Bivariate range functions with superior convergence order

2026-04-15

AI Summary

This work addresses the limitation of traditional bivariate interval functions, which exhibit only second-order convergence in verified computations for geometric modeling, computer graphics, and robotics, thereby constraining both accuracy and efficiency. Building upon the Cornelius–Lohner framework, the study systematically introduces higher-order interpolation schemes—namely Taylor, Lagrange, and Hermite—to construct novel interval functions achieving third- and fourth-order convergence. The proposed approach is efficiently implemented in Julia, and experimental results demonstrate its clear superiority over conventional methods, delivering significantly enhanced convergence rates while preserving rigorous reliability guarantees.

Technology Category

• Knowledge Representation and Reasoning: Geometric, Spatial, and Temporal Reasoning
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Reasoning under Uncertainty: Uncertainty Representations

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics
• Systems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deployments

Abstract

Range functions are a fundamental tool for certified computations in geometric modeling, computer graphics, and robotics, but traditional range functions have only quadratic convergence order ($m=2$). For ``superior'' convergence order (i.e., $m>2$), we exploit the Cornelius--Lohner framework in order to introduce new bivariate range functions based on Taylor, Lagrange, and Hermite interpolation. In particular, we focus on practical range functions with cubic and quartic convergence order. We implemented them in Julia and provide experimental validation of their performance in terms of efficiency and efficacy.

Problem

• range functions
• convergence order
• bivariate
• certified computations
• geometric modeling

Innovation

• bivariate range functions
• superior convergence order
• Cornelius–Lohner framework
• Taylor interpolation
• Hermite interpolation