From intervals to domains: towards a general description of validated uncertainty, with potential applications to geospatial and meteorological data

Abstract

When physical quantities xi are numbers, then the corresponding measurement accuracy can be usually represented in interval terms, and interval computations can be used to estimate the resulting uncertainty in y = f (x1,...,xn). In some practical problems, we are interested in more complex structures such as functions, operators, etc. Examples: we may be interested in how the material strain depends on the applied stress, or in how a physical quantity such as temperature or velocity of sound depends on a 3-D point. For many such structures, there are ways to represent uncertainty, but usually, for each new structure, we have to perform a lot of complex analysis from scratch. It is desirable to come up with a general methodology that would automatically produce a natural description of validated uncertainty for all physically interesting situations (or at least for as many such situations as possible). In this paper, we describe the foundations for such a methodology; it turns out that this problem naturally leads to the technique of domains first introduced by D. Scott in the 1970s. In addition to general domain techniques, we also describe applications to geospatial and meteorological data.