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Convergence rates of Gaussian ODE filters
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Kersting, Hans, Sullivan, T. J. and Hennig, Philipp (2020) Convergence rates of Gaussian ODE filters. Statistics and Computing, 30 (6). pp. 1791-1816. doi:10.1007/s11222-020-09972-4 ISSN 0960-3174.
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Official URL: https://doi.org/10.1007/s11222-020-09972-4
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 X, which is then iteratively conditioned on information about x˙. This article establishes worst-case local convergence rates of order q + 1 for a wide range of versions of this Gaussian ODE filter, as well as global convergence rates of order q in the case of q = 1 and an integrated Brownian motion prior, and analyses how inaccurate information on x˙ coming from approximate evaluations of f affects these rates. Moreover, we show that, in the globally convergent case, the posterior credible intervals are well calibrated in the sense that they globally contract at the same rate as the truncation error. We illustrate these theoretical results by numerical experiments which might indicate their generalizability to q ∈ {2, 3,...}.
| Item Type: | Journal Article | ||||||
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| Subjects: | Q Science > QA Mathematics | ||||||
| Divisions: | Faculty of Science, Engineering and Medicine > Science > Mathematics | ||||||
| Library of Congress Subject Headings (LCSH): | Numerical analysis, Differential equations, Gaussian processes, Markov processes | ||||||
| Journal or Publication Title: | Statistics and Computing | ||||||
| Publisher: | Springer | ||||||
| ISSN: | 0960-3174 | ||||||
| Official Date: | 12 September 2020 | ||||||
| Dates: |
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| Volume: | 30 | ||||||
| Number: | 6 | ||||||
| Page Range: | pp. 1791-1816 | ||||||
| DOI: | 10.1007/s11222-020-09972-4 | ||||||
| Status: | Peer Reviewed | ||||||
| Publication Status: | Published | ||||||
| Access rights to Published version: | Restricted or Subscription Access | ||||||
| Date of first compliant deposit: | 27 April 2021 | ||||||
| Date of first compliant Open Access: | 27 April 2021 |
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