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

TL;DR

This work delivers a complete validated algorithm for the End Enclosure Problem arising from initial value problems of autonomous ODEs ${\bf x}'={\bf f}({\bf x})$. It introduces a scaffold data structure and two novel subroutines, StepA and StepB, to iteratively extend enclosures while ensuring halting under the promise of IVP existence. A radical transformation paired with logarithmic norms enables tighter end- and full-enclosures by operating in a transformed space and pulling results back to the original space, with Euler-tube refinements providing robust containment guarantees. The approach is supported by a Banach-space foundation, interval arithmetic, and high-order Taylor expansions, and is supported by extensive local and global experiments showing favorable performance relative to existing validated IVP software. Overall, the paper advances automatic, hyperparameter-free, fully validated IVP computation and points to scalable future improvements including arbitrary-precision arithmetic and potential Lohner-style enhancements.

Abstract

We consider the first order autonomous differential equation (ODE) ${\bf x}'={\bf f}({\bf x})$ where ${\bf f}: {\mathbb R}^n\to{\mathbb R}^n$ is locally Lipschitz. For ${\bf x}_0\in{\mathbb R}^n$ and $h>0$, the initial value problem (IVP) for $({\bf f},{\bf x}_0,h)$ is to determine if there is a unique solution, i.e., a function ${\bf x}:[0,h]\to{\mathbb R}^n$ that satisfies the ODE with ${\bf x}(0)={\bf x}_0$. Write ${\bf x} ={\tt IVP}_{\bf f}({\bf x}_0,h)$ for this unique solution. We pose a corresponding computational problem, called the End Enclosure Problem: given $({\bf f},B_0,h,\varepsilon_0)$ where $B_0\subseteq{\mathbb R}^n$ is a box and $\varepsilon_0>0$, to compute a pair of non-empty boxes $(\underline{B}_0,B_1)$ such that $\underline{B}_0\subseteq B_0$, width of $B_1$ is $<\varepsilon_0$, and for all ${\bf x}_0\in \underline{B}_0$, ${\bf x}={\tt IVP}_{\bf f}({\bf x}_0,h)$ exists and ${\bf x}(h)\in B_1$. We provide a complete validated algorithm for this problem. Under the assumption (promise) that for all ${\bf x}_0\in B_0$, ${\tt IVP}_{\bf f}({\bf x}_0,h)$ exists, we prove the halting of our algorithm. This is the first halting algorithm for IVP problems in such a general setting. We also introduce novel techniques for subroutines such as StepA and StepB, and a scaffold datastructure to support our End Enclosure algorithm. Among the techniques are new ways refine full- and end-enclosures based on a {\bf radical transform} combined with logarithm norms. Our preliminary implementation and experiments show considerable promise, and compare well with current validated algorithms.

Figures (4)

• Figure 1: Volterra system (Eg1). The negative zone of the system is the region above the green parabola.
• Figure 2: The dashed lines in the figure form a $\delta$-tube around the red solid curve representing ${\boldsymbol{x}}(t)$. The segment $l(t)$ is a line segment inside this $\delta$-tube.
• Figure 3: A 7-step scaffold. The horizontal axis represents time, and the vertical axis represents ${\mathbb R}^n$. The red curve corresponds to ${\boldsymbol{x}}(t)$, the blue line segments represent end-enclosures, and the green boxes, represent full-enclosures.
• Figure 4: $3$-stage scaffold ${{\mathcal{S}}}$ with $\ell_1=2$, $\ell_2=1$ and $\ell_3=0$ in $G$.

Theorems & Definitions (15)

• Theorem 1: Picard-Lindelöf Theorem
• Theorem 2: Admissible Triple
• Lemma 3
• Theorem 4: Neumaier
• Corollary 5
• Lemma 6: Euler Tube Method
• Lemma 7
• Lemma 8: Correctness of StepA
• Lemma 9: Correctness of StepB
• Lemma 10: 1-Step Euler Enclosures with logNorm