Bayesian ODE solvers: the maximum a posteriori estimate

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.