Logarithmic Reduction of the Wrapping Effect with Application to Ordinary Differential Equations (ODEs) — Principles and Recent Advances

Abstract

Considering verified numerical computing, solving ordinary differential equations (ODEs) with guaranteed error bounds is a critical challenge. One significant obstacle in this process is the wrapping effect, which causes overestimation of reachable sets due to the compounding of conservative approximations over time. Recent developments in logarithmic reduction techniques have emerged as promising strategies to reduce the wrapping effect, especially within interval arithmetic and affine arithmetic frameworks. This article explores the wrapping effect, the principle of logarithmic reduction, and the latest methodological and computational advances applied to the rigorous integration of ODEs.

1. Introduction

Numerical integration of ODEs is fundamental in modeling physical, biological, and engineering systems. However, when solving these systems with guaranteed bounds (e.g., using interval arithmetic), a phenomenon known as the wrapping effect can significantly deteriorate the accuracy of the result. This effect arises from enclosing multidimensional sets (e.g., ellipsoids or rectangles) with simpler geometric shapes like intervals or boxes…