TPFA Finite Volume Approximation of Wasserstein Gradient Flows

Natale, Andrea; Todeschi, Gabriele

doi:10.1007/978-3-030-43651-3_16

Cited by 3 publications

(9 citation statements)

References 7 publications

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“…The fundamental tool is the solution of JKO steps, which requires the expensive problem of computing the Wasserstein distance. Following [12,30], we linearize the Wasserstein distance obtaining LJKO steps, a more affordable problem to solve. Remarkably, this approach preserves the second order accuracy in time of our time discretization.…”

Section: Finite Volume Discretizationmentioning

confidence: 99%

“…2 ), both leading to second order accurate schemes in space [30]. The former choice is possible only if x K ∈ K, which may not be always the case for arbitrary admissible meshes.…”

Section: Finite Volume Discretizationmentioning

confidence: 99%

“…We propose a discretization in the Eulerian framework of finite volumes. Specifically, we implement Two Point Flux Approximation (TPFA) finite volumes, which have been extensively analyzed lately for the discretization of optimal transport and Wasserstein gradient flows [21,17,31,12,30]. Following these last two works in particular, we propose a scheme in which the Wasserstein distance is locally linearized, at each step of the scheme, in order to decrease the computational complexity of the approach, without dropping the second-order accuracy in time.…”

Section: Introductionmentioning

confidence: 99%

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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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Section: Finite Volume Discretizationmentioning

confidence: 99%

“…2 ), both leading to second order accurate schemes in space [30]. The former choice is possible only if x K ∈ K, which may not be always the case for arbitrary admissible meshes.…”

Section: Finite Volume Discretizationmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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

Gallouët,

Natale,

Todeschi

2024

Math. Comp.

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“…On the other hand, using the arithmetic interpolation, I (r, s) = (r + s)/2, would not work directly since the solutions may become negative. In this case additional technical steps, like a Lagrange multiplicator as in [39], are necessary to obtain the evolution of a non-negative probability density. We use the more physical inspired upwind flux, which automatically ensures the positivity of the density.…”

Section: Graph Setting With General Interactionsmentioning

confidence: 99%

Nonlocal-Interaction Equation on Graphs: Gradient Flow Structure and Continuum Limit

Esposito

Patacchini

Schlichting

et al. 2021

Arch Rational Mech Anal

122

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We consider dynamics driven by interaction energies on graphs. We introduce graph analogues of the continuum nonlocal-interaction equation and interpret them as gradient flows with respect to a graph Wasserstein distance. The particular Wasserstein distance we consider arises from the graph analogue of the Benamou–Brenier formulation where the graph continuity equation uses an upwind interpolation to define the density along the edges. While this approach has both theoretical and computational advantages, the resulting distance is only a quasi-metric. We investigate this quasi-metric both on graphs and on more general structures where the set of “vertices” is an arbitrary positive measure. We call the resulting gradient flow of the nonlocal-interaction energy the nonlocal nonlocal-interaction equation (NL$$^2$$ 2 IE). We develop the existence theory for the solutions of the NL$$^2$$ 2 IE as curves of maximal slope with respect to the upwind Wasserstein quasi-metric. Furthermore, we show that the solutions of the NL$$^2$$ 2 IE on graphs converge as the empirical measures of the set of vertices converge weakly, which establishes a valuable discrete-to-continuum convergence result.

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“…The perturbation introduced by the barrier function can be tuned by multiplying it by a positive coefficient µ and the original solution is recovered via a continuation method for µ going to zero. The final algorithm is robust and can be easily generalized to similar problems (for example, we have already applied it successfully in [25] for the solution of Wasserstein gradient flows).…”

Section: Numerical Solutionmentioning

confidence: 99%

Computation of optimal transport with finite volumes

Natale

Todeschi

2021

ESAIM: M2AN

Self Cite

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We construct Two-Point Flux Approximation (TPFA) finite volume schemes to solve the quadratic optimal transport problem in its dynamic form, namely the problem originally introduced by Benamou and Brenier. We show numerically that these type of discretizations are prone to form instabilities in their more natural implementation, and we propose a variation based on nested meshes in order to overcome these issues. Despite the lack of strict convexity of the problem, we also derive quantitative estimates on the convergence of the method, at least for the discrete potential and the discrete cost. Finally, we introduce a strategy based on the barrier method to solve the discrete optimization problem.

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TPFA Finite Volume Approximation of Wasserstein Gradient Flows

Cited by 3 publications

References 7 publications

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

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

Nonlocal-Interaction Equation on Graphs: Gradient Flow Structure and Continuum Limit

Computation of optimal transport with finite volumes

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