Reducing the wrapping effect
When solving ordinary differential equations with interval methods, the main difficulty is the wrapping effect. This phenomenon has nothing to do with precision-controlled arithmetic, it rather is caused by bounding sets in \(\mathbb{R}^n\) componentwise. For instance the optimal componentwise bounds for the set \(A := \{x \mid x \in \mathbb{R}^2,\;|x_1|+ 2|x_2|\leq 2\}\) are provided by the interval \(I := \{x \mid x \in \mathbb{R}^2\), \(-2 \leq x_1 \leq 2\), \(-1 \leq x_2 \leq 1\}\). Though \(I\) is the smallest interval containing \(A\), the transition from \(A\) to \(I\) doubles the area. This is the wrapping effect. The author considers ordinary differential equations with tolerances in the input data. He describes an algorithm which, instead of intervals, uses a larger family of sets. The algorithm exhibits a very small wrapping effect, but it can be applied only to systems with two equations.
scientific article; zbMATH DE number 769258