Random set solutions to elliptic and hyperbolic partial differential equations

The past decades have seen increasing interest in modelling uncertainty by heterogeneous methods, combining probability and interval analysis, especially for assessing parameter uncertainty in engineering models. A unifying mathematical framework admitting the combination of a wide range of such methods is the theory of random sets, describing input and output of a structural model by set-valued random variables. The purpose of this paper is to highlight the mathematics behind this approach. The modelling and computational implications are discussed and demonstrated with the help of prototypical partial differential equations-a scalar elliptic equation from elastostatics and hyperbolic systems arising in elastodynamics.

Automated summary

In recent decades, there has been increasing interest in combining probability and interval analysis to model parameter uncertainty in engineering models. The theory of random sets provides a unified mathematical framework for describing the input and output of structural models using set-valued random variables. This paper aims to highlight the mathematical principles behind this approach and demonstrate its modeling and computational implications through prototypical partial differential equations in elastostatics and elastodynamics.