TABLE OF CONTENTS Michael Oberguggenberger
Institut f r Technische Mathematik, Geometrie und Bauinformatik, Universit t Innsbruck
Summary. This article discusses various mathematical theories that have been put forth as tools for modelling uncertainty, among them probability, interval arithmetic, random sets, and fuzzy sets. After recalling the definitions, we stress their interpretations (semantics), axioms, interrelations as well as numerical procedures and demonstrate how the concepts are applied in practice.
An adequate understanding of the influence of input parameter variability on the output of engineering computations requires that the uncertainty itself is captured in mathematical terms. This section serves to present a number of mathematical approaches, focused around generalizations of probability theory, that are able to formalize the state of knowledge about parameter uncertainty.
Scientific modelling in engineering has to deal with three facets. First, there is reality (with soils, materials etc.). Second, there is the model of reality (formulated in mathematical terms and containing physical laws and constitutive equations). Third, correspondence rules (prescribing how to translate one into the other) are needed. It is important to note that theory is prior to observation [77, Sect. 30]. In this context this simply means that the physical model decides what are the state variables and what are the material constants, the parameters to be observed. Once this has been decided, the values of the parameters have to be determined from information extracted from the real world and will serve as input in the physical model. This input in turn is processed numerically and should...
