Rigorous Inner Approximation of the Range of Functions

A basic problem of interval analysis is the computation of a superset of the image of an interval by a function, called an outer enclosure. Here we consider the computation of an inner enclosure, which is a subset of the image. Inner approximations are harder than the outer ones in general: proving that a box is inside the image is equivalent to proving existence of solutions for a collection of systems of equations. Based on this remark, a new construction of the inner approximation is proposed that is particularly efficient for small domains. Then, it is shown than one can apply these ideas in the context of ordinary differential equations, hence providing some tools of potential interest for the theory of shadowing in dynamical systems.