The Worst Scenario Method: A Red Thread Running Through Various Approaches to Problems with Uncertain Input Data

Three ingredients constitute mathematical models dependent on parameters whose value is uncertain: a convex compact set Uad of admissible parameters a, a state problem A(a)u = f with the a-dependent state u ≡ u(a), and a continuous quantity of interest Ψ (a) = Φ(a,u(a)). In the worst scenario method, the maximum of Ψ over Uad is identified. If the minimum is also found, then the range Ψ |Uad is determined. Solving this range determination problem is also the fundamental step in other methods analyzing the propagation of uncertainty through a model; namely in the Dempster-Shafer theory and fuzzy set theory. In this paper, elements of these non-stochastic approaches to uncertainty modeling are presented with the emphasis on uncertain functions appearing in differential equations as their coefficients.