Convergence rates of Gaussian ODE filters
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Convergence rates of Gaussian ODE filters Hans Kersting1
· T. J. Sullivan2,3 · Philipp Hennig1
Received: 20 March 2020 / Accepted: 27 August 2020 © The Author(s) 2020
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 a priori as a Gauss–Markov process X, which is then iteratively conditioned on information about 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 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 ∈ {2, 3, . . .}. Keywords Probabilistic numerics · Ordinary differential equations · Initial value problems · Numerical analysis · Gaussian processes · Markov processes Mathematics Subject Classification 65L20 · 37H10 · 68W20 · 93E11
1 Introduction A solver of an initial value problem (IVP) outputs an approximate solution xˆ : [0, T ] → Rd of an ordinary differential equation (ODE) with initial condition: x (1) (t) :=
dx (t) = f (x(t)) , dt x(0) = x0 ∈ Rd .
∀t ∈ [0, T ], (1)
(Without loss of generality, we simplify the presentation by restricting attention to the autonomous case.) The numerical
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Hans Kersting [email protected] T. J. Sullivan [email protected]; [email protected] Philipp Hennig [email protected]
1
University of Tübingen and Max Planck Institute for Intelligent Systems, Maria-von-Linden-Straße 6, 72076 Tübingen, Germany
2
University of Warwick, Coventry CV4 7AL, United Kingdom
3
Zuse Institute Berlin, Takustraße 7, 14195 Berlin, Germany
solution xˆ is computed by iteratively collecting information on x (1) (t) by evaluating f : Rd → Rd at a numerical estimate x(t) ˆ of x(t) and using these approximate evaluations of the time derivative to extrapolate along the time axis. In other words, the numerical solution (or estimator) xˆ of the exact solution (or estimand) x is calculated based on evaluations of the vector field f (or data). Accordingly, we treat xˆ itself as an estimator, i.e., a statistic that translates evaluations of f into a probability distribution over C 1 ([0, T ]; Rd ), the space of continuously differentiable functions from [0, T ] to Rd . This probabilistic interpretation of numerical computations of tractable from intractable quantities as statistical inference of latent from observable quantities applies to all numerical problems and has been repeatedly recommended in the p
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