Conservative-dissipative approximation schemes for a generalized Kramers equation

Manh Hong Duong; Mark A. Peletier; Johannes Zimmer

doi:10.1002/mma.2994

Conservative-dissipative approximation schemes for a generalized Kramers equation

Manh Hong Duong, Mark A. Peletier, Johannes Zimmer

Eindhoven University of Technology

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Abstract

We propose three new discrete variational schemes that capture the conservative-dissipative structure of a generalized Kramers equation. The first two schemes are single-step minimization schemes, whereas the third one combines a streaming and a minimization step. The cost functionals in the schemes are inspired by the rate functional in the Freidlin-Wentzell theory of large deviations for the underlying stochastic system. We prove that all three schemes converge to the solution of the generalized Kramers equation.

Original language	English
Pages (from-to)	2517-2540
Number of pages	24
Journal	Mathematical Methods in the Applied Sciences
Volume	37
Issue number	16
Early online date	10 Oct 2013
DOIs	https://doi.org/10.1002/mma.2994
Publication status	Published - 15 Nov 2014

Keywords

Gradient flows
Hamiltonian flows
Kramers equation
Optimal transport
Variational principle

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10.1002/mma.2994

Revised DPZ_M2AS
This is the pre-peer reviewed version of the following article: Duong, M. H., Peletier, M. A., & Zimmer, J. (2014). Conservative-dissipative approximation schemes for a generalized Kramers equation. Mathematical Methods in the Applied Sciences, 37(16), 2517-2540. 10.1002/mma.2994, which has been published in final form at http://dx.doi.org/10.1002/mma.2994. This article may be used for non-commercial purposes in accordance with Wiley Terms and Conditions for Self-Archiving."
Submitted manuscript, 296 KB

Cite this

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N2 - We propose three new discrete variational schemes that capture the conservative-dissipative structure of a generalized Kramers equation. The first two schemes are single-step minimization schemes, whereas the third one combines a streaming and a minimization step. The cost functionals in the schemes are inspired by the rate functional in the Freidlin-Wentzell theory of large deviations for the underlying stochastic system. We prove that all three schemes converge to the solution of the generalized Kramers equation.

AB - We propose three new discrete variational schemes that capture the conservative-dissipative structure of a generalized Kramers equation. The first two schemes are single-step minimization schemes, whereas the third one combines a streaming and a minimization step. The cost functionals in the schemes are inspired by the rate functional in the Freidlin-Wentzell theory of large deviations for the underlying stochastic system. We prove that all three schemes converge to the solution of the generalized Kramers equation.

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KW - Hamiltonian flows

KW - Kramers equation

KW - Optimal transport

KW - Variational principle

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JF - Mathematical Methods in the Applied Sciences

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Conservative-dissipative approximation schemes for a generalized Kramers equation

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Keywords

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