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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Vydané v:	Statistics and computing Ročník 31; číslo 3
Hlavní autori:	Tronarp, Filip, Särkkä, Simo, Hennig, Philipp
Médium:	Journal Article
Jazyk:	English
Vydavateľské údaje:	New York Springer US 01.05.2021 Springer Nature B.V
Predmet:	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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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.
AbstractList	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. 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 $$\nu $$ 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 $$\nu +1$$ 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 $$\nu $$ ν times differentiable prior process obtains a global order of $$\nu $$ ν , which is demonstrated in numerical examples. 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.
ArticleNumber	23
Author	Hennig, Philipp Tronarp, Filip Särkkä, Simo
Author_xml	– sequence: 1 givenname: Filip orcidid: 0000-0002-1102-7706 surname: Tronarp fullname: Tronarp, Filip email: filip.tronarp@uni-tueingen.de organization: University of Tübingen – sequence: 2 givenname: Simo surname: Särkkä fullname: Särkkä, Simo organization: Aalto University – sequence: 3 givenname: Philipp surname: Hennig fullname: Hennig, Philipp organization: University of Tübingen and MPI for Intelligent Systems
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Snippet	There is a growing interest in probabilistic numerical solutions to ordinary differential equations. In this paper, the maximum a posteriori estimate is... There is a growing interest in probabilistic numerical solutions to ordinary differential equations. In this paper, the maximum a posteriori estimate is...
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SubjectTerms	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
Title	Bayesian ODE solvers: the maximum a posteriori estimate
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