Advances in High-Order Sensitivity Analysis

The high-order sensitivities of model responses with respect to model parameters are notoriously difficult to compute for large-scale models involving many parameters. The neglect of higher-order response sensitivities leads to substantial errors in predicting the moments (expectation, variance, skewness, kurtosis) of the model responseâs distribution in the phase-space of model parameters.

The mathematical/computational models of physical systems comprise parameters, independent and dependent variables. Since the physical processes themselves are seldom known precisely and since most of the modelâs parameters stem from experimental procedures that are also subject to imprecision and/or uncertainties, the results predicted by these models are also imprecise, being affected by the uncertainties underlying the respective model.

In the particular case of sensitivity analysis using conventional methods, the number of large-scale computations increases exponentially in the phase-space of âmodel parametersâ as the order of sensitivities increases. For large-scale models involving many parameters, even the first-order sensitivities are computationally very expensive to determine accurately by conventional methods. Furthermore, the âcurse of dimensionalityâ prohibits the accurate computation of higher-order sensitivities by conventional methods.

The âadjoint methodâ of sensitivity analysis conceived by Cacuci (1981a, b) is the most efficient method for computing exactly first-order sensitivities, since it requires a single large-scale (adjoint) computation, independently of the number of model parameters. The mathematical underpinnings of the first-order adjoint sensitivity analysis, along with representative applications to large-scale systems, are detailed in Sensitivity and Uncertainty Analysis, Volume: Theory (Cacuci, CRC Press, 2003), and Sensitivity and Uncertainty Analysis, Volume II: Applications to Large-Scale Systems (Cacuci, et al., CRC Press, 2005). The most important uses of these sensitivities are for data assimilation and model calibration, as illustrated in the book Computational Methods for Data Evaluation and Assimilation (Cacuci, et al., CRC Press, 2014).

Additional books by Dan G. Cacuci, all published by CRC Press: The Second-Order Adjoint Sensitivity Analysis Methodology (2018) and Advances in High-Order Predictive Modeling Methodologies and Illustrative Problems (2025).