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 ν times differentiable linear time-invariant Gauss–Markov priors, which can be computed with an iterated extended Kalman...

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Veröffentlicht in:Statistics and computing Jg. 31; H. 3
Hauptverfasser: Tronarp, Filip, Särkkä, Simo, Hennig, Philipp
Format: Journal Article
Sprache:Englisch
Veröffentlicht: New York Springer US 01.05.2021
Springer Nature B.V
Schlagworte:
Artificial Intelligence
Convergence
Differential equations
Fields (mathematics)
Hilbert space
Mathematics and Statistics
Nonlinear analysis
Ordinary differential equations
Probability and Statistics in Computer Science
Smoothness
Sobolev space
Statistical Theory and Methods
Statistics
Statistics and Computing/Statistics Programs
Maximum a posteriori estimation
Kernel methods
Probabilistic numerical methods
ISSN:0960-3174, 1573-1375
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Zusammenfassung: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.
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ISSN:0960-3174
1573-1375
DOI:10.1007/s11222-021-09993-7