Abstract
We study connections between ordinary differential equation (ODE) solvers and probabilistic regression methods in statistics. We provide a new view of probabilistic ODE solvers as active inference agents operating on stochastic differential equation models that estimate the unknown initial value problem (IVP) solution from approximate observations of the solution derivative, as provided by the ODE dynamics. Adding to this picture, we show that several multistep methods of Nordsieck form can be recasted as Kalman filtering on q-times integrated Wiener processes. Doing so provides a family of IVP solvers that return a Gaussian posterior measure, rather than a point estimate. We show that some such methods have low computational overhead, nontrivial convergence order, and that the posterior has a calibrated concentration rate. Additionally, we suggest a step size adaptation algorithm which completes the proposed method to a practically useful implementation, which we experimentally evaluate using a representative set of standard codes in the DETEST benchmark set.
Highlights
Numerical algorithms estimate intractable quantities from tractable ones
The results presented in preceding sections pertain to the estimation of local extrapolation errors. It is a well-known aspect of ordinary differential equation (ODE) solvers (Hairer et al 1987, §III.5) that the global error can be exponentially larger than the local error
We proposed a probabilistic inference model for the numerical solution of ODEs and showed the connections with established methods
Summary
Numerical algorithms estimate intractable quantities from tractable ones. It has been pointed out repeatedly (Poincaré 1896; Diaconis 1988; O’Hagan 1992) that this process is structurally similar to statistical inference, where the tractable computations play the role of data in statistics, and the intractable quantities relate to latent, inferred quantities. Several models and methods have been proposed for the solution of initial value problems (IVPs) (Skilling 1992; Chkrebtii et al 2016; Schober et al 2014a; Conrad et al 2017; Kersting and Hennig 2016; Teymur et al 2016). These probabilistic algorithms have no immediate connection to the extensive literature on this. In the context of a larger pipeline of empirical studies and numerical computations, the framework of probability modeling provides a common language to analyze the epistemic confidence in its result (Cockayne et al 2017). In the framework of Cockayne et al (2017), the code provides approximate Bayesian uncertainty quantification (Sullivan 2015) at low computational overhead and almost complete backwards compatibility to the MATLAB IVP solver suite
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