Sensitivity analysis for chemical models.

Chemists routinely create models of reaction systems to understand reaction mechanisms, kinetic properties, process yields under various operating conditions, or the impact of chemicals on man and the environment. As opposed to concise physical laws, these models are attempts to mimic the system by hypothesizing, extracting, and encoding system features (e.g. a potentially relevant reaction pathway versus another plausible one), within a process that can hardly be formalized scientifically.1 The model will hopefully help to corroborate or falsify a given description of reality, e.g. by validating a reaction scheme for a photochemical process in the atmosphere, and possibly to influence it, e.g. by allowing the identification of optimal operating conditions for an industrial process or suggesting mitigating strategies for an undesired environmental impact. These models are customarily built in the presence of uncertainties of various levels, in the pathway, in the order of the kinetics associated to the pathway, in the numerical value of the kinetic and thermodynamic constants for that pathway, and so on. Propagating via the model all these uncertainties onto the model output of interest, e.g. the yield of a process, is the job of uncertainty analysis. Determining the strength of the relation between a given uncertain input and the output is the job of sensitivity analysis.2 Mathematical sensitivities (in the form of model output derivatives) are a straightforward implementation of this sensitivity concept. If the model output of interest is Y, its sensitivity to an input factor Xi is simply Y′ Xi ) ∂Y/∂Xi. This measure tells how sensitive the output is to a perturbation of the input. If a measure independent from the units used for Y and Xi is needed, SXi r ) (Xh i/Yh )(∂Y/∂Xi) can be used, where Xh i is the nominal (or central, if a range is known) value of factor Xi and Yh is the value taken by Y when all input factors are at their nominal value. If factors are uncertain within a known or hypothesized range, then the measure SXi σ ) (σXi/σY)(∂Y/∂Xi) can be of use, where the standard deviations σXi, σY are uncertainty analysis’ input and output, respectively, in the sense that σXi comes from the available knowledge on Xi, while σY must be inferred using the model. These sensitivity measures can be efficiently computed by an array of techniques, ranging from automated differentiation (where the computer program that implements the model is modified so that the sensitivities are computed with a modicum of extra execution time3) to direct methods (where the differential equations describing the model are solved directly in terms of species concentrations and their derivatives4). There is a vast amount of literature on these sensitivity measures,5-11 which shall be referred to as local in the following. The majority of sensitivity analyses met with in chemistry and physics are local and derivative-based. Local sensitivities are useful for a variety of applications, such as the solution of inverse problems, e.g. relating macroscopic observables of a system, such as kinetic constants, to the quantum mechanics properties of the system,6 or the analysis of runaway and parametric sensitivity of various types of chemical reactors.8 Contexts where local sensitivity has been widely used are as follows: (1) to understand the reaction path, mechanism, or rate-determining steps in a detailed kinetic model with a large number of elementary reactions, e.g. in photochemistry or in combustion chemistry,4,7,9 (see ref 12 for an alternative approach in this context), (2) to extract important elementary reactions from a complex kinetic model to obtain a reduced model (e.g. a minimal reaction scheme) with equivalent predictive power7 or to select important reactions for further analysis,13,14 (3) to estimate the output of a * Corresponding author. Telephone: +39 0332 789686. Fax: +39 0332 785733. E-mail: [email protected]. 2811 Chem. Rev. 2005, 105, 2811−2827