of model responses to model parameters while overcoming the curse of dimensionality. The components of the TP-dimensional column vector α ∈ RTP are considered to include imprecisely known geometrical parameters that characterize the physical system’s boundaries in the phase-space of the model’s independent variables. The computation by conventional methods of the nth-order functional derivatives (called “sensitivities” in the field of sensitivity analysis) of a response with respect to the TP-parameters it depends on would require at least O(TPn) large-scale computations. The relation provided in Equation (23) can be used to determine which order of approximation would need to be used to obtain an approximate result which would be within a predetermined error by comparison to the exact result
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