Abstract
There is a growing interest in probabilistic numerical solutions to ordinary differential equations. In this paper, the maximum a posteriori estimate is studied under the class of nu times differentiable linear time-invariant Gauss–Markov priors, which can be computed with an iterated extended Kalman smoother. The maximum a posteriori estimate corresponds to an optimal interpolant in the reproducing kernel Hilbert space associated with the prior, which in the present case is equivalent to a Sobolev space of smoothness nu +1. Subject to mild conditions on the vector field, convergence rates of the maximum a posteriori estimate are then obtained via methods from nonlinear analysis and scattered data approximation. These results closely resemble classical convergence results in the sense that a nu times differentiable prior process obtains a global order of nu , which is demonstrated in numerical examples.
Highlights
Let T = [0, T ], T < ∞, f : T × Rd → Rd, y0 ∈ Rd and consider the following ordinary differential equation (ODE): Dy(t) = f (t, y(t)), y(0) = y0, (1)where D denotes the time derivative operator
In applications where a numerical solution is sought as a component of a larger statistical inference problem, it is desirable that the error can be quantified with the same semantic, that is to say, probabilistically (Hennig et al 2015; Oates and Sullivan 2019)
Probabilistic ODE solvers can roughly be divided into two classes, sampling based solvers and deterministic solvers
Summary
Probabilistic ODE solvers can roughly be divided into two classes, sampling based solvers and deterministic solvers The former class includes classical ODE solvers that are stochastically perturbed (Teymur et al 2016; Conrad et al 2017; Teymur et al 2018; Abdulle et al 2020; Lie et al 2019), solvers that approximately sample from a Bayesian inference problem (Tronarp et al 2019b), and solvers that perform Gaussian process regression on stochastically generated data (Chkrebtii et al 2016). It is fruitful to select the Gaussian process prior to be Markovian (Kersting and Hennig 2016; Magnani et al 2017; Schober et al 2019; Tronarp et al 2019b), as
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