Bayesian ode solvers: The maximum a posteriori estimate
(2021) In Statistics and Computing 31.- 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 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... (More)
- 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. (Less)
Please use this url to cite or link to this publication:
https://lup.lub.lu.se/record/0e6bf722-56df-43b3-8f5f-ade11288c527
- author
- Tronarp, Filip LU ; Särkkä, Simo and Hennig, Philipp
- publishing date
- 2021
- type
- Contribution to journal
- publication status
- published
- subject
- keywords
- Probabilistic numerical methods, Maximum a posteriori estimation, Kernel methods
- in
- Statistics and Computing
- volume
- 31
- article number
- 23
- pages
- 18 pages
- publisher
- Springer
- external identifiers
-
- scopus:85102124866
- ISSN
- 0960-3174
- DOI
- 10.1007/s11222-021-09993-7
- language
- English
- LU publication?
- no
- id
- 0e6bf722-56df-43b3-8f5f-ade11288c527
- date added to LUP
- 2023-08-20 22:38:02
- date last changed
- 2025-04-04 14:33:50
@article{0e6bf722-56df-43b3-8f5f-ade11288c527,
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 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.}},
author = {{Tronarp, Filip and Särkkä, Simo and Hennig, Philipp}},
issn = {{0960-3174}},
keywords = {{Probabilistic numerical methods; Maximum a posteriori estimation; Kernel methods}},
language = {{eng}},
publisher = {{Springer}},
series = {{Statistics and Computing}},
title = {{Bayesian ode solvers: The maximum a posteriori estimate}},
url = {{http://dx.doi.org/10.1007/s11222-021-09993-7}},
doi = {{10.1007/s11222-021-09993-7}},
volume = {{31}},
year = {{2021}},
}