QoI

QoI stands for Quantity of Interest. In mathematical modeling and uncertainty quantification, it denotes the specific output of a model that is of interest for analysis, prediction, or decision-making. It is obtained by applying a QoI operator Q to the model state or inputs: y = Q(u, θ), or more simply y = f(θ) if the state is parameterized. The QoI can be scalar or vector-valued and may refer to a single feature or a set of features; common examples include a pointwise displacement, an integral quantity like total energy, a maximum stress, a pollutant concentration in a region, or a time-to-failure.

Choice of QoI is problem-specific and guides both the computational workflow and the interpretation of results. It influences how input uncertainties are propagated (yielding a distribution for the QoI), how model error is assessed, and how experimental data are used for calibration. In Bayesian inference and data assimilation, the QoI often corresponds to observed data that the model is conditioned on; the likelihood is built from the discrepancy between predicted QoI and measurements. In forward UQ, one propagates uncertainty in inputs to the QoI to quantify predictive uncertainty. In inverse problems, the goal is to infer inputs or parameters that produce QoI-consistent outputs.

QoI selection also affects the use of surrogate models, reduced-order models, and sensitivity analysis. By focusing on the QoI, practitioners can perform efficient design of experiments, compute sensitivity indices, and manage computational cost.

Common examples include structural engineering (displacement at a point, maximum stress), fluids (lift or drag coefficients), environmental modeling (mean pollutant concentration), and reliability studies (time to threshold). The term QoI is distinct from information-theoretic quantities and should be interpreted within modeling and decision contexts.