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 $\boldsymbol{X}$, which is then iteratively conditioned on information about $\dot{x}$. We prove worst-case local convergence rates of order $h^{q+1}$ for a wide range of versions of this Gaussian ODE filter, as well as global convergence rates of order $h^q$ in the case of $q=1$ and an integrated Brownian motion prior, and analyse how inaccurate information on $\dot{x}$ coming from approximate evaluations of $f$ affects these rates. Moreover, we present explicit formulas for the steady states and show that the posterior confidence intervals are well calibrated in all considered cases that exhibit global convergence---in the sense that they globally contract at the same rate as the truncation error.
| Author(s): | Hans Kersting and T. J. Sullivan and Philipp Hennig |
| Journal: | arXiv preprint 2018 |
| Volume: | arXiv:1807.09737 [math.NA] |
| Year: | 2018 |
| Month: | July |
| Project(s): |
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| BibTeX Type: | Article (article) |
| URL: | http://arxiv.org/abs/1807.09737 |
| Electronic Archiving: | grant_archive |
BibTeX
@article{KerstingSullivanHennig18,
title = {Convergence Rates of Gaussian ODE Filters},
journal = {arXiv preprint 2018},
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 $\boldsymbol{X}$, which is then iteratively conditioned on information about $\dot{x}$. We prove worst-case local convergence rates of order $h^{q+1}$ for a wide range of versions of this Gaussian ODE filter, as well as global convergence rates of order $h^q$ in the case of $q=1$ and an integrated Brownian motion prior, and analyse how inaccurate information on $\dot{x}$ coming from approximate evaluations of $f$ affects these rates. Moreover, we present explicit formulas for the steady states and show that the posterior confidence intervals are well calibrated in all considered cases that exhibit global convergence---in the sense that they globally contract at the same rate as the truncation error. },
volume = {arXiv:1807.09737 [math.NA]},
month = jul,
year = {2018},
author = {Kersting, Hans and Sullivan, T. J. and Hennig, Philipp},
url = {http://arxiv.org/abs/1807.09737},
month_numeric = {7}
}