A BDF2-approach for the non-linear Fokker-Planck equation

Plazotta, Simon

doi:10.3934/dcds.2019120

Cited by 4 publications

(3 citation statements)

References 39 publications

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“…This has been achieved by means of the superposition principle (see [36], and also [6,Theorem 8.2.1], [8, Theorem 7.1], and [5,Theorem 5.2]), which provides the uniqueness of Eulerian solutions [5,Theorem 5.3]. Furthermore, the Lagrangian formulation has been used to propose discretization schemes to solve the nonlinear PDE numerically [15,25,26,29,35]. Moreover, the Lagrangian point of view has been used in [5] to provide a heuristic derivation of the nonlinear continuity equation arising as the mean-field limit of the spatially inhomogeneous replicator dynamics.…”

Section: Overview Of the Problem And State Of The Artmentioning

confidence: 99%

A multi-step Lagrangian scheme for spatially inhomogeneous evolutionary games

2021

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A multi-step Lagrangian scheme at discrete times is proposed for the approximation of a nonlinear continuity equation arising as a mean-field limit of spatially inhomogeneous evolutionary games, describing the evolution of a system of spatially distributed agents with strategies, or labels, whose payoff depends also on the current position of the agents. The scheme is Lagrangian, as it traces the evolution of position and labels along characteristics, and is a multi-step scheme, as it develops on the following two stages: First, the distribution of strategies or labels is updated according to a best performance criterion, and then, this is used by the agents to evolve their position. A general convergence result is provided in the space of probability measures. In the special cases of replicator-type systems and reversible Markov chains, variants of the scheme, where the explicit step in the evolution of the labels is replaced by an implicit one, are also considered and convergence results are provided.

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Section: Overview Of the Problem And State Of The Artmentioning

confidence: 99%

A multi-step Lagrangian scheme for spatially inhomogeneous evolutionary games

2021

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“…This approach has been proposed by Matthes and Plazotta [29,33], who proved equivalent versions of Theorem 1.2 and 1.3. Even if in the Euclidean setting the analogue problems to (1.16) and (1.8) yield the same solutions, one can check that this is not the case in the Wasserstein space (see, e.g., the example in Figure 2).…”

Section: Introductionmentioning

confidence: 99%

From geodesic extrapolation to a variational BDF2 scheme for Wasserstein gradient flows

Gallouët,

Natale,

Todeschi

2024

Math. Comp.

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We introduce a time discretization for Wasserstein gradient flows based on the classical Backward Differentiation Formula of order two. The main building block of the scheme is the notion of geodesic extrapolation in the Wasserstein space, which in general is not uniquely defined. We propose several possible definitions for such an operation, and we prove convergence of the resulting scheme to the limit PDE, in the case of the Fokker-Planck equation. For a specific choice of extrapolation we also prove a more general result, that is convergence towards EVI flows. Finally, we propose a variational finite volume discretization of the scheme which numerically achieves second order accuracy in both space and time.

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“…This has been achieved by means of the superposition principle (see [33], and also [5,Theorem 8.2.1], [7,Theorem 7.1], and [4,Theorem 5.2]), which provides the uniqueness of Eulerian solutions [4,Theorem 5.3]. Furthermore, the Lagrangian formulation has been used to propose discretization schemes to solve the nonlinear PDE numerically [13,22,23,26,32]. Moreover, the Lagrangian point of view has been used in [4] to provide a heuristic derivation of the nonlinear continuity equation arising as the mean-field limit of the spatially inhomogeneous replicator dynamics.…”

Section: Introductionmentioning

confidence: 99%

An alternate Lagrangian scheme for spatially inhomogeneous evolutionary games

Almi¹,

Morandotti²,

Solombrino³

2020

Preprint

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An alternate Lagrangian scheme at discrete times is proposed for the approximation of a nonlinear continuity equation arising as a mean-field limit of spatially inhomogeneous evolutionary games, describing the evolution of a system of spatially distributed agents with strategies, or labels, whose payoff depends also on the current position of the agents. The scheme is Lagrangian, as it traces the evolution of position and labels along characteristics and is alternate, as it consists of the following two steps: first the distribution of strategies or labels is updated according to a best performance criterion and then this is used by the agents to evolve their position. A general convergence result is provided in the space of probability measures. In the special cases of replicator-type systems and reversible Markov chains, variants of the scheme, where the explicit step in the evolution of the labels is replaced by an implicit one, are also considered and convergence results are provided.

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A BDF2-approach for the non-linear Fokker-Planck equation

Cited by 4 publications

References 39 publications

A multi-step Lagrangian scheme for spatially inhomogeneous evolutionary games

A multi-step Lagrangian scheme for spatially inhomogeneous evolutionary games

From geodesic extrapolation to a variational BDF2 scheme for Wasserstein gradient flows

An alternate Lagrangian scheme for spatially inhomogeneous evolutionary games

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