QoIs

A quantity of interest (QoI) is a scalar or vector value derived from a mathematical model that represents the outcome or feature of the system deemed most relevant for analysis, decision making, or further computation. In modeling contexts, the QoI is the observable or physically meaningful quantity that results from the state of the system and the model inputs or parameters.

Mathematically, a QoI is a mapping from the model state and inputs to a real-valued or vector-valued quantity. If the model depends on uncertain inputs, the QoI becomes a random variable or random vector whose distribution reflects uncertainty propagation through the model. The QoI is often the object of study in forward uncertainty quantification, inverse problems, optimization, and model reduction.

Common examples include scalar QoIs such as lift or drag coefficients in aerodynamics, peak temperature at a location, maximum stress in a structure, or time to failure in reliability analyses. Vector QoIs can consist of an entire field of values, such as temperature or pressure distributions at selected locations, or multiple components of a velocity field. The choice of QoI is problem-dependent and should reflect what is measurable, controllable, or decision-relevant in practice.

Properties of a good QoI include physical interpretability, stability with respect to input variations and discretization, and computational tractability. The QoI should capture the essential behavior of the system while reducing dimensionality where possible. In practice, QoIs are used to guide optimization, quantify uncertainty, calibrate models against data, and compare alternative models or designs.