Bayesian ODE solvers: the maximum a posteriori estimate

Filip Tronarp*, Simo Särkkä, Philipp Hennig

*Corresponding author for this work

Research output: Contribution to journalArticleScientificpeer-review

14 Citations (Scopus)
125 Downloads (Pure)


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 ν 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 ν+ 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 ν times differentiable prior process obtains a global order of ν, which is demonstrated in numerical examples.

Original languageEnglish
Article number23
Number of pages18
Issue number3
Publication statusPublished - 3 Mar 2021
MoE publication typeA1 Journal article-refereed


  • Kernel methods
  • Maximum a posteriori estimation
  • Probabilistic numerical methods


Dive into the research topics of 'Bayesian ODE solvers: the maximum a posteriori estimate'. Together they form a unique fingerprint.

Cite this