Introductory remarks: Mathematical models of uncertainty

Abstract As indicated in the preface to this volume, it is essential in the engineering modelling process that the uncertainty of input parameters is captured and formulated in mathematical terms, so that it can be propagated through numerical computations. This is a necessary requirement for a proper assessment of the output variability. This introduction serves to discuss a number of mathematical approaches, focused around generalizations of probability theory, that are able to formalize the state of knowledge about parameter uncertainty. It is a very brief introduction to the realm of what is now commonly termed imprecise probabilities [2,4]. The papers collected in this volume all use one or more of these approaches; the purpose of this introduction is to point out some unifying ideas, remarks on interpretations (correspondence rules between model and reality, or semantics), and rudiments of a hierarchical classification. To keep the presentation simple, we shall consider a single input parameter called (upper case) A , while the specific values it may take (realizations) will be denoted by lower case letters. Further, the computational engineering model produces an output as a function F ( A ) of the input A . Uncertainty of multivariate models, correlations and stochastic field variables have their counterparts in all the approaches discussed below; for this we refer the reader to the literature cited at the end.