End Cover for Initial Value Problem: Complete Validated Algorithms with Complexity Analysis

2026-01-29

AI Summary

This work proposes a complete and verified algorithm for computing an $\varepsilon$-accurate covering of the endpoint set of solutions to initial value problems of autonomous ordinary differential equations at a time step $h$. The method innovatively constructs a finite box cover based on the boundary of the endpoint set, integrating interval analysis, reliable numerical computation, and complexity theory. It is the first algorithm to provide both completeness and rigorous accuracy guarantees for this problem, accompanied by a thorough complexity analysis. Theoretical proofs establish its correctness and completeness, while experimental results demonstrate its efficiency and practical feasibility.

Technology Category

• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Security and Privacy: Browser and end-point security
• Systems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deployments

Abstract

We consider the first-order autonomous ordinary differential equation \[ \mathbf{x}'= \mathbf{f}(\mathbf{x}), \] where $\mathbf{f} : \mathbb{R}^n \to \mathbb{R}^n$ is locally Lipschitz. For a box $B_0 \subseteq \mathbb{R}^n$ and $h>0$, we denote by $\mathrm{IVP}_{\mathbf{f}}(B_0,h)$ the set of solutions $\mathbf{x} : [0,h] \to \mathbb{R}^n$ satisfying \[ \mathbf{x}'(t) = \mathbf{f}(\mathbf{x}(t)), \qquad \mathbf{x}(0) \in B_0 . \] We present a complete validated algorithm for the following \emph{End Cover Problem}: given $(\mathbf{f}, B_0, \varepsilon, h)$, compute a finite set $\mathcal{C}$ of boxes such that \[ \mathrm{End}_{\mathbf{f}}(B_0,h) \;\subseteq\; \bigcup_{B \in \mathcal{C}} B \;\subseteq\; \mathrm{End}_{\mathbf{f}}(B_0,h) \oplus [-\varepsilon,\varepsilon]^n , \] where \[ \mathrm{End}_{\mathbf{f}}(B_0,h) = \left\{ \mathbf{x}(h) : \mathbf{x} \in \mathrm{IVP}_{\mathbf{f}}(B_0,h) \right\}. \] Moreover, we provide a complexity analysis of our algorithm and introduce a novel technique for computing the end cover $\mathcal{C}$ based on covering the boundary of $\mathrm{End}_{\mathbf{f}}(B_0,h)$. Finally, we present experimental results demonstrating the practicality of our approach.

Problem

• Initial Value Problem
• End Cover
• Ordinary Differential Equations
• Validated Computation
• Reachability

Innovation

• validated algorithms
• initial value problem
• end cover
• complexity analysis
• boundary covering