Uncertainty quantification in the numerical solution of coupled systems by involutive completion

Abstract

We address the issue of epistemic uncertainty quantification in the context of constrained differential systems. To illustrate our approach we have chosen a certain chromatographic adsorption model which is a coupled system of partial differential, ordinary differential and algebraic equations. The difficulty in solving this type of a system is that typically certain unknowns lack a natural time evolution equation. The standard approach in such cases is to devise specific numerical schemes which somehow try to take into account the implicit structure of the system. In our approach, we complete the system by finding the appropriate missing evolution equations. This makes the system overdetermined and more complicated in some way but, on the other hand, the completed system provides extra information useful for error estimation and uncertainty quantification. We will also show that reducing the epistemic uncertainties also leads to better estimations of aleatory uncertainties.