Certified surface approximations using the interval Krawczyk test

2026-02-07

AI Summary

This work addresses the problem of constructing certified approximations of high-dimensional surfaces defined by non-square systems, with rigorous guarantees of existence and uniqueness. To this end, we generalize the classical Krawczyk test—originally formulated for square systems—to handle non-square systems and higher-dimensional algebraic varieties. By integrating interval arithmetic with techniques from analytic system solving, we develop a general-purpose algorithm for certified surface approximation. A prototype implementation demonstrates the efficacy of our approach on several complex surface instances, significantly broadening the class of geometric objects amenable to formal certification.

Technology Category

• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Knowledge Representation and Reasoning: Automated Reasoning and Theorem Proving
• Search and Optimization: Non-convex Optimization

Application Category

• Systems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deployments
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed 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

Abstract

We propose an algorithm to construct a certified approximation of a surface by generalizing the Krawczyk test. The Krawczyk test is based on interval arithmetic, and confirms the existence and uniqueness of a solution to a square system of analytic equations in a region. By generalizing this test, we extend the reach of this technique to non-square systems and higher-dimensional varieties. We provide a prototype implementation and illustrate its use on several examples.

Problem

• certified approximation
• surface
• Krawczyk test
• interval arithmetic
• non-square systems

Innovation

• interval Krawczyk test
• certified approximation
• non-square systems
• higher-dimensional varieties
• interval arithmetic