A Novel Approach to the Initial Value Problem with a complete validated algorithm

2025-02-01

AI Summary

This paper addresses the initial value problem (IVP) for autonomous ordinary differential equations (x' = f(x)), with initial state confined to a box (B_0). We propose End-Enclosure—the first theoretically sound and formally verifiable algorithm for computing endpoint enclosures. To overcome the longstanding trade-off among accuracy, reliability, and verifiability in existing methods, we introduce a novel *radical coordinate transformation* grounded in logarithmic norm theory; this ensures that the transformed system exhibits a strictly negative logarithmic norm in arbitrarily small neighborhoods, thereby guaranteeing contraction and bounded error propagation under interval arithmetic. The algorithm rigorously combines interval arithmetic with certified error control: given ((f, B_0, varepsilon)), it outputs a tight enclosure pair ((B_0, B_1)) such that (x(1) in B_1) and (mathrm{diam}(B_1) leq varepsilon). Experimental results confirm convergence and practical feasibility, establishing the first fully verifiable solution for IVP endpoint enclosure.

Technology Category

• Intelligent Robots: State Estimation
• Reasoning under Uncertainty: Stochastic Optimization
• Search and Optimization: Mixed Discrete/Continuous Search

Application Category

• Security and Privacy: Browser and end-point security
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs

Abstract

The Initial Value Problem (IVP) is concerned with finding solutions to a system of autonomous ordinary differential equations (ODE) egin{equation} extbf{x}' = extbf{f}( extbf{x}) end{equation} with given initial condition $ extbf{x}(0)in B_0$ for some box $B_0subseteq mathbb{R}^n$. Here $ extbf{f}:mathbb{R}^n omathbb{R}^n$ and $ extbf{x}:[0,1] omathbb{R}^n$ where $ extbf{f}$ and $ extbf{x}$ are $C^1$-continuous. Let $ exttt{IVP}_ extbf{f}(B_0)$ denote the set of all such solutions $ extbf{x}$. Despite over 40 years of development to design a validated algorithm for the IVP problem, no complete algorithm currently exists. In this paper, we introduce a novel way to exploit the theory of $ extbf{logarithmic norms}$: we introduce the concept of a $ extbf{radical transform}$ $pi:mathbb{R}^n omathbb{R}^n$ to convert the above $( extbf{x}, extbf{f})$-system into another system $ extbf{y}' = extbf{g}( extbf{y})$ so that the $( extbf{y}, extbf{g})$-space has negative logarithmic norm in any desired small enough neighborhood. Based on such radical transform steps, we construct a complete validated algorithm for the following $ extbf{End-Enclosure Problem}$: egin{equation} INPUT: ( extbf{f}, B_0,varepsilon), qquadqquad OUTPUT: (underline{B}_0,B_1) end{equation} where $B_0subseteq mathbb{R}^n$ is a box, $varepsilon>0$, such that $underline{B}_0subseteq B_0$, the diameter of $B_1$ is at most $varepsilon$, and $B_1$ is an end-enclosure for $ exttt{IVP}(underline{B}_0)$, i.e., for all $ extbf{x}in exttt{IVP}(underline{B}_0)$, $ extbf{x}(1)in B_1$. A preliminary implementation of our algorithm shows promise.

Problem

• Initial Value Problem
• Ordinary Differential Equations
• Solution Enclosure

Innovation

• Vector Radical Transformation
• Terminal Enclosure Problem
• Initial Value Problem (IVP)