Toward Definition of Systematic Criteria for the Comparison of Verified Solvers for Initial Value Problems

Abstract

Solving initial value problems for ordinary differential equations is a common task in many disciplines. Over the last decades, several different verified techniques have been developed to compute enclosures of the exact result numerically. The obtained bounds are guaranteed to contain the corresponding solution to the initial value problem. Ideally, we want to calculate tight enclosures over sufficiently long time intervals for systems with uncertainties in both the initial conditions and system parameters. However, the existing solvers are not always equal in attaining this goal. On the one hand, the quality of the obtained results depends strongly on the types of ordinary differential equations that describe a given dynamical system. On the other hand, a great influence of the considered uncertainties can be observed. Our general aim is to provide assistance in choosing an appropriate verified initial value problem solver with its most suitable ‘tuning parameters’ for the application at hand. In this paper, we make first steps toward setting up a framework for the fair comparison of the different approaches. We suggest criteria, benchmark scenarios, and typical applications which can be used for the quantification of the efficiency of verified initial value problem solvers.