Random set solutions to elliptic and hyperbolic partial differential equations

TL;DR

This paper explores the use of random set theory as a unifying framework for modeling and computing uncertainty in elliptic and hyperbolic partial differential equations, with applications in engineering models.

Contribution

It provides a mathematical foundation for random set methods in PDEs and discusses their computational implications through practical examples.

Findings

Random set theory effectively models parameter uncertainty in PDEs.

The approach unifies probability and interval analysis methods.

Demonstrations include elliptic and hyperbolic PDE applications.

Abstract

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.