Operator-splitting schemes for degenerate, non-local, conservative-dissipative systems

Manh Hong Duong; Daniel Adams; Goncalo dos Reis

doi:10.3934/dcds.2022109

Operator-splitting schemes for degenerate, non-local, conservative-dissipative systems

Manh Hong Duong^*, Daniel Adams, Goncalo dos Reis

^*Corresponding author for this work

Mathematics

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Abstract

In this paper, we develop a natural operator-splitting variational scheme for a general class of non-local, degenerate conservative-dissipative evolutionary equations. The splitting-scheme consists of two phases: a conservative (transport) phase and a dissipative (diffusion) phase. The first phase is solved exactly using the method of characteristic and DiPerna-Lions theory while the second phase is solved approximately using a JKO-type variational scheme that minimizes an energy functional with respect to a certain Kantorovich optimal transport cost functional. In addition, we also introduce an entropic-regularisation of the scheme. We prove the convergence of both schemes to a weak solution of the evolutionary equation. We illustrate the generality of our work by providing a number of examples, including the kinetic Fokker-Planck equation and the (regularized) Vlasov-Poisson-Fokker-Planck equation.

Original language	English
Pages (from-to)	5453-5486
Number of pages	34
Journal	Discrete and Continuous Dynamical Systems - Series A
Volume	42
Issue number	11
Early online date	31 Aug 2022
DOIs	https://doi.org/10.3934/dcds.2022109
Publication status	Published - Nov 2022

Keywords

Wasserstein gradient flows
degenerate diffusions
variational principle
operator-splitting methods
non-local partial differential equations
optimal transport
entropic regularisation

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10.3934/dcds.2022109

DuongM2022Operator
This article has been published in a revised form in Discrete and Continuous Dynamical Systems - Series A [http://dx.doi.org/10.3934/dcds.2022109]. This version is free to download for private research and study only. Not for redistribution, re-sale or use in derivative works.
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Cite this

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title = "Operator-splitting schemes for degenerate, non-local, conservative-dissipative systems",

abstract = "In this paper, we develop a natural operator-splitting variational scheme for a general class of non-local, degenerate conservative-dissipative evolutionary equations. The splitting-scheme consists of two phases: a conservative (transport) phase and a dissipative (diffusion) phase. The first phase is solved exactly using the method of characteristic and DiPerna-Lions theory while the second phase is solved approximately using a JKO-type variational scheme that minimizes an energy functional with respect to a certain Kantorovich optimal transport cost functional. In addition, we also introduce an entropic-regularisation of the scheme. We prove the convergence of both schemes to a weak solution of the evolutionary equation. We illustrate the generality of our work by providing a number of examples, including the kinetic Fokker-Planck equation and the (regularized) Vlasov-Poisson-Fokker-Planck equation.",

keywords = "Wasserstein gradient flows, degenerate diffusions, variational principle, operator-splitting methods, non-local partial differential equations, optimal transport, entropic regularisation",

author = "Duong, {Manh Hong} and Daniel Adams and {dos Reis}, Goncalo",

year = "2022",

month = nov,

doi = "10.3934/dcds.2022109",

language = "English",

volume = "42",

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T1 - Operator-splitting schemes for degenerate, non-local, conservative-dissipative systems

AU - Duong, Manh Hong

AU - Adams, Daniel

AU - dos Reis, Goncalo

PY - 2022/11

Y1 - 2022/11

N2 - In this paper, we develop a natural operator-splitting variational scheme for a general class of non-local, degenerate conservative-dissipative evolutionary equations. The splitting-scheme consists of two phases: a conservative (transport) phase and a dissipative (diffusion) phase. The first phase is solved exactly using the method of characteristic and DiPerna-Lions theory while the second phase is solved approximately using a JKO-type variational scheme that minimizes an energy functional with respect to a certain Kantorovich optimal transport cost functional. In addition, we also introduce an entropic-regularisation of the scheme. We prove the convergence of both schemes to a weak solution of the evolutionary equation. We illustrate the generality of our work by providing a number of examples, including the kinetic Fokker-Planck equation and the (regularized) Vlasov-Poisson-Fokker-Planck equation.

AB - In this paper, we develop a natural operator-splitting variational scheme for a general class of non-local, degenerate conservative-dissipative evolutionary equations. The splitting-scheme consists of two phases: a conservative (transport) phase and a dissipative (diffusion) phase. The first phase is solved exactly using the method of characteristic and DiPerna-Lions theory while the second phase is solved approximately using a JKO-type variational scheme that minimizes an energy functional with respect to a certain Kantorovich optimal transport cost functional. In addition, we also introduce an entropic-regularisation of the scheme. We prove the convergence of both schemes to a weak solution of the evolutionary equation. We illustrate the generality of our work by providing a number of examples, including the kinetic Fokker-Planck equation and the (regularized) Vlasov-Poisson-Fokker-Planck equation.

KW - Wasserstein gradient flows

KW - degenerate diffusions

KW - variational principle

KW - operator-splitting methods

KW - non-local partial differential equations

KW - optimal transport

KW - entropic regularisation

U2 - 10.3934/dcds.2022109

DO - 10.3934/dcds.2022109

M3 - Article

SN - 1078-0947

VL - 42

SP - 5453

EP - 5486

JO - Discrete and Continuous Dynamical Systems - Series A

JF - Discrete and Continuous Dynamical Systems - Series A

IS - 11

ER -

Operator-splitting schemes for degenerate, non-local, conservative-dissipative systems

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Keywords

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