Asymmetric Metric: An Application to Dealing with Uncertainty

The uncertainty in mathematical models is often represented via set-valued data, parameters or solutions. We propose a new approach for dealing with such uncertainty, which combines features of validated computing (wrapping the set via a set of computer representable type, e.g., intervals, zonotopes, ellipsoids) and point approximation accompanied by relevant error analysis. More precisely, we consider approximation by a set which is not necessarily an enclosure. The mathematical theory is based on the theory of asymmetric metric spaces, where the metric gives an estimation of the error, while the order induced by the metric provides means for estimating the size of the approximating set.