Convergence Rates of Gaussian ODE Filters

TL;DR

This paper analyzes the convergence rates of Gaussian ODE filters, establishing local and global convergence orders, and examines how approximation errors influence these rates, supported by numerical experiments.

Contribution

It provides the first rigorous convergence rate analysis for Gaussian ODE filters, including effects of approximation errors and credible interval calibration.

Findings

Local convergence rate of order q+1 for Gaussian ODE filters

Global convergence rate of order q for q=1 with Brownian motion prior

Credible intervals contract at the same rate as truncation error

Abstract

A recently-introduced class of probabilistic (uncertainty-aware) solvers for ordinary differential equations (ODEs) applies Gaussian (Kalman) filtering to initial value problems. These methods model the true solution $x$ and its first $q$ derivatives \emph{a priori} as a Gauss--Markov process $\boldsymbol{X}$, which is then iteratively conditioned on information about $\dot{x}$. This article establishes worst-case local convergence rates of order $q+1$ for a wide range of versions of this Gaussian ODE filter, as well as global convergence rates of order $q$ in the case of $q=1$ and an integrated Brownian motion prior, and analyses how inaccurate information on $\dot{x}$ coming from approximate evaluations of $f$ affects these rates. Moreover, we show that, in the globally convergent case, the posterior credible intervals are well calibrated in the sense that they globally contract at the same rate as the truncation error. We illustrate these theoretical results by numerical experiments which might indicate their generalizability to $q \in \{2,3,\dots\}$.