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

TL;DR

This paper introduces a complete validated algorithm for solving the initial value problem (IVP) for autonomous differential equations, ensuring existence and enclosure of solutions with proven halting, novel techniques, and promising preliminary results.

Contribution

It presents the first halting validated algorithm for the IVP in a general setting, with new techniques and data structures for enclosures and solution refinement.

Findings

Algorithm guarantees halting under certain conditions.

New techniques for solution enclosure refinement.

Preliminary implementation shows competitive performance.

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.