On the complexity of solving initial value problems

TL;DR

This paper proves that solving initial value problems with polynomial vector fields can be done in polynomial time relative to key parameters, regardless of domain bounds or Lipschitz conditions, providing guaranteed complexity and precision.

Contribution

It introduces a polynomial-time algorithm for initial value problems with polynomial vector fields that works without domain restrictions or Lipschitz assumptions.

Findings

Algorithm runs in polynomial time relative to T, μ, and solution bounds.

Works for any polynomial vector field over bounded or unbounded domains.

Guarantees precision and complexity without fixed polynomial or compact domain assumptions.

Abstract

In this paper we prove that computing the solution of an initial-value problem $\dot{y}=p(y)$ with initial condition $y(t_0)=y_0\in\R^d$ at time $t_0+T$ with precision $e^{-\mu}$ where $p$ is a vector of polynomials can be done in time polynomial in the value of $T$, $\mu$ and $Y=\sup_{t_0\leqslant u\leqslant T}\infnorm{y(u)}$. Contrary to existing results, our algorithm works for any vector of polynomials $p$ over any bounded or unbounded domain and has a guaranteed complexity and precision. In particular we do not assume $p$ to be fixed, nor the solution to lie in a compact domain, nor we assume that $p$ has a Lipschitz constant.